Flatband generators arXiv:2011.04710v1 [cond-mat.str-el] 9 ...

Ph.D. Thesis

Flatband generators

Wulayimu Maimaiti

Basic Science

UNIVERSITY OF SCIENCE AND TECHNOLOGY

December 2019

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Flatband generators

Wulayimu Maimaiti

A Dissertation Submitted in Partial Fulfillment of RequirementsFor the Degree of Doctor of Philosophy

December 2019

UNIVERSITY OF SCIENCE AND TECHNOLOGYMajor of: Basic Science

Supervisor: Sergej FlachCo-supervisor: Alexei Andreanov

We hereby approve the Ph.D.thesis of "Wulayimu Maimaiti".

December 2019

UNIVERSITY OF SCIENCE AND TECHNOLOGY

ACKNOWLEDGEMENT

Professor Sergej Flach and Professor Alexei Andreanov have been great advisors

during these years, and I am really fortunate to work with them. I thank them for pro-

viding me with nice ideas and great suggestions when I met difficult problems. They

showed great persistance in supporting me with patient explanations when I didn’t

understand some points. The broad vision and accurate hints from Sergej, and the

mathematical skills and smart ideas from Alexei were highly valuable to me, while at

the same time I was given the freedom to think and work independently.

I thank all my colleagues and friends who supported me during my PhD studies.

I am really lucky to work in the Center for Theoretical Physics of Complex Systems,

where I received great support from everyone. I had nice discussions with my office-

mates Dr. Ajith Ramachandran and Dr. Carlo Danieli over the years; these discussions

were a great help. I would also like to thank Professor Barbara Dietz for her support

during our collaboration. And I thank Professor Joel Rasmussen for his help to revise

my thesis and strengthen the language.

These years of PhD studies have been very challenging for me. I have gone through

difficult and stressful times, and during such times, my colleagues and friends gave a

lot of support. Sergej always offered great guidance, and I especially thank my friend

Dr. Sadiq Seytniyaz who provided me great emotional support during his time in

Korea. Their encouragement always gave me energy and courage.

I acknowledge the financial assistance of the Institute for Basic Science, and the

great support and outstanding research environment of the Center for Theoretical Physics

of Complex Systems.

At the end, and most importantly, I thank my parents and all my family members

for their endless support from the very beginning. I also thank my wife, who has

always been my supporter, for all her help during my studies. My family has been my

pillar to lean on, and without their support, it would be impossible for me to reach this

stage. I ascribe my achievements to my parents and my family. Thank you.

9

ABSTRACT

Flatband generators

Flatbands (FBs) are dispersionless energy bands in the single-particle spectrum of

a translational invariant tight-binding network. The FBs occur due to destructive inter-

ference, resulting in macroscopically degenerate eigenstates living in a finite number of

unit cells, which are called compact localized states (CLSs). Such macroscopic degen-

eracy is in general highly sensitive to perturbations, such that even slight perturbation

lifts the degeneracy and leads to various interesting physical phenomena.

In this thesis, we develop an approach to identify and construct FB Hamiltonians

in 1D, 2D Hermitian, and 1D non-Hermitian systems. First, we introduce a systematic

classification of FB lattices by their CLS properties, and propose a scheme to generate

tight-binding Hamiltonians having FBs with given CLS properties—a FB generator.

Applying this FB generator to a 1D system, we identify all possible FB Hamiltoni-

ans of 1D lattices with arbitrary numbers of bands and CLS sizes. Extending the 1D

approach, we establish a FB generator for 2D FB Hamiltonians that have CLSs occu-

pying a maximum of four unit cells in a 2 × 2 plaquette. Employing this approach in

the non-Hermitiaon regime, we realize a FB generator for a 1D non-Hermitian lattice

with two bands. Ultimately, we apply our methods to propose a tight-binding model

that explains the spectral properties of a microwave photonic crystal.

Our results and methods in this thesis further our understanding of the properties

of FB lattices and their CLSs, provide more flexibility to design FB lattices in experi-

ments, and open new avenues for future research.

∗A thesis submitted to committee of the University of Science and Technology in a partial ful-

fillment of the requirement for the degree of Doctor of Philosophy conferred in February 2020.

11

Contents

1 Motivation and outline 1

2 Introduction: Flatbands in discrete systems 5

2.1 Tight-binding model for discrete systems . . . . . . . . . . . . . . . 6

2.1.1 Tight-binding model . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2 Bloch theorem and band structure . . . . . . . . . . . . . . . 9

2.2 Flatbands, compact localized states, and macroscopic degeneracy . . . 11

2.3 Flatband construction methods . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Geometrical methods . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Flatband generation from local unitary transformations . . . . 17

2.3.3 Flatband construction from symmetry . . . . . . . . . . . . . 19

2.4 Applications and experimental realizations of flatband systems . . . . 22

2.4.1 Perturbation as a playmaker . . . . . . . . . . . . . . . . . . 23

2.4.2 Experimental realizations and applications . . . . . . . . . . 25

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Classification and construction of flatbands by compact localized states 33

3.1 Classification of flatband networks by compact localized states (CLSs) 33

3.2 Properties of CLSs . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Relation between CLSs and Bloch wave functions . . . . . . 35

3.2.2 Irreducibility condition of CLSs . . . . . . . . . . . . . . . . 36

3.2.3 Completeness of CLSs . . . . . . . . . . . . . . . . . . . . . 39

3.2.4 Relation between CLS completeness and band touching . . . 41

3.3 CLS existence conditions and the flatband generator . . . . . . . . . . 42

3.3.1 Block matrix representation of the tight-binding model . . . . 43

3.3.2 CLS existence conditions and flatband tester . . . . . . . . . 45

3.3.3 Flatband Generator . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

i

4 Flatband generator in one dimension 514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Two-band problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 U=1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.2 U=2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Arbitrary number of bands . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 The generator . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.3 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.4 Network constraints . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Flatband generator in two dimensions 695.1 The eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Classification of CLSs in 2D . . . . . . . . . . . . . . . . . . . . . . 70

5.3 The flatband generator . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Nearest neighbor hoppings . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 Next nearest neighbor hoppings . . . . . . . . . . . . . . . . . . . . 77

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Non-Hermitian flatband generator 816.1 Overview of non-Hermitian physics and flatbands in non-Hermitian

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2 Non-Hermitian Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 82

6.3 The generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.4.1 Completely flat case . . . . . . . . . . . . . . . . . . . . . . 84

6.4.2 Partially flat case . . . . . . . . . . . . . . . . . . . . . . . . 87

6.4.3 Modulus-flat case . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7 Flatband in a microwave photonic crystal 917.1 Photonic crystals and Dirac billiards . . . . . . . . . . . . . . . . . . 91

7.2 Tight-binding model for a microwave photonic crystal . . . . . . . . 93

7.3 The honome lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7.4.1 Positioning van Hove singularities . . . . . . . . . . . . . . . 98

ii

7.4.2 Deviations from experiment . . . . . . . . . . . . . . . . . . 99

7.4.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.5.1 Fitting with three hoppings . . . . . . . . . . . . . . . . . . . 102

7.5.2 Fitting with five hoppings . . . . . . . . . . . . . . . . . . . 102

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8 Conclusions and outlook 107

Bibliography 111

A Supplementary materials for CLS properties 129

A.1 Linear dependence and reducibility of CLSs in 1D (U=3 case) . . . . 129

B Supplementary materials for the flatband generator in 1D 133

B.1 On the linear independence of CLSs . . . . . . . . . . . . . . . . . . 133

B.2 Generator and band structure for two-band U = 2 FB networks . . . . 134

B.2.1 Real H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.2.2 Complex H1 . . . . . . . . . . . . . . . . . . . . . . . . . . 135

B.2.3 Degenerate H0 . . . . . . . . . . . . . . . . . . . . . . . . . 136

B.2.4 FB energy equals one of the eigenvalues of H0: Reduction to

U = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.3 Generalized sawtooth chain . . . . . . . . . . . . . . . . . . . . . . . 137

B.4 Inverse eigenvalue problem: A toy example and the solution of the

U = 2 CLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.4.1 Toy example . . . . . . . . . . . . . . . . . . . . . . . . . . 140

B.4.2 U=2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B.4.3 Bipartite lattices and chiral symmetry . . . . . . . . . . . . . 143

B.5 Resolving the non-linear constraints . . . . . . . . . . . . . . . . . . 145

B.5.1 U=2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

B.5.2 U=3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

B.6 Examples for FB generators . . . . . . . . . . . . . . . . . . . . . . 148

B.6.1 ν = 3, U = 2 case . . . . . . . . . . . . . . . . . . . . . . . 148

B.6.2 ν = 3, U = 3 case . . . . . . . . . . . . . . . . . . . . . . . 151

B.7 Network constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B.7.1 U=2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

B.7.2 U=3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

iii

C Supplementary materials for the flatband generator in 2D 159C.1 Two-band problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

C.2 More than two bands with nearest neighbor hoppings . . . . . . . . . 160

C.2.1 U=(2,1,0) case . . . . . . . . . . . . . . . . . . . . . . . . . 160

C.2.2 U=(2,2,1) case . . . . . . . . . . . . . . . . . . . . . . . . . 161

C.2.3 Special case for U=(2,2,1) with nearest neighbor hoppings . . 167

C.2.4 U=(2,2,0) case with three bands . . . . . . . . . . . . . . . . 169

C.3 Next nearest neighbor hoppings . . . . . . . . . . . . . . . . . . . . 174

C.3.1 U=(2,2,1) case . . . . . . . . . . . . . . . . . . . . . . . . . 174

C.3.2 U=(2,1,0) case . . . . . . . . . . . . . . . . . . . . . . . . . 177

D Supplementary materials for the non-Hermitian flatband generator 181D.1 CLS-based generator . . . . . . . . . . . . . . . . . . . . . . . . . . 181

D.1.1 U = 1 case . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

D.1.2 U=2 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

D.1.3 U=3 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

D.1.4 Inverse eigenvalue method for CLS approach . . . . . . . . . 186

D.2 Solving completely flat bands . . . . . . . . . . . . . . . . . . . . . . 187

D.2.1 Both bands are completely flat . . . . . . . . . . . . . . . . . 187

D.2.2 One band is completely flat . . . . . . . . . . . . . . . . . . . 189

D.3 Solving partially flat bands . . . . . . . . . . . . . . . . . . . . . . . 191

D.3.1 Real parts of both bands are flat . . . . . . . . . . . . . . . . 191

D.3.2 Real part of one band is flat . . . . . . . . . . . . . . . . . . 195

D.3.3 Imaginary parts of both bands are flat . . . . . . . . . . . . . 197

D.3.4 Imaginary part of one band is flat . . . . . . . . . . . . . . . 200

D.3.5 Modulus of a band is flat . . . . . . . . . . . . . . . . . . . . 201

iv

vi

List of Figures

1.1 Flatband systems and perturbations . . . . . . . . . . . . . . . . . . . 2

1.2 Flatbands in different setups . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Lattices with isolated unit cells and interacting unit cells . . . . . . . 12

2.2 Cross-stitch lattice with one flatband . . . . . . . . . . . . . . . . . . 13

2.3 Kagome lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Example of a cell construction method – sawtooth chain . . . . . . . 16

2.5 Coupling an isolated lattice to a dispersive chain . . . . . . . . . . . . 17

2.6 Generalization of the entangling procedure . . . . . . . . . . . . . . . 19

2.7 Flatband construction by local symmetry partitioning . . . . . . . . . 22

2.8 Dice lattice and its band structure . . . . . . . . . . . . . . . . . . . . 24

2.9 Wannier–Stark ladder of a dice lattice . . . . . . . . . . . . . . . . . 25

2.10 Experimental realizations of flatbands in engineered atomic lattices

and optical lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.11 Photonic crystal waveguide of a kagome lattice. . . . . . . . . . . . . 28

2.12 Optical waveguide Lieb lattice and exciton-polariton stub lattice. . . . 30

3.1 1D examples of U classification . . . . . . . . . . . . . . . . . . . . 34

3.2 2D examples of U classification . . . . . . . . . . . . . . . . . . . . 35

3.3 Relationship between reducible and irreducible CLSs, and linear de-

pendence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Reduction of U class CLS to U − 1 . . . . . . . . . . . . . . . . . . 40

3.5 A cross-stitch lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Schematic of a CLS in a two-band lattice. . . . . . . . . . . . . . . . 46

4.1 Canonical ν = 2 chain, generalized sawtooth, ST1, and ST2 chains . . 53

4.2 Parameter space of a flatband in a 1D two-band network . . . . . . . 56

4.3 Band width vs. band gap in a 1D two-band FB network . . . . . . . . 57

4.4 Examples of flatband Hamiltonians with CLSs of class U = 2, ν = 3 . 62

vii

4.5 Examples of 1D U = 3 flatband lattices . . . . . . . . . . . . . . . . 65

4.6 Example of 1D bipartite flatband lattice . . . . . . . . . . . . . . . . 67

4.7 Examples of flatband construction under network constraints . . . . . 68

5.1 Simple example of nearest neighbors and next nearest neighbors in 2D 70

5.2 The U-classification of various 2D FB lattices . . . . . . . . . . . . . 71

5.3 Classification of different CLSs and destructive interference in 2D . . 72

5.4 Classification of CLSs in 2D lattices with next nearest neighbor hoppings 73

7.1 Photograph of the basin plate of a microwave Dirac billiard . . . . . . 92

7.2 Schematic view of the triangular lattice structure of the billiard . . . . 93

7.3 Comparison of the experimental DOS with the computed band structure 94

7.4 The honome lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.5 Relative positions of peaks and Dirac points . . . . . . . . . . . . . . 100

7.6 Plain Monte Carlo fitting with three hoppings . . . . . . . . . . . . . 102

7.7 Importance sampling fitting with three hoppings . . . . . . . . . . . . 103

7.8 Plain Monte Carlo fitting with five hoppings . . . . . . . . . . . . . . 103

7.9 Importance sampling fitting with five hoppings . . . . . . . . . . . . 104

viii

List of Tables

7.1 Relative positions (RP) of van Hove singularities (VHS) extracted from

experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

ix

x

Chapter 1

Motivation and outline

Localization is widespread in condensed matter physics, such as Anderson localization

in the presence of disorder, states localized to the edges of topological insulators, and

so on. In general, disorder is needed to ensure localization, yet there is a special type of

localization in translational invariant tight-binding networks where lattice geometry or

symmetry leads to destructive interference, which results in the compact localization

of wave functions in a finite number of lattice sites. Such states are called compact

localized states (CLSs), and their most striking property is their ability to generate dis-

persionless energy bands, i.e. flatbands (FB), which are macroscopically degenerate.

Systems with macroscopic degeneracies are rare in physical systems, since the high

degree of symmetry or fine-tuning needed to support them is easily destroyed by weak

perturbations. However, this fragility is the very reason that makes macroscopic de-

generacies attractive. Small perturbations in such a fine-tuned system will typically

lift the degeneracy and yield uniquely defined eigenstates, thereby leading to exotic

physical phenomena that may qualitatively differ for different perturbations (see Fig.

1.1). Thus, macroscopic degeneracies could host endpoints of various phase transi-

tion lines, and promise rich physics in their neighborhood. Therefore, searching for

FB lattice models has been the focus of a wide range of theoretical and experimental

research. Nowadays, manufacturing technologies are approaching the realization of

such fine-tunings—perhaps not with exact precision, but with some level of control.

Studies of FBs started in the early 1990s, when Mielke showed that a special type

of lattices (known as line graphs in mathematics) can have a ferromagnetic ground

state if the lowest band is flat [105]. Subsequently, the same type of ferromagnetism

associated with FBs was found in further types of lattices by Tasaki [174]. Flatband

models have also been applied in Heisenberg spin models to show the existence of fully

polarized anti-ferromagnetic ground states under an external magnetic field [140, 149,

1

2 Motivation and outline

Flatband

Flatbandferromagnetism

AndersonlocalizationMultifractality

DisorderinducedtopologicalphasetransitionLandau-Zener-Bloch

oscillationsWannierStarkbands

Superfluidity

Heizenberganti-ferromagnet

Mobilityedge

Compactbreathers

Disorder

Figure 1.1: Various physical phenomena in flatband systems under different perturba-tions.

150]. Currently, FBs are being studied in various fields of condensed matter physics,

and have been experimentally realized in 1D, 2D, and 3D systems (see Fig. 1.2). For

example, on the atomic scale and in the ultra low temperature region, FBs have been

shown in cold atoms in optical lattices [5, 168]. In the room temperature regime, FBs

are studied theoretically and experimentally in photonic crystal waveguide networks

and optical waveguide arrays [111, 151, 183]. Flatbands have also been observed in

electronic systems such as superconducting wire networks and nano-engineered atomic

lattices on metallic surfaces [41, 115, 161]. In cavity quantum electrodynamics (QED)

setups, FBs have been realized in exciton–polariton condensates [10, 86].

Figure 1.2: Flatbands in various setups on different scales and in different fields.

Theoretically, many different approaches have been devised to construct FB lat-

tices, such as Mielke’s line graph approach [105], Tasaki’s cell construction method [174],

3

origami rules [37], local unitary transformation [49], local symmetry partitioning [142],

chiral symmetry [133], repetition of mini arrays [109], etc. Most of these construction

methods are limited to either exploiting specific symmetries, like chiral symmetry, or

considering specific geometric properties, like those in line graphs. Therefore, known

FB models, in most cases, correspond to some highly symmetric points in the param-

eter space of FB Hamiltonians. By adjusting system parameters though, it becomes

possible to achieve FBs without the support of symmetry or special geometry, leading

to increased flexibility in both model design and experimental realization. In terms of

CLSs, while they play a decisive role in the behaviour of FB models, little is known

about how they are linked to lattice properties, or how they determine the response to

various perturbations, etc. In this regard, systematically classifying and constructing

FB Hamiltonians based on CLSs is highly desirable.

In this thesis, we consider a single particle (non-interacting) Schrödinger equation

(Hamiltonian) for a discrete translational invariant tight-binding network that possesses

at least one FB. We explore the properties of the CLSs and systematically construct

FB Hamiltonians that have the required CLS properties. The FB generator is a scheme

that generates all possible FB Hamiltonians possessing CLSs with specific sizes and

shapes. This involves solving a set of eigenvalue problems subjected to some con-

straints.

In Chapter 2, we introduce the basic concepts used in the rest of the thesis. Start-

ing with a discrete translational invariant system, we review the tight-binding model,

Bloch theorem, and band structure. Then, using a simple 1D example, we elucidate

the origins of FBs, CLSs, and macroscopic degeneracy. Next, we cover the existing

FB construction methods and discuss the related applications and experimental real-

izations.

In Chapter 3, we introduce a method to classify FB lattices according to their CLSs.

Then we show various properties of CLSs, and propose a block matrix representation

of tight-binding Hamiltonians, through which we mathematically formulate the condi-

tions for the existence of CLSs. Based on these conditions, we present our core idea

of FB generation.

In Chapter 4, we study FB generators in 1D systems. Starting from a two-band

problem, we show the full parameterization of FB Hamiltonians and identify the pa-

rameter space in which we can obtain FBs. Moving to a higher number of bands, we

introduce an inverse eigenvalue method that yields analytic and numerical solutions

for FB Hamiltonians.

In Chapter 5, we extend the 1D FB generator to 2D. We introduce a classification

4 Motivation and outline

scheme for CLSs that occupy a maximum of 4 unit cells. Borrowing the methods

from the 1D case, we derive analytic solutions for FB Hamiltonians for different CLS

classes. Our results cover all known examples, and a vast amount of new models can

be generated from our solutions.

In Chapter 6, we extend the idea of FB generation to the non-Hermitian regime,

where we only consider a 1D non-Hermitian lattice with two bands. Using k-space

analysis, we achieve the full parameterization of non-Hermitian FB Hamiltonians with

both complete and partial FBs, as well as bands in which the modulus is flat.

In Chapter 7, we apply our methodologies for FB generation to the design of a

tight-binding model that explains the experimental results from a microwave photonic

crystal. We develop a numerical algorithm to fit the experimental data, and explain the

singularity in the experimentally observed spectrum using the FB in our tight-binding

model.

We conclude the thesis with a summary of our results along with their significance,

before discussing interesting open problems and future research directions.

Chapter 2

Introduction: Flatbands in discretesystems

Systems with translational symmetry are common throughout physics, for example

periodic arrangements of atoms in a solid crystal. A single particle in such a system

can be described by a Schrödinger equation with periodic potential, as(−~2∇2

2m+ V (~r)

)ψ(~r) = Eψ(~r), (2.1)

where the potential V is periodic with periodicity ~R, such that

V (~r + ~R) = V (~r) . (2.2)

For a 1D system, ~R is an integer multiple of the smallest periodicity. For higher di-

mensions, ~R is a vector sum of the integer multiples of the smallest periodicities in

each spacial dimension. Periodic systems are described by a Bloch wave [8, 84, 159],

which is the solution of Eq. (2.1).

In Bloch representation, the energies of the system depend on the wave vector,

which is termed dispersion relation and forms an infinite number of energy bands.

In practice though, physicists are interested in finite numbers of bands that can be

addressed separately from the rest of the bands (e.g. gapped bands). When a fi-

nite number of bands is considered, the system is well-described by the tight-binding

model [8, 160]. In this model, particles are considered to be localized around a discrete

set of periodically arranged points in space, which is called a lattice. In general, lat-

tices have translational invariance, with the periodically arranged points referred to as

lattice sites. We focus on a special type of lattice that possesses dispersionless energy

bands, which we call flatband (FB) lattices.

5

6 Introduction: Flatbands in discrete systems

This thesis is dedicated to identifying FB Hamiltonians in discrete translational in-

variant systems under tight-binding approximation, i.e. lattice systems. In this chapter,

we cover the fundamentals of band structure in such systems as well as FBs. We begin

with the tight-binding model and the Bloch theorem in Section 2.1, before turning to

FB lattices in Section 2.2 where we introduce FBs, compact localized states (CLSs),

and macroscopic degeneracy. Then, in Section 2.3, we briefly describe FB construc-

tion methods, and in Section 2.4 we review applications of FBs in various fields, from

solid-state physics to photonics. The chapter is summarized in Section 2.5.

2.1 Tight-binding model for discrete systems

Generally, in discrete translational invariant lattice systems, the lattice sites are well

separated such that the following approximation is commonly applied.

Definition 2.1. Tight-binding approximation: When the overlaps of the wave functions

of all neighboring sites are small, the overlaps can be neglected. As a result, the wave

functions are considered to be well-localized around their lattice sites.

In this approximation, the wave functions live in discrete points in space, and thus

can be labeled by the index of their lattice sites.

2.1.1 Tight-binding model

The time-independent Schrödinger equation for a translational invariant lattice is writ-

ten as

H|Ψ〉 = E|Ψ〉, (2.3)

where H is the Hamiltonian matrix of the system and |Ψ〉 is a state vector. If there are

N lattice sites, then H is an N ×N matrix and |Ψ〉 is an N -component vector.

As previously stated, the overlaps between the wave functions of neighboring sites

are zero in the tight-binding approximation. This allows us to introduce a basis vector

|n〉 =

...

0

1

0...

, (2.4)

2.1 Tight-binding model for discrete systems 7

which represents the particle living in the nth lattice site, where 1 is the nth element

of |n〉. Therefore, |n〉 is a position eigenstate, whose dimension is equal to the total

number of sites in the lattice. Then the set {|n〉} forms a complete basis of the Hilbert

space of the lattice as

〈n|m〉 = δn,m ,∑n

|n〉〈n| = 1. (2.5)

Since the wave functions are localized in the tight-binding approximation, transport

only occurs via hoppings between neighboring lattice sites. Hopping can be understood

as a quantum tunneling effect, and is defined as a matrix element of the tight-binding

Hamiltonian:

tnm = 〈n|H|m〉. (2.6)

This equation gives the hopping strength between the nth and mth sites, where H is

the Hamiltonian of the lattice. Therefore, the Hamiltonian in Eq. (2.3) can be written

in tight-binding form as

H =∑n

εn|n〉〈n|+∑n,m

tnm|n〉〈m| , (2.7)

where εn = 〈n|n〉 is the onsite energy of the nth lattice site. In this tight-binding

Hamiltonian, if only nearest neighbor hoppings are considered, m runs over all nearest

neighboring sites.

We can also write the state vector |Ψ〉 of the entire lattice as linear combinations of

the basis |ψn〉 as

|Ψ〉 =∑n

φn|n〉 =1√N

...

φn−1

φn

φn+1

...

,∑n

|φn|2 = 1 (2.8)

where φn is the wave function of the nth site. Thus, the probability of finding a particle

in the nth site is |φn|2.

Putting Eqs. (2.7) and (2.8) into Eq. (2.3), we get single-particle wave function φnat the nth site that satisfies

εnφn +∑m

tnmφm = Eφn. (2.9)


If there are ν sites per unit cell, it is convenient to label each site with a unit cell

index. The basis vectors then become

|n, j〉 =

...

0

1

0...

, 〈n, j|m, j′〉 = δn,mδj, j

′,∑n,j

|n, j〉〈n, j| = 1 , (2.10)

which represents the particle living in the jth site of the nth unit cell. Then it is helpful

to introduce a state vector for a single unit cell, as

|Φn〉 =∑j

φn,j|n, j〉 =

...

0

~ψn

0...

, (2.11)

where the single unit cell wave function ~ψn is a ν-component vector

~ψn =

φn,1

φn,2...

φn,ν

, (2.12)

whose component φn,j=1,...,ν is the wave function of the jth site in the nth unit cell. As

before, then the probability of finding the particle in the jth site of the nth unit cell is

|φn,j|2.

The set {|Φn〉} now forms the orthonormal basis

〈Φn|Φm〉 = δn,m,∑n

|Φn〉〈Φn| = 1 . (2.13)

The state vector of the entire lattice can be written in terms of the single unit cell state

2.1 Tight-binding model for discrete systems 9

vectors |Φn〉 as

|Ψ〉 =∑n

|Φn〉 =

...~ψn−1

~ψn~ψn+1

...

. (2.14)

Then, the tight-binding Hamiltonian reads

H =∑n,j

εj|n, j〉〈n, j|+∑

tn,j,m,j′ |n, j〉〈m, j′|, (2.15)

where

tn,j,m,j′ = 〈n, j|H|m, j′〉 (2.16)

is the hopping between the jth site of the nth unit cell and the j′th site of the mth unit

cell. Note that tn,j,n,j′ is intracell hopping, or the hoppings between sites inside the

same unit cell.

Putting Eqs. (2.14) and (2.15) into Eq. (2.3), we arrive at the eigenvalue problem

for the tight-binding model:

εjφn,j +∑m,j′

tn,j,m,j′φm,j′ = Eφn,j, j = 1, . . . , ν. (2.17)

We will use this notation extensively throughout the thesis.

2.1.2 Bloch theorem and band structure

The Bloch theorem states that the solution of the Schrödinger equation (2.3) for dis-

crete translational invariant systems with ν sites per unit cell can be written as Bloch

waves (plane waves multiplied by a periodic function with a periodicity equaling lattice

periodicity) as~ψn(~k) = ~u(~k)ei

~k·~Rn , (2.18)

where ~ψn and Bloch function ~u(~k) are ν-component vectors, ~Rn is the lattice trans-

lation vector for the nth unit cell, and ~k is the wave vector. The Bloch function is

invariant under lattice translations, i.e. ~u(~k, ~R0) = ~u(~k, ~R0 + ~Rn).

Since single-site wave functions form a complete basis, the extended Bloch func-

tion can be approximated by a linear combination of single-site wave functions. For


simplicity, we take a lattice with a single site per unit cell, then

u(~k) =1√A

∑n

φnei~k·~Rn , (2.19)

where ~Rn is the lattice translation vector to the nth site, A is a normalization factor,

and φn is the wave function of the nth site. Note that Eq. (2.19) is a Fourier transform

of φn. Then, φn can also be written as an inverse Fourier transform of u(~k) as

φn =1√B

∑~k

u(~k)e−i~k·~Rn , (2.20)

where B is a normalization factor. Putting Eq. (2.20) into Eq. (2.9) and canceling out

the normalization factor and∑

~k u(~k)e−i~k·~Rn , we get

E(~k) = ε+∑m

tnme−i~k·~Rm−n , (2.21)

where ε is onsite energy, ~Rm−n = ~Rm − ~Rn, and we use e−i~k·~Rm = e−i~k·~Rne−i

~k·~Rm−n .

In Eq. (2.21), E(~k) for all ~ks forms the energy band of the lattice. For example, if we

have a 1D lattice with nearest neighbor hopping t, then Eq. (2.21) gives

E(~k) = ε+ 2t cos(k). (2.22)

When there are multiple ν sites per unit cell, one can also rewrite Eqs. (2.19) and

(2.20) in terms of the single unit cell state vector from Eq. (2.11). In this case, the

Bloch function for each band also becomes a ν-component vector, ~u(~k). The Bloch

function ~uµ=1,...,ν(~k) of the µth band can be written as a Fourier transform of single

unit cell wave functions by

~uµ(~k) =1√A

∑n,j

~ψnei~k·~Rn , (2.23)

where ν-component vector ~ψn is the single unit cell wave function of the nth unit cell.

Now, ~ψn can be written as an inverse Fourier transform of ~uµ(~k),

~ψn =1√B

∑~k

uµ(~k)e−i~k·~Rn , µ = 1, . . . , ν. (2.24)

2.2 Flatbands, compact localized states, and macroscopic degeneracy 11

Putting Eq. (2.24) into Eq. (2.17), we obtain the following equation for every ~k:

Eµ(~k)uµj(~k) =∑j

εjuµj(~k) +∑m,j′

tn,j,m,j′uµj′(~k)e−i~k·~Rm−n , j = 1, . . . , ν , (2.25)

where uµj(~k) is the jth component of ~uµ(~k), εj is onsite energy of the jth site in a unit

cell, and ~Rm−n = ~Rm − ~Rn. The eigenvalue Eµ(~k) of Eq. (2.25) for all ~k forms the

µth energy band of the lattice, with µ = 1, . . . , ν.

For finite lattices, only the set of wave vectors ~k is allowed, determined by Born–

von Karman periodic boundary conditions, which are written as

~ψn(~k) = ~u(~k)ei~k·~Rn = ~u(~k)ei(

~k·~Rn+Nj~aj) , (2.26)

where ψn(~k) is the Bloch wave, j runs over the dimensions of the lattice, ~aj are the

primitive vectors of the lattice, and Nj are integers (assuming the lattice has N cells

where N = N1N2N3). This implies that

eiNj~k·~aj = 1 , (2.27)

and so the allowed wave vectors are

~k = 2π3∑j=1

mj

Njaj, mj = 1, 2, . . . Nj . (2.28)

The set of allowed ~ks forms a space called reciprocal space or k-space, which is also

defined as the Fourier transform of a direct lattice. As it can be seen from Eq.(2.28),

there are as many number of allowed ~ks as the number of unit cells in the lattice.

2.2 Flatbands, compact localized states, and macroscopic

degeneracy

Flatbands are dispersionless energy bands (Eµ(k) = const) of translational invariant

tight-binding networks, which are the consequence of destructive interference. In this

section, we take a 1D cross-stitch lattice as an example to show how FBs are formed.

We start from the most trivial case of isolated lattice sites, i.e. all hoppings are

zero. If the unit cell contains a single site, then each site is described by the following

Schrödinger equation,

Haφa = εφa, (2.29)


where ε is onsite energy and φa is the local wave function. The Hamiltonian of this

system is a diagonal matrix with a highly degenerate eigenvalue ε. Therefore, the

whole system has only one energy E = ε that gives a dispersionless energy band—a

flatband. Note that this is the only possibility where one can get a FB in a single band

system. Similarly, lattices with isolated unit cells with ν sites per unit cell give ν FBs.

For example, Fig. 2.1 (a) shows a lattice with isolated unit cells of two sites, giving

two trivial FBs. When the unit cells get closer and the hoppings between them are

non-zero due to tunneling, as in Fig. 2.1 (b), the bands generally become dispersive.

Figure 2.1: Lattices and corresponding band structures for (a) isolated unit cells and(b) interacting unit cells. The curved lines below the lattices illustrate the potentialaround the sites. In (a), the band structure is obtained with ε = 0, 1 and intracellhopping 1. In (b), the band structure is obtained with ε = 0, 1, intra-cell hoppings 1,horizontal hoppings −2, and diagonal hoppings 1.

Besides this trivial case of isolated unit cells, there are lattices with non-zero hop-

pings between unit cells that still have FBs. In such systems, due to lattice geometry

or symmetry, wave functions hopping from neighboring sites possess opposite phases

with equal amplitudes, and so interfere destructively at certain sites—this is called de-

structive interference. As a result, the wave functions are trapped in a finite number

of lattice sites and are strictly zero elsewhere (see Fig. 2.2). Such strictly localized

states are called compact localized states (CLS), which cause net hoppings to vanish,

giving an analogous effect of isolated unit cells. We explain this situation in a simple

example below.

Consider a 1D cross-stitch lattice, as shown in Fig. 2.2 (a). If we set onsite energies

to zero and tune the hoppings as

tnm = −tδ|n−m|,1 , (2.30)

2.2 Flatbands, compact localized states, and macroscopic degeneracy 13

Figure 2.2: A cross-stitch lattice with one FB. The filled circles show the locations ofthe CLSs, and the empty circles show the sites with destructive interference where thewave function is strictly zero. j is the unit cell index, and aj, bj are the wave amplitudesat each lattice site inside the jth unit cell. (a) The CLS is located in the jth unit cell,and its band structure corresponds to t = 1, aj = −bj = 1. (b) Superposition of twolattice translations of the CLS in (a).

then Eq. (2.17) gives the following eigenvalue problem

Eaj = εajaj − taj−1 − taj+1 − tbj−1 − tbj+1 − tbj,

Ebj = εaj bj − tbj−1 − tbj+1 − taj−1 − taj+1 − taj.(2.31)

This has one FB at energy EFB = t. As shown in Fig. 2.2 (a), one possible eigenstate

of the tight-binding Hamiltonian corresponding to this FB energy is

|Ψj〉 =

...

0

1

−1

0...

=

...

0

~ψjCLS

0...

, (2.32)

where ~ψjCLS = (aj, bj)T = (1,−1)T , and index j refers to the location of the non-zero

amplitudes at the jth unit cell.

In Fig. 2.2 (a), the CLS (in Eq. (2.32)) is located in the jth unit cell and is the

consequence of destructive interference in the neighboring sites. More precisely, the

sum of the hopping amplitudes from aj, bj to the neighboring sites is zero, i.e. aj+1 =

−taj − tbj = 0, which indicates that wave amplitudes at the jth unit cell must satisfy


aj = −bj in order to have a CLS.

The lattice translations of the CLS in Eq. (2.32) are also solutions of Eq. (2.31);

i.e. for ∀j, these lattice translations are eigenstates of the tight-binding Hamiltonian

(2.15) corresponding to EFB = t. Equation (2.3) then reads

H|Ψj〉 = EFB|Ψj〉 → for ∀j. (2.33)

The superposition of all lattice translations of |Ψj〉 is also an eigenstate of the tight-

binding Hamiltonian corresponding to EFB = t (see Fig. 2.2 (b)), as

H|Ψ〉 = EFB|Ψ〉, |Ψ〉 =∑j

cj|Ψj〉. (2.34)

Therefore, the whole system has one k-independent energy at EFB = t, which forms

the FB.

Now suppose there are N unit cells in this example lattice, giving N lattice transla-

tions of the CLS which are all eigenstates of the FB. The superpositions of these copies

are also eigenstates, thereby forming macroscopic degeneracy.

We note that a given FB eigenstate could be a linear combination of smaller eigen-

states, as shown in Eq. (2.34). However, one can always define an irreducible eigen-

state for a given FB lattice that cannot be decomposed into the superposition of smaller

eigenstates. For example, in the above 1D example, Eq. (2.32) is an irreducible eigen-

state. In the next chapter, we will discuss this irreducibility in detail, as it determines

most of the properties of FB lattices.

More generally, the parameters in the example shown in Fig. 2.2 are not the only

parameters that give a FB in a cross-stitch lattice. For example, let’s suppose the hor-

izontal hoppings are t1, diagonal hoppings are t2, and t1 6= t2. In order to achieve

destructive interference in the aj+1 site, we need t1aj + t2bj = 0. In the same way,

we can achieve destructive interference at all other neighboring unit cells such that

the state is strictly localized at the aj, bj sites (i.e. the jth unit cell). Therefore, FB

networks are fine-tuned such that we can control the hoppings as well as the wave am-

plitudes to achieve destructive interference and thus the FB. This is one of the reasons

why we aim in this thesis at identifying all possible FB Hamiltonians for a given type

of lattice. For example, in Chapter 4 we identify all possible 1D FB Hamiltonians with

two bands.

Above, we took a simple 1D example to explain the origin of FBs, which naturally

extends to higher lattice dimensions, where FBs also originate from the same mech-

anism (destructive interference). Our explanations on the concepts of CLSs, macro-

2.3 Flatband construction methods 15

scopic degeneracy, and fine-tuning also apply to higher lattice dimensions.

2.3 Flatband construction methods

In this section, we briefly review existing methods for designing FB lattices. The

central idea in modeling FB lattices is to achieve destructive interference that supports

CLSs.

2.3.1 Geometrical methods

One direct approach to achieving destructive interference is through lattice geometry.

This approach is shown through the following four examples.

Figure 2.3: (a) Kagome lattice and (b) its band structure. The original honeycomblattice is shown in blue dotted lines and blue circles, and the kagome lattice is shownin black solid lines and circles. Filled and empty black circles correspond to latticesites with non-zero and zero wave amplitudes, respectively. The non-zero amplitudesof the CLS are located in an elementary cycle with alternating "+" and "–" signs, andarrows indicate the two wave functions interfering destructively. The band structurehas one FB (red).

.

• Line graph approach: The line graph is a special type of mapping of an orig-

inal lattice, which is basically a bond-site exchange. More precisely, given an

original lattice, a line graph is obtained by first assigning a site for each of the

bonds in the original lattice, and then connecting the sites in the line graph if the

corresponding bonds in the original lattice share a common site. Mielke [104,

104, 105, 107] pointed out that destructive interference is a common feature of

line graphs. Compact localized states are located in a vertex-disjoint elementary

cycle with alternating signs of amplitudes [106, 110], which gives destructive


interference (see Fig. 2.3). The vertex-disjoint cycles in graph theory are cy-

cles that do not share common sites. From 1991–1993, Mielke [104, 105, 107]

showed that, in line graphs with Hubbard interaction, these CLSs can form a

highly degenerate ferromagnetic ground state for a certain electron density. One

typical example is a 2D Kagome lattice, which is a line graph of a honeycomb

lattice, as shown in Fig. 2.3.

• Cell construction: As proposed by Tasaki [173, 174, 176], cell construction

starts from an elemental cell consisting of a single internal site and two or more

external sites. These cells are then assembled to form a lattice by sharing the

external sites, where destructive interference occurs. One typical example of the

cell construction method is a 1D sawtooth chain, in which the elemental cell is

a triangle, as shown in Fig. 2.4. These triangular elemental cells are arranged in

such a way that they share the external sites with neighboring elemental cells.

Figure 2.4: (a) Elemental cell, (b) 1D sawtooth chain, and (c) its band structure. Theelemental cell has two external sites (gray circles) and one internal site (black circle).The sawtooth chain is constructed by assembling elemental cells with shared externalsites (gray circles). When onsite energies are the same and the ratio between diagonalhoppings (solid lines) and baseline hoppings (dashed line) is

√2, the sawtooth lattice

has one FB, as shown in (c).

• Origami rules: Traditionally a technique to fold paper into beautiful shapes,

origami was popularized in mathematics by Fusè through "modular" or unit

origami [52, 53]. In 2015, Dias et al. [37] introduced a method to construct

localized eigenstates in a Hubbard model using origami rules. In this method,

they started from a plaquette or a set of plaquettes with a higher symmetry than

that of the whole lattice, with compact localized modes living on these plaque-

ttes that result in FBs. Then a simple set of rules were applied to divide, fold,

or unfold the tight-binding localized states in such plaquette(s) to new plaquette

geometries.


• Repetition of miniarrays: In 2016, Morales-Inostroza et al. [109] introduced

a simple way to construct FB lattices using a repetition of miniarrays. This

method starts from a miniarray and consecutively adds new miniarrays using a

connector site to form 1D and 2D lattices having single or multiple FBs. Using

lattice geometry and by tuning the hoppings and wave functions of each site,

destructive interference is achieved at the connector sites.

A common feature among the above methods is that destructive interference takes

place at the corner sites in lattices composed of corner-sharing triangles. As a result,

CLSs surrounded by these triangles are formed that give rise to FBs.

2.3.2 Flatband generation from local unitary transformations

For FB lattices with a CLS occupying the U = 1 unit cell, it is always possible to

detangle FB eigenstates from the dispersive part of the spectrum [49]. Reverting this

procedure leads to one of the most generic FB construction methods for a CLS occu-

pying the U = 1 unit cell—the entangling method [49]. More precisely, the method

starts from a lattice with isolated sites in each unit cell, then applies a unitary transfor-

mation to couple these sites to the rest of the lattice. This procedure preserves the FB

that is formed by these isolated sites [49]. Following Ref. [49], we present an example

to illustrate this method.

Figure 2.5: Coupling (a) an isolated site to a dispersive chain to achieve (b) a cross-stitch lattice with a FB.

Consider a 1D chain with one isolated site nearby each lattice site in the chain, as

shown in Fig. 2.5 (a). If we set the hoppings between nearest neighboring sites to be

−1, then the eigenvalue problem reads

Epj = εppj − pj−1 − pj+1,

Efj = εffj .(2.35)


It is convenient to introduce the following hopping matrices,

H0 =

(εp 0

0 εf

), H1 =

(−1 0

0 0

), ~ψj =

(pj

fj

), (2.36)

where H0 gives the hoppings and onsite energies inside the unit cell, H1 gives the

hoppings between nearest neighboring unit cells, and ~ψj is wave function of the jth

unit cell. The eigenvalue problem in Eq. (2.35) becomes

H0~ψj +H†1

~ψj−1 +H1~ψj+1 = E~ψj. (2.37)

The lattice described by this equation has one dispersive band formed by the connected

sites and one FB formed by the isolated sites, as

EFB = εf , E(k) = εp − 2 cos(k) . (2.38)

We can apply unitary transformation to the system, which keeps band structure

unchanged. More precisely, if we apply local rotation in the space {pn, fn},

H0 = UH0U†, H1 = UH1U

†, ψj = U ~ψj

U =

(cos(θ) − sin(θ)

sin(θ) cos(θ)

),

(2.39)

the eigenvalue problem after unitary transformation reads,

Eψj = H0ψj + H†1ψj−1 + H1ψj+1, (2.40)

which gives the same band structure as in Eq. (2.38).

If we choose θ = π/4, we have

H0 =1

2

((εf + εp) (εp − εf )(εp − εf ) (εf + εp)

), H1 =

1

2

(−1 −1

−1 −1

), ψj =

1√2

(pj + fj

pj − fj

).

(2.41)

If we make the following variable replacements,

aj =1√2

(pj + fj), bj =1√2

(pj − fj),

ε = εf + εp t = εp − εf ,(2.42)

we get a cross-stitch lattice as shown in Fig. 2.5 (b).


This procedure transformed a 1D chain with isolated sites into a cross-stitch lattice

using local unitary operator U , while keeping the same band structure, and thus the

FB. A similar procedure can be applied to other lattices with CLS size U = 1; this

procedure can also be generalized to any dimension, as described below.

Figure 2.6: Generalization of the entangling procedure. (a) A 2D square lattice withisolated sites, and (b) rotated version. Bonds show only the connectivity, not the actualvalues. This figure is taken from [49].

Entangling method for flatband construction: Consider a d-dimensional disper-

sive lattice with m bands (m sites per unit cell), then assign n decoupled sites to each

unit cell. By applying local unitary transformation, we can couple these isolated sites to

the remaining sites and form a lattice with n FBs, which has a CLS of size U = 1 [49].

The graphical illustration of the simplest transformation in 2D with m = n = 1 is

shown in Fig. 2.6 (b).

This method is the most generic method to construct FB lattices with a CLS oc-

cupying a single unit cell. For CLS sizes larger than 1, one cannot decouple the FB

into isolated sites because the CLSs do not form an orthogonal basis [49]; this is dis-

cussed in the next chapter. Therefore, the inverse procedure (entangling procedure) is

not possible for CLS size U ≥ 2. In later chapters, we introduce other methods for

constructing FB lattices with CLS size U ≥ 2.

2.3.3 Flatband construction from symmetry

As mentioned previously, in most known cases, the existence of FBs depends on spe-

cial lattice geometry or symmetry, and we have already reviewed the lattice geometry-

based FB construction methods. In this section, we discuss symmetry-based FB con-

struction methods.

Chiral symmetry

Bipartite lattices host FBs that are protected by chiral sysmmetry, and a systematic way

to construct chiral FBs has been proposed by Ramachandran et al. [133]. Based on this


work, we discuss this chiral FB-generating principle.

Bipartite lattice: A lattice consisting of two sublattices A and B, such that lattice

sites in A are coupled only to sites in B, and vice versa.

Chiral symmetry: In lattice systems, chiral symmetry refers to both particle-hole

and time-reversal symmetry. This concept originated from quantum chromodynamics

(QCD) [79, 153, 154]. The consequence of chiral symmetry is: if {ψA, ψB} is an

eigenvector of eigenenergy E, then there is an eigenvector {±ψA,∓ψB} correspond-

ing to eigenenergy −E.

A bipartite lattice with chiral symmetry is called a chiral lattice. From Lieb’s the-

orem [96], it can be inferred that chiral lattices with an odd number of bands always

possess at least one chiral FB at energy E = 0. More precisely, if the number NA of

sublattice A sites is larger than the corresponding numberNB of sublattice B sites, then

there are at leastN = |NA−NB| states {ψA, 0} at energyE = 0 [96, 164], which only

live in sublattice A. The sublattice A (B) with larger (smaller) number of sites is called

the majority (minority) sublattice, with non-zero amplitudes of the eigenstates of en-

ergy E = 0 occupying the majority sublattice only. This result leads to a systematic

classification of chiral FBs.

Consider a d-dimensional translational invariant bipartite lattice with odd number

ν = µA +µB of sites per unit cell, with µA sites belonging to sublattice A and µB sites

belonging to sublattice B. Suppose sublattice A is the majority sublattice and B is the

minority sublattice, such that 1 < µB < µA < ν. The µA sites in any unit cell are only

connected to the remaining µB sites (some of them possibly belonging to other unit

cells) by nonzero hopping terms tlm. Then, the eigenvalue problem reads

EψA,Bl = −∑m

tlmψB,Am , (2.43)

and chiral symmetry leads to a symmetric band structure, i.e. any band Eµ(k) which

does not cross E = 0 is either positive or negative, and has a symmetry-related part-

ner band Eµ′(k) = −Eµ(k). Due to the odd number of bands, there is at least one

band without a symmetric partner band. Therefore, under chiral symmetry action, this

unpaired band must transform into itself, becoming a FB at energy E = 0.

Since ν is odd, the difference in the number of sites on sublattices A and B is

∆N = Nuc(2µA − ν) 6= 0, where Nuc = Ld is the number of unit cells, and L is

the linear dimension. This indicates a macroscopic degeneracy at E = 0, which is

possible only when there are (2µA − ν) FBs at E = 0. Classification of chiral FBs by

the imbalance of minority and majority sites follows from this observation, and can be


used to generate chiral FBs [133]. While this work considered Hermitian systems, the

concepts can apply to non-Hermitian systems as well.

Local symmetry partitioning

Schmelcher et al. [142] proposed a framework to design CLSs using local symmetry

properties of a discrete Hamiltonian, which can be used to construct tunable FB lat-

tices in 1D and 2D. In this framework, the Hamiltonian of the system can be block

partitioned using two recent theorems from graph theory [12, 50, 51]. Such partition-

ing of the Hamiltonian is induced by the symmetry of a given system under local site

permutations. Based on Ref. [142], we briefly discuss this local symmetry partitioning

method for constructing FB lattices.

Consider a three-site system as shown in Fig. 2.7 (a). The Hamiltonian of this

system is invariant under the permutation of sites 2 and 3. When the wave functions

at sites 2 and 3 have opposite phases, they interfere destructively at site 1, leading to

the localization of wavefunctions at sites 2 and 3. If an arbitrary system is connected

to site 1, as shown in Fig. 2.7 (b), the wave functions still localize at sites 2 and 3. The

eigenvalue corresponding to this localized state in the enlarged system is the same as

the original three-site system. Sites 2 and 3 together are called a symmetric subsystem,

and site 1 with the attached system is called a non-symmetric subsystem. Then, the

Hamiltonian of the system can be partitioned into blocks.

Now, if a dimer is added to a dispersive chain, with the hoppings and onsite en-

ergies as shown in Fig. 2.7 (c), such that the dimer has permutation symmetry, there

will be a CLS living in the dimer. Then the system Hamiltonian can be partitioned

into blocks corresponding to the symmetric part (dimer) and the non-symmetric part

(dispersive chain). When the system parameters are tuned such that the spectrum of

the non-symmetric part of the Hamiltonian coincides with the spectrum of the original

dispersive chain, the spectrum of the system consists of the band structure of the un-

perturbed chain augmented by the energy of the CLS. Since, in the present case, the

CLS does not interact with the extended states, one can add such dimers periodically

to form a lattice, as shown in Fig. 2.7 (d), whose spectrum consists of a FB from the

eigenenergy of the CLS.

As discussed in Ref. [142], the local symmetry (permutation) operators can be iden-

tified as commutative and non-commutative, depending on the commutation of these

operators with the system Hamiltonian. As in the above case, generally, the Hamiltoni-

ans of such systems with local symmetry can be partitioned into blocks corresponding

to symmetric and non-symmetric subsystems. Following a similar construction method


Figure 2.7: Flatband construction by local symmetry partitioning. (a) A three-sitesystem, which is symmetric under the permutation of sites 2 and 3. (b) The extendedsystem with an arbitrary subsystem (gray) attached to site 1 (which is fixed under per-mutation). (c) A dispersive chain locally perturbed by a symmetric dimer defect withindicated onsite and hopping elements. (d) A diamond chain formed by periodicallyadding dimers to the dispersive chain in (c). (e) A 2D FB lattice constructed by thelocal symmetry partitioning method and its band structure having two FBs at E1, E2.Figures (a)–(c) and (e) are taken from Ref. [142].

as above, more complicated FB lattices can be designed in 1D and 2D, as shown in Fig.

2.7 (e).

2.4 Applications and experimental realizations of flat-

band systems

Flatband lattices are fine-tuned, and the eigenstates of a FB are macroscopically de-

generate. Fine-tuning of the parameters of the FB Hamiltonian suggests their extreme

sensitivity to perturbations, which might change drastically the properties of the sys-

2.4 Applications and experimental realizations of flatband systems 23

tem and even trigger phase transitions leading to a plethora of interesting physical

phenomena. However as is demonstrated in the following chapters there are actually

many perturbations that preserve the flatness of the band. These perturbation induced

reach physical phenomena has led to FB lattices being the focus of many experiments

in different scales, different disciplines, and different energy scales as well. In this

section, we provide a brief review of such phenomena and experimental realizations.

2.4.1 Perturbation as a playmaker

Amidst the search for different FB models, other theoretical studies have been focusing

on the effects of different perturbations in FB lattices. In one early example, the effect

of repulsive Coulomb interaction in FB models was studied by Mielke and Tasaki,

who independently showed that some FB Hubbard models host a ferromagnetic ground

state [104, 105, 107, 174]. Flatband ferromagnetism has been an interesting topic of

various studies [101, 173, 175]; as discussed in the previous section, their approaches

also provide a method for constructing FB models.

Disorder is another important class of perturbation that has been widely studied in

FB systems. Except for symmetry-protected FBs, such as chiral FBs, the majority of

them are sensitive to disorder. Adding onsite disorder to a FB lattice will induce An-

derson localization, which breaks the FB and causes the CLS to become exponentially

localized. In Ref. [49], the authors added onsite disorder to FB lattices, and found

energy-dependent localization length scaling in terms of the Fano resonances. When

locally correlated disorder and quasiperiodic potentials are added in FB lattices, they

show vanishing localization lengths for arbitrarily weak disorder and mobility edges

for quasiperiodic perturbations [24, 35].

In higher dimensions, more interesting phenomena have been found. In 2D, Chalker

et al. reported multifractality of the FB eigenstates in the weak disorder limit [29],

which is induced by the long-range decay of the projected interaction. In 3D disor-

dered FBs, Goda et al. [60, 118] numerically demonstrated an inverse Anderson transi-

tion, where all states are localized at first, then delocalize at a critical disorder strength

before localizing again when a second critical disorder strength is reached. Such tran-

sitions are also seen in the level spacing statistics of certain 2D FBs [157]. Recently,

a number of new phenomena in disordered FB systems are attracting more attention,

including FBs under nonquenched (evolving) disorder [131], disorder-induced topo-

logical phase transitions [31], and the temporal dynamics of disordered FB states [59].

The effect of external fields has also been studied. Khomeriki et al. reported that

FB lattices under an external field show sinusoidal Bloch oscillations in the band struc-


tures. This study was conducted on a 1D diamond chain [82], and it was observed that

the FB almost completely stops the Bloch oscillations for a substantial time when the

wavefunction is trapped in the original FB part of the unperturbed bands, made possi-

ble by Landau–Zener tunneling [82].

Figure 2.8: (a) Dice lattice and (b) its band structure without an external DC field.The unit cell consists of three sites denoted by A, B, and C. The red and green circlesrepresent the amplitudes +1/6 and −1/6, respectively, of a CLS. The sites in the red boxindicate an uncoupled trimer in the limit of infinitely strong DC bias [87].

More interestingly, there are cases where FBs can be partially preserved under

an external DC field. Kolovsky et al. [87] showed that, under such field, the energy

spectrum of a 2D dice lattice (see Fig. 2.8) forms a new band structure consisting

of the periodic repetition of 1D energy band multiplets, with one of them being flat

(see Fig. 2.9). Without an external field, the dice lattice has a FB supported by a CLS.

When an external field is applied, the FB in each multiplet is supported by noncompact

exponentially localized states [87].

As it is known, adding nonlinearity lifts the spectral band structure of linear net-

works. Interestingly though, it has recently been shown that local Kerr nonlinearity

may preserve the destructive interference of a FB network, yielding compact breather

solutions. These solutions are periodic in time and compact in space, and they have

been discovered in several sample nonlinear FB networks [13, 76]. Moreover, com-

pact breathers have also been observed in the framework of ultra-cold atoms in a

diamond chain lattice network, where nonlinear terms and spin-orbit coupling are

simultaneously present [58]. Likewise, Perchikov and Gendelman studied compact

time-periodic solutions in nonlinear mechanical cross-stitch networks [125]. In 2018,

Danieli et al. established the necessary and sufficient conditions for the continuabil-

ity of linear CLSs as compact breathers in the nonlinear regime of FB networks [34].

Proven to be linearly stable once their frequency is tuned off-resonance with linear


Figure 2.9: (a) Wannier–Stark (WS) band ladder of the biased dice lattice in Fig.2.8 with F along the y axis. The thick lines highlight the irreducible triplet of threeWS bands. (b)–(d) Irreducible triplets for different field strengths [87]. This figure iscourtesy of Ramachandran et al. [87]

dispersive states and overlapping linear CLS, compact breather solutions have been

recently shown to act as Fano scatterers for dispersive waves [134].

Further, recent studies have shown that, if local attractive interactions are consid-

ered, FB systems exhibit superfluidity[77, 124, 179, 180]. In 2015, Peotta et al. [124]

demonstrated the promising applications of topologically nontrivial FBs to increase

the critical temperature of the superconducting transition. In 2016, Julku et al. [77]

studied the transport properties of the Lieb lattice FB in the presence of an attractive

Hubbard interaction, and showed that topologically trivial FBs also have the potential

to achieve high-Tc superconductivity.

2.4.2 Experimental realizations and applications

The fine-tuned nature of FB lattices requires special configurations of lattice geometry

and hoppings. Therefore, real materials with macroscopically degenerate FBs are rare

in nature, which makes the search for such materials an attractive and ambitious task

for many experimentalists. With the development of fabrication technologies, artifi-

cial lattices with FBs have been designed and tested in various systems. The effort to

observe FBs in laboratory was started soon after theoretical models for FB ferromag-

netism were first established. Here, we briefly review experimental schemes to realize

FBs in different frameworks; more details regarding experimental realizations of FBs

in artificial lattices are given in a review paper by Leykam et al. [92].

As mentioned, the earliest efforts to realize FBs systems started from ferromag-

netism. Even though no direct observation of a FB ferromagnet has been reported so

far, it is worth bringing up some important theoretical proposals and experiments that


demonstrate the possibility of realizing FB ferromagnetism. Tamura et al.[171] the-

oretically showed the potential to observe FB ferromagnetism in semiconductor dot

arrays using existing fabrication technology. Other theoretical models have also sug-

gested quantum dot arrays and quantum atomic wires formed on solid surfaces for

this purpose [72, 83, 156]. Experimental realization remains challenging though be-

cause these setups require ultra-cold temperatures. Later, the theoretical proposal to

experimentally realize organic FB ferromagnets based on polymers [4, 6, 7, 165] also

turned out to be very challenging. In 2015, FBs were observed in the electronic struc-

ture of silicene [67], with which FB ferromagnetism may be possible. More recently,

hole-doped pyrochlore oxides Sn2Nb2O7 and Sn2Ta2O7 were reported by I. Hase et

al. [66] to be candidates to realize FB ferromagnetism. For more information about

FBs in spin systems, readers can refer to the review paper by Derzhko et al. [36].

Although no FB ferromagnet has yet been found, experimental FBs have been re-

alized in many other platforms, such as electronic systems [41, 130, 161, 167], optical

lattices [3, 5, 122, 169], photonic lattices [111, 183, 189], and exciton-polariton con-

densates [10, 73, 103].

Flatbands in electronic systems: In 1998, Vidal et al. [184] found a completely flat

spectrum induced by a critical magnetic field strength in certain periodic electronic

networks. This flat spectrum is the result of an interference effect, analogous to the

Aharonov–Bohm (AB) effect [2], which is induced by a π magnetic flux threaded

through each unit cell. This effect is thus referred to as AB caging, and the paper

suggested the the possibility of observing this effect in superconducting wire networks.

Even though the direct observation of single-particle band structure is not possible

in such networks, and measurements are limited to transport properties, the indirect

observation of AB caging was reported by Abilio et al. [1]. Later, in 2001, indications

of AB caging were observed in the magnetoresistance oscillations of a low-temperature

(30 mK) normal metal dice lattice [115], where a lattice period of ≈ µm was used to

reach the required magnetic field.

Progress in nano-fabrication methods, such as lithography and atomic manipulation

techniques [130, 167], have enabled experimentalists to devise artificial FB lattices

for electrons on the nano-scale. Some works have employed a scanning tunnelling

microscope (STM) to embed a 2D Lieb lattice onto a substrate surface. In one example,

Slot et al. [161] designed an electronic Lieb lattice formed by the surface state electrons

of Cu(111), where they used an array of carbon monoxide molecules positioned with

an STM to confine the electrons and observed the predicted characteristic electronic

structure of the Lieb lattice (see Fig. 2.10 (b)). Using a low-temperature STM, Drost


et al. [41] created a vacancy lattice by removing atoms from a chlorine monolayer, and

designed analogues of the poly-acetylene (dimer) chain and Lieb lattice.

Figure 2.10: Experimental realization of FBs in an engineered atomic lattice andoptical lattice as taken from Refs. [161, 169]. (a) Schematics of a Lieb lattice formedfrom surface state electrons of Cu(111) (red and blue circles) confined by an array ofcarbon monoxide molecules (black circles) from [161], with (b) corresponding bandstructure. Due to next nearest neighboring interactions, the FB become dispersive. (c)Experimental realization of a Lieb optical lattice from [169]. Black arrows indicate thepolarizations of the lattice beams. (d) Lattice potential profile, and (e) band structuresfor different lattice potentials.

Flatbands in optical lattices: The development of laser cooling techniques, ion trap-

ping, and optical lattices has provided a new platform for the realization of FB lattices.

Tunability of the potential in these techniques has made it possible to devise FB lat-

tices by arranging atoms and tuning the hoppings. Lieb lattices are the most-studied FB

model in optical lattices due to their relatively simple geometry and novel properties

under interactions.

An optical Lieb lattice was first realized by Shen et al. [155] in order to experimen-

tally observe massless Dirac fermionic behavior in the vicinity of the Dirac cone. In

optical lattices with relatively shallow optical potential, next nearest neighbor hoppings

are unavoidable, leading to a non-zero width of their ’flat’ band. The exact Bloch wave

spectrum for such optical lattices has been computed numerically by Apaja et al. [5],

who found that a ’flat’ band with a width of only 1.5% of the total bandwidth could

be achieved. Later, Takahashi et al. [169] reported the experimental realization of an


optical Lieb lattice with bosonic condensation (see Fig. 2.10 (c–e). By manipulating

the optical potential, they were able to engineer the population and phase of each lat-

tice site, which enabled them to coherently transfer atoms into the FB and observe the

CLS. Manipulating system parameters enabled them to control the delocalization tran-

sition of the CLS, and detect the presence of FB-breaking perturbations. As predicted

by Apaja et al. [5], they observed interactions that induced a decay of the condensate

into the lower dispersive band.

Later, in 2017, Ozawa et al. [122] found that FB energy at the edge of the Brillouin

zone is shifted by interactions, leading to a non-zero dispersion. These measurements

were in accordance with the tight-binding limit of the Gross–Pitaevski equation. Sub-

sequently, Taie et al. [168] used fermionic cold atoms in a Lieb lattice to demonstrate

the transport of particles adiabatically between the horizontal and vertical rim sites

through a dark state.

Aside from Lieb lattices, optical lattices have been realized in the form of 1D saw-

tooth chains by An et al. in [3].

Figure 2.11: (a) Scanning electron microscope image of an air-bridge kagome-latticephotonic crystal waveguide. (b) Experimental transmission spectrum of the waveguideshown in (a). (c) Experimental (solid green) and theoretical (dashed light-blue) groupindex spectra. The corresponding band structure is shown in (d). This figure is fromSchulz et al. [151].


Flatbands in photonic systems: In photonics, FBs have significant applications in

the technologically important concept of slow light, which could be advantageous for

the buffering and time-domain processing of optical signals. From 2001, it has been

known that slow light can be generated by strong dispersion close to the photonic band

edge of a photonic crystal waveguide (PCW) [91, 120]. However, slow light in PCWs

has two issues to be considered: the frequency bandwidth of the effect and higher-order

dispersion. To begin with, a balance must always be achieved between reducing group

velocity and maintaining a useful operation bandwidth. Furthermore, higher-order

dispersion must be avoided, as it severely distorts optical signals. Flatbands can solve

higher-order dispersion, making them valuable in PCW-based slow light applications.

An early proposal for achieving FBs in photonic crystals came from Takeda et

al. [170]. Their model was composed of circular rods, with electromagnetic waves

localizing at the rods that formed photonic FBs. This proposal was not successfully

pursued though due to fabrication difficulties. Instead, in slow light schemes, previous

research was mostly focused on 1D photonic waveguide arrays or 2D lattices designed

through defects in triangular photonic crystals and optimized by numerical or intuitive

approaches [9, 95, 196]. In 2017, Schulz et al. presented an alternative to the standard

triangular-structured waveguides, based on the kagome lattice, that demonstrated an

improved reduction of group velocity and even stopped light [151] (see Fig. 2.11).

The kagome-lattice waveguides inspired further slow light engineering in PCWs [46,

47, 78, 114, 119].

Another active approach to realize FBs is through the use of the femtosecond laser

writing technique to fabricate optical waveguide networks [166]. The advantages of

this technique over other photonic systems is the long propagation distance and ar-

bitrarily tunable coupling between waveguides, which allows for the exploration of

time-periodic lattice dynamics, and Bloch oscillations induced by weak potential gra-

dients. In 2014, Guzmán-Silva et al. [65] were the first to fabricate a Lieb lattice based

on phonic waveguide arrays, and studied the effect of the FB on the transport proper-

ties of the lattice. However, the presence of the FB could only be deduced indirectly,

because a superposition of all bands is excited by the single waveguide input in this

original experiment. In subsequent studies, Vicencio and Mukherjee et al. [111, 183]

used a multi-waveguide input beam to directly excite the CLS and observe nondiffract-

ing photons (see Fig. 2.12 (a)). Weimann et al. [189] studied the transport properties of

a photonic waveguide sawtooth lattice. The optical induction technique [42, 99, 152]

was used by Xia et al. [194] to create photonic Lieb lattices, where they demonstrated

distortion-free image transmission using the CLS. Further related studies are ongoing.


Figure 2.12: (a) Optical waveguide Lieb lattice fabricated by the femtosecond laserwriting technique and the observed compact localized modes. This figure is from [183].(b) A 1D stub lattice made by exciton-polariton condensate and its band structure. Thisfigure is from [10].

Flatbands in exciton-polariton condensates: Microcavity exciton-polariton con-

densates provide another way of achieving FB lattices. In 2012, Masumoto et al.

[103] realized a 2D kagome lattice with exciton-polariton condensates and observed

a weakly dispersive band. Due to limitations in fabrication techniques, the potential

shell was shallow. Therefore, a balance needs to be found between smaller lattice peri-

odicity and the minimum next nearest neighbor hoppings in order to resolve the single

band while keeping its flatness. Furthermore, condensation in the FB could not be

achieved, because the FB was not the lowest band.

Later, improved fabrication techniques such as micropillar cavity etching made it

possible to experimentally achieve ideal FBs. In 2014, Jacqmin et al. [73] was the

first to report polariton FBs in a honeycomb array of micropillar cavities. Then, also

using micropillar optical cavities, a 1D stub lattice was realized in 2016 by Baboux et

al. [10]; they were able to achieve polariton condensation into a FB (see Fig. 2.12 (b)).

More recently, exciton-polariton condensation into a FB was also demonstrated in a

2D Lieb lattice of micropillars [86, 190].

2.5 Summary 31

Future perspective: Beyond these examples, while a number of new platforms have

been theoretically proposed, many of them have yet to be experimentally demonstrated.

One of them is FB localization in an opto-mechanical system with controllable photons

and phonons, which might bring wide applications in information storage, transfer, and

the engineering solid-state quantum devices. In 2017, Wan et al. [188] reported the

appearance of a FB in a general bipartite opto-mechanical lattice, and in the same year,

Wan et al. [187] also observed photon and phonon localization in an opto-mechanical

Lieb lattice.

Another platform where FBs have been proposed is cavity QED systems at mi-

crowave and optical frequencies [71]. Many novel quantum correlations of the FB

states, which are induced by dissipations and interactions, have been reported [21, 27,

121, 143, 148, 197]. Moreover, the effect of multi-photon interactions on FBs can be

explored in optical waveguide arrays [141]. Finally, FBs in graphene systems have

also been reported [90, 102, 126].

With the development of fabrication technology, artificial FB systems will become

a more interesting topic of study. The previously mentioned approaches will advance

with improved precision and quality, as will the application of FB physics into new

systems including micro- and nano-scale devices.

2.5 Summary

In this chapter, we have introduced the concept of the flatband and how it forms. Start-

ing from the Bloch theorem, we introduced the tight-binding model for discrete trans-

lational invariant networks. The origin of FBs has been explained in detail, and the

concepts of CLSs and macroscopic degeneracy were introduced. We reviewed exist-

ing FB construction methods, applications, and experimental realizations in various

systems.

As discussed, existing methods of constructing FBs are limited to intuitive geo-

metric or symmetry-based procedures. However, the fine-tuned nature of FB networks

allows for the freedom to adjust system parameters, such as hopping strength, onsite

energy, and wave amplitude, to achieve FBs. Consequently, there are a vast number

of uncovered FB lattices, and moreover, the properties of CLSs and their decisive role

in the behaviour of FB lattices have yet to be systematically studied. Therefore, it

is desirable to establish the systematic construction and complete classification of FB

Hamiltonians. In the next chapter, we will present our systematic approach to generat-

ing FB lattices based on CLS properties.


Chapter 3

Classification and construction offlatbands by compact localized states

Compact localized states (CLSs) play an important role in the physical characteristics

of flatband (FB) lattices. Despite this, little has been known about CLS properties,

which we focus on at the beginning of this chapter. First, we introduce the classifica-

tion of FB lattices by CLSs and discuss their various properties. Then we develop a

matrix representation—the main FB generator tool. At the end, we introduce destruc-

tive interference conditions and CLS existence conditions, and the core ideas of our

FB generator.

3.1 Classification of flatband networks by compact lo-

calized states (CLSs)

In this section, we discuss how to classify FB lattices by the size and shape of their

CLSs. First, we introduce shape vector U that specifies these characteristics.

Definition 3.1. Shape vector U of a CLS is a vector whose components are integers

that give the span of the CLS along each lattice dimension. The CLS size U is the total

number of unit cells occupied by a CLS.

For example, in 1D, U is an integer, i.e. U = U , which is the number of unit cells

occupied by the CLS (see Fig. 3.1). In 2D, we can write U = (U1, U2), which tells

that the CLS is occupying the unit cells in a U1 × U2 rectangular area, as seen in Fig.

3.2.

Given a CLS of a FB, the linear combination of its lattice translations is also an

eigenstate of the same FB. It is therefore important to identify the CLS that cannot be

33

34 Classification and construction of flatbands by compact localized states

Figure 3.1: Various examples of 1D FB lattices and their U classification. (a) Cross-stitch, U = 1, (b) diamond chain, U = 1, (c) 1D pyrochlore, U = 1, (d) 1D Lieb,U = 2, (e) stub, U = 2, and (f) sawtooth chain, U = 2. This figure is courtesy of S.Flach et al. [49].

decomposed into smaller CLSs.

Definition 3.2. An irreducible CLS is a flatband eigenstate that occupies the smallest

possible number of adjacent unit cells.

From here on, if we do not specify, the notation CLS implies the irreducible one.

An important observation is that all known FB models have a unique irreducible

CLS. This leads to the following conjecture.

Conjecture 3.1. The irreducible CLS of a given flatband is unique.

Consequently, all eigenstates of FB energy can be formed by the linear combination

of lattice translations of a unique irreducible CLS. Additionally, since the irreducible

CLS of a FB is unique, it can be used to identify different FBs. We can therefore

classify FB lattices by the shape vector U of their irreducible CLSs, which we call U

classification.

Definition 3.3. A class U CLS is an irreducible CLS with shape vector U. If the

irreducible CLS of a flatband lattice is a class U CLS, then we call this flatband a

class U flatband, and call the lattice a class U flatband lattice for lattices with a single

flatband.

In 1D, there is only one shape, and thus FB lattices are classified by their CLS size

U . For example, a 1D sawtooth chain is a class U = 2 FB lattice, because its CLS

3.2 Properties of CLSs 35

Figure 3.2: Classifications of some known 2D FB lattices. The shaded areas representthe shapes of the CLSs, and the red boxes show the unit cell. In each case, the CLSstretches two unit cells along both x, y directions, and so U = (2, 2). (a) Lieb, (b)checkerboard, (c) kagome, and (d) dice.

occupies two unit cells. Figure 3.1 illustrates some examples of different class U FB

lattices in 1D. Examples of the U classification of some of the known 2D FB lattices

are shown in Fig. 3.2.

Flatband lattices of different U classes have different properties; this makes U

classification useful to identify these properties. For example, as we discussed in Sec-

tion 2.3.2, the U = 1 FB eigenstate can be detangled from the dispersive part of the

lattice [49], but the U > 1 FB eigenstate cannot be detangled [49].

3.2 Properties of CLSs

In this section, we discuss CLS properties mainly in the context of 1D Hermitian sys-

tems with ν sites per unit cell and nearest neighbor hoppings. Some of the results here

hold in higher dimensions, as we will discuss.

3.2.1 Relation between CLSs and Bloch wave functions

Consider a 1D flatband lattice with ν sites per unit cell. Suppose the ν component vec-

tor ~ψj is the wave function of the jth unit cell, and the components give wave functions

at the lattice sites in the unit cell. Then, the CLS is given by ~Ψl =(. . . 0, 0, ~ψ1, ~ψ2, . . . , ~ψU , 0, 0, . . .

),

and ~ψ1 is located in the lth unit cell. Since all lattice translations of the CLS share the

same eigenenergy, we can construct a Bloch eigenstate (up to normalization, see Sec-

tion 2.1) with

~u(k) ∼∞∑

l=−∞

eikl~Ψl ∼U∑l=1

~ψlei(U−1)k . (3.1)

Note that, in above equation, the infinite sum in l is truncated due to the compactness

of the CLS wave function, ~ψ1≤l≤U = 0. The inverse of (3.1) is also true if the Bloch


function for a certain band can be expressed as

~u(k) = N(k)U∑l=1

~ψlei(U−1)k, (3.2)

where N(k) is a common prefactor.

A similar but more involved decomposition can be written for CLSs in higher di-

mensions, where size alone does not fix the shape of the CLS.

3.2.2 Irreducibility condition of CLSs

As previously mentioned, a CLS of a given FB could be the linear combination of

the lattice translations of the irreducible CLS. So how do we know if a given CLS is

irreducible? The answer is to check for linear dependence of the CLS components.

Figure 3.3: Relationship between reducible and irreducible CLSs, with linear depen-dence of CLS components.

Theorem 3.4. Given CLS ΨU = (~ψ1, ~ψ2, · · · , ~ψU) occupying U unit cells, if CLS

components ~ψi=1,...,U are linearly independent, then ΨU is irreducible.

Theorem 3.4 can also be stated as follows: If the matrix (~ψ1~ψ2 · · · ~ψU) is full rank,

then CLS ΨU = (~ψ1, ~ψ2, · · · , ~ψU) is irreducible.

Below we prove the following equivalent statement of Theorem 3.4: If a CLS is

reducible, then its components are linearly dependent. According to Conjecture 3.1,

we assume that if a CLS is reducible then it should be the linear combination of the

unique irreducible CLS (i.e., it cannot be written as the linear combination of different

class irreducible CLSs). Here, we prove a 1D case and sketch how it can be extended

to higher dimensions.


Proof. Consider a 1D CLS ΨU which is composed of the lattice translations of irre-

ducible CLS ΦU−1 = (~φ1, ~φ2, · · · , ~φU−1), whose components are linearly independent.

Then,

ΨU =

~ψ1

~ψ2

· · ·~ψU

= α

~φ1

~φ2

· · ·~φU−1

0

+ β

0

~φ1

~φ2

· · ·~φU−1

, α, β 6= 0. (3.3)

If ΨU can be decomposed into an even smaller-sized irreducible CLS, then there will

be more terms in Eq. (3.3); the following procedure can apply to this case as well.

From Eq. (3.3), we can see that~ψ1 = α~φ1

~ψU = β~φU−1

~ψl = α~φl + β~φl−1, 2 ≤ l ≤ U − 2.

(3.4)

If Theorem 3.4 is true, then the components of ΨU must be linearly dependent. More

precisely, there should exist a non-zero set {cj} which satisfies

U∑j=1

cj ~ψj =U−1∑i=1

(ciα + ci+1β)~φi = 0. (3.5)

In the above equation, we paired the non-zero components in Eq. (3.3). The index i

above runs from 1 to U − 1, because there are U − 1 pairs in the sum.

Since the components of ΦU−1 are linearly independent, the only solution of Eq.

(3.5) is

ciα + ci+1β = 0, for all 1 ≤ i ≤ U − 1, (3.6)

in which the U coefficients and U − 1 equations give an under-determined problem,

so we can always find a set {cj} that satisfies the linear dependence condition (Eq.

(3.5)). This shows that if a CLS is the linear combination of smaller CLSs, then its

components are linearly dependent. This proves Theorem 3.4.

This proof can be extended to higher dimensions once the more complicated shapes

of the CLSs are taken into account.

Note that the inverse statement of Theorem 3.4, which is: "If CLS components

are linearly dependent, then the CLS is reducible", does not hold (see Fig. 3.3). For

example, the irreducible CLS of a Lieb lattice has linearly dependent components.


This result, together with the observation that CLS size completely fixes the shape

in 1D , leads to the following conjecture.

Conjecture 3.2. In 1D, linear dependence of CLS components implies the reducibility

of the CLS.

This conjecture can be rigorously proved for the U = 2 and U = 3 cases, and it holds

for all the other known d = 1 examples. We present here the proof for the U = 2 case,

and put the proof for the U = 3 case in Appendix A.1, which uses the block matrix

representation that we introduce later in Section 3.3.

Proof. Consider the U = 2 class CLS ~Ψ =(. . . , 0, ~ψ1, ~ψ2, 0, . . .

), with nearest neigh-

bor unit cell hoppings. Suppose two components ~ψ1, ~ψ2 are linearly dependent, such

that ~ψ1 = α~ψ2. From the eigenvalue problem in Eq. (2.17), we have

εjφ1,j +∑j′

tj,j′φ2,j′ = Eφ1,j, j, j′ = 1, . . . , ν

, εjφ2,j +∑j′

tj,j′φ1,j′ = Eφ2,j, j, j′ = 1, . . . , ν.(3.7)

From the linear combination ~ψ1 = α~ψ2, we have φ1,j = αφ2,j , and plugging this into

Eq. (3.7) we get

εjφ1,j + α∑j′

tj,j′φ1,j′ = Eφ1,j, j, j′ = 1, . . . , ν

, αεjφ1,j +∑j′

tj,j′φ1,j′ = αEφ1,j, j, j′ = 1, . . . , ν.(3.8)

The above equations only involve φi,j . For a non-zero α, the above equations hold only

when ∑j′

tj,j′φ1,j′ =∑j

tj,j′φ1,j = 0 , (3.9)

which means that there are no hoppings to neighboring unit cells. As a result, we have

a U = 1 CLS.

Assuming Conjecture 3.2 is true, it leads to the following conjecture.

Conjecture 3.3. In 1D, the CLS class is U ≤ ν.

Proof. Suppose there is a CLS of size U > ν; then there are U > ν CLS compo-

nents that are ν-dimensional vectors. In other words, there are U > ν vectors in

ν-dimensional vector space, which always leads to the linear dependence of these vec-

tors. As the result, according to Conjecture 3.2, the CLS is reducible as long as U > ν.


Therefore, the maximum size of the irreducible CLS is U = ν, or in other words, the

CLS class in 1D is U ≤ ν.

As discussed in Section 2.3.2, unitary transformation does not change the band

structure while modifying CLS components. Moreover, depending on the choice of

unit cell, CLS size can differ. In some cases, one can reduce CLS size by unitary

transformation and redefining the unit cell.

Theorem 3.5. In 1D, a CLS occupying U unit cells with ~ψ1 ⊥ ~ψU can be reduced to

the U − 1 class.

Proof. Suppose we have CLS ~ψcls = (~ψ1, ~ψ2, . . . , ~ψU)T of size U with ~ψ1 ⊥ ~ψU .

Then, we can apply unitary transformationR to the CLS such that ~φi=1,...,U = R~ψi=1,...,U

and

~φ1 =

1

0...

0

, ~φ2 =

φ2,1

φ2,2

...

φ2,ν

, . . . , ~φU =

0

φU,2...

φU,ν

, (3.10)

where ν is the number of sites per unit cell. Due to the unitary transformation R, the

eigenvalue problem (Eq. (2.17)) does not change. Next, we redefine the unit cell as

illustrated in Fig. 3.4 as

~ϕ1 =

1

φ2,2

...

φ2,ν

, ~ϕ2 =

φ2,1

φ3,2

...

φ3,ν

, . . . , ~ϕU−1 =

φU−1,1

φU,2...

φU,ν

,

and ~ϕU = 0. Therefore, after the unitary transformation R and redefinition of the unit

cell, the class of the CLS reduces to U − 1. Illustration of this procedure are shown in

Fig. 3.4.

3.2.3 Completeness of CLSs

The Bloch eigenstate of a FB can always form a complete basis for the Hilbert space

of the FB. Can CLSs also form such a complete basis? In this section, we discuss this

question.

Given a finite-size FB lattice with N sites, lattice translations of its CLS form

(N − 1) different copies. If these N copies are linearly independent, they form a


Figure 3.4: (Color online) Schematics showing how a CLS of class U = 4 reducesto U = 3, when ~ψ1 ⊥ ~ψ4. Each rectangle stands for one unit cell. The filled circlesstand for non-zero wave function components, and the open circles for the zero wavefunction components.

complete basis for the Hilbert space of the FB. Then, what is the condition for the

linear independence of the set of N CLSs?

Suppose the CLS of a given FB lattice occupies U unit cells, and we write the CLS

whose first component is located in the lth unit cell as ~Ψl =(. . . , 0, ~ψl, ~ψl+1, . . . , ~ψl+U−1, 0, . . .

).

The CLS component ~ψj=l,l+1,...,l+U−1 is a ν-dimensional vector, with each component

giving the wave function of each site in the jth unit cell. In this case, all lattice trans-

lations form a set {~Ψl=1,...,N}, which leads to the following theorem.

Theorem 3.6. If the set of CLS components {~ψj=l,...,l+U−1} is linearly independent,

then the set of CLS translations {~Ψl=1,...,N} is also linearly independent, and forms a

complete basis to span the Hilbert space of the flatband.

For convenience, in this thesis, if the set of lattice translations of a CLS forms a

complete basis, then we refer to the CLS as complete.

Theorem 3.6, which applies in any dimension, can be equivalently stated as: If the

set of CLS translations {~Ψl=1,...,N} is linearly dependent, then the set of CLS com-

ponents {~ψj=l,...,l+U−1} is also linearly dependent. Here, we prove this equivalent

statement for a 1D case, and sketch an extension for higher dimensions.

Proof. Consider a linearly dependent set of CLS translations {~Ψl=1,...,N} of a 1D FB

lattice, such thatN∑l=1

αl~Ψl = 0, l ∈ Z, (3.11)

where ~Ψl =(. . . , 0, ~ψl, ~ψl+1, . . . , ~ψl+U−1, 0, . . .

), and ~ψj=l,l+1,...,l+U−1 are ν-component

vectors whose components are the wave functions on all lattice sites in j = l, l +


1, . . . , l + U − 1th unit cells. A necessary condition for Eq. (3.11) is

l+U−1∑j=l

αj+l ~ψj = 0 , (3.12)

and thus set {~ψj=l,...,l+U−1} has to be linearly dependent. Therefore, if CLS compo-

nent set {~ψj=l,...,l+U−1} is linearly independent, then CLS translation set {~Ψl=1,...,N} is

linearly independent as well. This proves Theorem 3.6 in 1D.

The proof for higher dimensions can be performed by a similar procedure, but CLS

shape has to be first taken into account.

When a CLS is not complete, i.e. the number of linearly independent CLS trans-

lations is less than the dimension of the Hilbert space of its FB, there must be missing

states. These can be explained by the band touching or crossing properties of the FB,

which we discuss in the following section.

3.2.4 Relation between CLS completeness and band touching

In this section, we discuss how FBs interact with dispersive bands, which in turn is

related to CLS properties. When FBs are gapped from dispersive bands, their CLSs

are complete [139]; however, when FBs touch or cross dispersive bands, their CLSs

may not be complete.

Consider the simplest CLS classU = 1. As we discussed in Section 2.3.3, we know

that such a CLS and its dispersive states can be decoupled [49]. Then, band touching

or crossing can be removed by shifting the FB without disturbing its flatness, and thus

the CLSs are complete. This type of band touching or crossing is called removable

band touching. Therefore, we may write the following theorem for U = 1 CLSs.

Theorem 3.7. The U = 1 class CLSs in any dimension are always complete as the

band touchings or crossings between U = 1 flatbands and dispersive bands are re-

movable.

In the U = 2 case, our analytic results for 1D FB lattices with two bands [97] show

that such FBs are always gapped from dispersive bands (see Chapter 4). Likewise, in

our later study on 1D FB generators [98], all U > 1 examples show gapped FBs (see

Chapter 4). This leads to the following conjecture.

Conjecture 3.4. In 1D, U > 1 flatbands are gapped from dispersive bands, and their

CLSs are complete.


Combining Theorem 3.7 and Conjecture 3.4, we conclude the following: Band

touchings or crossings of 1D FBs are always removable or gapped, and thus their

CLSs are always complete.

For higher dimensions d > 1, there are some types of band touchings that cannot

be removed without destroying the FB. We call such band touchings as irremovable,

and these can be identified through the linear dependence of the CLS components. The

CLS of a FB with irremovable band touchings is not complete [139]. Then, Theorem

3.6 leads to the following corollary.

Corollary 3.8. In d > 1, if the components of a CLS are linearly independent, then the

CLS is complete and the flatband is either gapped or has removable band touching.

To the best of our knowledge, if the components of a CLS are linearly dependent,

then the set of the CLS translations is linearly dependent (there is no counter example

for this), which leads to the following conjuncture.

Conjecture 3.5. A CLS with linearly dependent components is not complete, and so

the corresponding flatband has irremovable band touching.

For example, Lieb and kagome lattices have CLSs with linearly dependent compo-

nents and their CLSs are not complete, while flatbands in these lattices have irremov-

able band touchings.

3.3 CLS existence conditions and the flatband genera-

tor

In FB lattices, there should be at least two sites per unit cell in order to achieve de-

structive interference, because a FB in a single-band system only exists when the lat-

tice sites are isolated. Therefore, we will only deal with multiple sites per unit cell,

where it is convenient to represent the single-particle quantum states of the system in

terms of single unit cell state vectors (Eq. (2.11)) and single unit cell wave functions

(Eq. (2.12)). This single unit cell representation leads to the decomposition of the

tight-binding Hamiltonian into a block matrix form, that we introduce in this section.

The FB generator, the main concern of this thesis, is also based on matrix formalism

and will be covered below.

3.3 CLS existence conditions and the flatband generator 43

3.3.1 Block matrix representation of the tight-binding model

For simplicity, we first consider a 1D translational invariant tight-binding network with

ν sites per unit cell. This is followed by extension to higher dimensions, for which one

has to consider hoppings in different directions.

We introduce ν × ν matrix H0 to describe onsite energies and hoppings inside a

unit cell, as well as ν × ν matrix Hm to describe hoppings between mth neighboring

unit cells. Then, we can write the eigenvalue problem (2.17) of the lattice in matrix

form as

H0~ψn +

∞∑m=−∞

Hm~ψn+m = E~ψn, m 6= 0 , (3.13)

where the single unit cell wave function ~ψn of the nth unit cell is given by Eq. (2.12).

For Hermitian systems, H0 is Hermitian and Hχ,−m = H†χ,m. For non-Hermitian

systems, there are no restrictions for either H0 or Hm. In the case of finite-range

hoppings, −mc ≤ m ≤ mc, where mc is the maximum hopping range. Note that, in

matrix representation, neighbor refers only to neighboring unit cells, which is different

from conventional definitions; we use this notation throughout this thesis unless stated

otherwise.

Band structure can be achieved using k-space (reciprocal space) representation.

Putting the Bloch wave from Eq. (2.20) into Eq. (3.13) we get

∑k

(H0 +

mc∑m=1

H†meimk +

mc∑m=1

Hme−imk

)~u(k)e−ink = E

∑k

~u(k)e−ink , (3.14)

where we take the lattice constant to be 1. Canceling out e−ink in Eq. (3.14) we get the

following, for ∀k,(H0 +

mc∑m=1

H†meimk +

mc∑m=1

Hme−imk

)~u(k) = E~u(k) . (3.15)

Therefore, we can write the k-space Hamiltonian as

H(k) = H0 +mc∑m=1

H†meimk +

mc∑m=1

Hme−imk. (3.16)

The eigenvalues of H(k) give the band structure.

Most of the time we consider nearest neighbor hoppings mc = 1, in which case

H1 describes the hoppings between nearest neighboring unit cells. Then Eq. (3.13)


becomes

H0~ψn +H†1

~ψn−1 +H1~ψn+1 = E~ψn . (3.17)

Therefore, the Hamiltonian matrix for 1D nearest neighbor unit cell hoppings is a tri-

diagonal block matrix,

H =

. . . . . . 0 . . .

. . . H0 H1 0 . . .

0 H†1 H0 H1 0

. . . 0 H†1 H0. . .

. . . 0. . . . . .

, (3.18)

where H1 and H0 are ν × ν matrices. Then the k-space Hamiltonian (Eq.(3.16)) for

nearest neighbor hoppings becomes

H(k) = H0 +H†1eik +H1e

−ik. (3.19)

Figure 3.5: Schematic of a cross-stitch lattice.

Take a simple 1D lattice with two sites per unit cell and nearest neighbor hoppings,

as shown in Fig. 3.5. Then the wave function of the nth unit cell and hopping matrices

are

~ψn =

(ψ1,n

ψ2,n

), H0 =

(ε1 t

t ε2

), H1 =

(t1,1 t1,2

t2,1 t2,2

). (3.20)

The eigenvalue problem in Eq. (3.17) now becomes

ε1ψ1,n + tψ2,n + t1,1ψ1,n+1 + t1,1ψ1,n−1 + t1,2ψ2,n+1 + t1,2ψ2,n−1 = Eψ1,n

ε2ψ2,n + tψ1,n + t2,1ψ1,n+1 + t2,1ψ1,n−1 + t2,2ψ2,n+1 + t2,2ψ2,n−1 = Eψ1,n,(3.21)


and the k-space Hamiltonian becomes

H(k) = H0 +H†1eik +H1e

−ik =

(ε1 + 2 cos(k)t1,1 t+ e−ikt1,2 + eikt2,1

t+ eikt1,2 + e−ikt2,1 ε2 + 2 cos(k)t2,2

).

(3.22)

Then, the energy bands are given by the eigenvalues of H(k).

The block matrix representation naturally extends to higher dimensions, which

contain multiple hopping directions, either along the primitive lattice translation vec-

tors or any other lattice translation vectors (as linear combinations of primitive lattice

translation vectors). If we consider nearest neighbor hoppings along each hopping di-

rection, we can introduce a hopping matrix for each hopping direction. Suppose Hχ is

the nearest neighbor hopping matrix for the χth direction, and then the tight-binding

eigenvalue problem from Eq. (2.17) reads

H0~ψn +

∑χ

H†χ~ψl′χ +

∑χ

Hχ~ψlχ = E~ψn , (3.23)

where lχ and l′χ are the indices of the nearest neighboring unit cells along the χth

direction. Then, the k-space Hamiltonian can be written as

H(k) = H0 +∑χ

H†χei ~Rχ·~k +

∑χ

Hχe−i ~Rχ·~k , (3.24)

where ~Rχ is the lattice vector of the nearest neighbor along the χth hopping direction.

3.3.2 CLS existence conditions and flatband tester

As learned in Section 2.2, FBs are the consequence of destructive interference. Ac-

cordingly, achieving CLSs via destructive interference is the key to constructing FB

lattices. In this section, we cover CLS existence conditions (or destructive interference

conditions) and a FB tester before introducing our FB generator in the next subsection.

For simplicity, we consider a 1D FB lattice with nearest neighbor hoppings. Sup-

pose the lattice has a CLS occupying U unit cells, as shown in Fig. 3.6. Here, the

hopping from the first unit cell to the left and hoppings from the U th unit cell to the

right must be zero, which gives the following destructive interference conditions (see

Fig. 3.6)

H1~ψ1 = 0, H†1

~ψU = 0 . (3.25)

This tells us that, in 1D, the hopping matrix H1 must have eigenvalues equaling zero.


……..#

……..#

CLS#Figure 3.6: (color online) Schematics of a compact localized state with nearest neigh-bor hopping (mc = 1) and ν = 2.

We can then write the following theorem.

Theorem 3.9. In a 1D lattice with nearest neighbor hoppings, if the lattice has a CLS,

then the nearest neighbor hopping matrix must be singular:

detH1 = 0. (3.26)

The destructive interference conditon in Eq. (3.25) and Theorem 3.9 imply that the

eigenvalue problem (Eq. (3.17)) of a class U FB lattice must have a solution in the

following form~ΦU = (. . . , 0, ~ψ1, . . . , ~ψU , 0, . . . ) . (3.27)

Then, the eigenvalue problem can be written as

HU~ΨU = EFB~ΨU , (3.28)

where EFB is FB energy, and HU is a U × U tri-diagonal block matrix

HU =

H0 H1 0 . . . 0 0

H†1 H0 H1 0 . . . 0

0. . . . . . . . . 0

...... . . .

. . . . . . . . . 0

0 . . . 0 H†1 H0 H1

0 0 . . . 0 H†1 H0

, (3.29)


and~ψU = ( ~ψ1, . . . , ~ψU) . (3.30)

Given a FB Hamiltonian, we can test the class of the FB using the following pro-

cedure. For simplicity, we consider a 1D case.

Definition 3.10. Flatband tester: Given a 1D FB Hamiltonian with nearest neighbor-

ing unit cell hoppings, we can find ~ψ1 and ~ψU from H1~ψ1 = H†1

~ψU = 0, and the

eigenvalue problem (3.28) has a solution in the form (3.30) corresponding to eigen-

value EEB. Starting from U = 1, we solve the eigenvalue problem. If there is no

solution for U = 1, we successively increase to U = 2, 3, . . . until we obtain a solu-

tion. In this way, we get the smallest U that gives an irreducible CLS; this U is the

class of the FB lattice.

A similar procedure can be employed for higher dimensions, with more complexity.

From this test procedure, we can infer the following necessary and sufficient con-

ditions for the existence of a CLS in a given 1D lattice Hamiltonian, as stated in the

theorem below.

Theorem 3.11. Given a 1D lattice with intracell hoppings H0 and nearest neighbor

hoppings H1, if the tri-diagonal block matrix (Eq. (3.29)) has an eigenvector (Eq.

(3.30)) that corresponds to eigenvalue EFB and satisfies the destructive interference

conditions (Eq. (3.25)), then a CLS of size U exists in this lattice.

Note that, in this theorem, U is not necessarily the smallest one; if needed, the

tester 3.10 can be used to find the smallest U .

Theorem 3.11 can be extended to higher dimensions using Eq. (3.23) and by taking

into account the shapes of the CLSs, which complicates the destructive interference

conditions. This is stated in the following theorem.

Theorem 3.12. Given a d > 1 dimensional lattice with intracell hoppings H0 and

nearest neighbor hoppings Hχ in the χth direction, the necessary and sufficient condi-

tions for the existence of a CLS of size U are (i) the eigenvalue problem

H0~ψl +

∑χ

H†χ~ψl′χ +

∑χ

Hχ~ψlχ = E~ψl l = 1, . . . , U , (3.31)

having an eigenvector ΨU =(~ψ1, ~ψ2, . . . , ~ψU

)Twith eigenvalue EFB, and (ii) an

eigenvector satisfying the destructive interference conditions

∑χ,m

Hχψm = 0,∑χ,m′

H†χψm′ = 0 (3.32)


at the boundaries, i.e. where m,m′ runs over the boundary unit cells.

The destructive interference conditions (Eq. (3.32)) vary depending on the shape

of the CLS, as we discuss later in Chapter 5.

These necessary and sufficient conditions are formulated in real space. Other work

studying the necessary and sufficient conditions for the existence of FBs was based on

k-space representation [178].

Now, by inverting the procedure in the FB tester 3.10, assuming the lattice has a

CLS of class U and asking what is the Hamiltonian that satisfies Eq. (3.28), we arrive

at our core idea of the FB generator.

3.3.3 Flatband Generator

Consider a 1D translational invariant tight-binding lattice (or network) with ν sites per

unit cell, with intracell hoppings H0 and nearest neighbor hoppings H1. We want to

get a FB of class U ≤ ν. We therefore assume the lattice has an irreducible CLS

ΨU = (~ψ1, ~ψ2, · · · , ~ψU), and look for the matrices H0, H1, and FB energy EFB that

satisfy the necessary and sufficient conditions for CLS existence (Eqs. (3.25) and

(3.28)):H0

~ψ1 +H1~ψ2 = E~ψ1,

H0~ψl +H†1

~ψl−1 +H1~ψl+1 = E~ψl l = 2, . . . , U − 1,

H0~ψU +H†1

~ψU−1 = E~ψU

H1~ψ1 = 0

H†1~ψU = 0 .

(3.33)

This is a set of non-linear equations that are in general difficult but solvable, as we

demonstrate in later chapters. We may now formally define our FB generator as fol-

lows.

Definition 3.13. A flatband generator is a scheme that generates a set of all possi-

ble hopping matrices {H0, H1}, flatband energy EFB, and irreducible CLSs ΨU that

satisfies the eigenvalue problem and destructive interference conditions (Eq.(3.33)).

This scheme systematically generates all possible FB Hamiltonians that possess

CLSs of class U .

Extension of the above FB generator to higher dimensions is straightforward, where

one can generate FB Hamiltonians by solving Eqs. (3.31) and (3.32). Depending on

the dimension and shapes of the CLSs, the solutions vary (see later chapters). In gen-

eral, the control parameters of the FB generator are as follows:

3.4 Summary 49

• Lattice dimension d: In this thesis, d = 1, 2.

• Hopping range mc: In this thesis, we focus on nearest neighboring unit cell

hoppings, i.e. mc = 1.

• The number of sites per unit cell = the number of bands ν.

• For d = 1: CLS size = the number unit cells U occupied by the irreducible CLS.

• For d ≥ 2 and U ≥ 2: size U and CLS shape.

We aim to generate FB Hamiltonians for given lattice dimension d, number of bands

ν, CLS size U , and CLS shape (for d ≥ 2).

3.4 Summary

In this chapter, we first introduced U-classification of FB lattices according to their

CLS class. Next, we discussed the properties of CLSs, including the conditions for

irreducibility, band touching or crossing properties of FBs of different classes, and

completeness of CLSs. We then introduced block matrix representation of the tight-

binding Hamiltonian. At the end, we formulated CLS existence conditions, a FB tester,

and our FB generator. In the following chapters, we apply the FB generator to different

lattice dimensions and different systems, starting from a 1D system.


Chapter 4

Flatband generator in one dimension

In the previous chapters, we introduced U classification of CLSs and FB lattices, a FB

generator for U = 1, and FBs with U = ∞ as well as chiral FBs. In this chapter, we

expand the idea presented in the last chapter and discuss the U > 1 FB generator in 1D.

Starting from a two-band problem, we use CLS existence conditions and the FB gen-

erator to achieve full parameterization of FB Hamiltonians in 1D two-band networks.

Extending the approach used in the two-band case to an arbitrary number of bands, we

introduce the inverse eigenvalue method, and obtain analytical and numerical solutions

for the FB Hamiltonians.

4.1 1D tight-binding Hamiltonian

We consider a 1D translational invariant lattice with ν > 1 lattice sites per unit cell.

The time-independent Schrödinger equation on such a network is given by

∞∑m=−∞

Hm~ψn+m = E~ψn , (4.1)

where the ν × ν matrices Hm = H†−m describe the hopping (tunneling) between sites

from unit cells at distance m. As mentioned in the previous chapter, H0 is Hermitian

while Hm with m 6= 0 are not in general. We further classify networks according to

the largest hopping range mc: Hm ≡ 0 for |m| > mc ≥ 1. Note that H0 describes

intracell connectivities and Hm6=0 intercell links.

As introduced in Section 2.2, an irreducible CLS is a solution of Eq. (4.1) with~ψn 6= 0 only on the smallest possible finite number U of adjacent unit cells, and zero

everywhere else [49]. The corresponding eigenenergy is denoted as EFB. If such an

eigenstate exists, then its translations along the lattice are also eigenstates, leading to

51

52 Flatband generator in one dimension

a macroscopic degeneracy of EFB. The resulting band is flat, i.e. Eµ(k) = EFB is

independent of k. As discussed in Section 3.3.3, for 1D, the control parameters that

classify FB networks are the hopping range mc, the number of bands ν, and the CLS

class U .

The existence of CLSs in a FB lattice can now be used to design a simple local test

routine as to whether a given network has a FB of class U or not. Consider the U × Ublock matrix

HU =

H0 H1 H2 H3 . . . HU

H†1 H0 H1 H2 . . . HU−1

... . . . . . . . . . . . . ...

......

H†U−1 . . . H†2 H†1 H0 H1

H†U . . . H†3 H†2 H†1 H0

, (4.2)

and an eigenvector (~ψ1, ~ψ2, . . . , ~ψU) with eigenvalue EFB such that

mc∑m=−mc

Hm~ψp+m = 0 , ~ψl≤0 = ~ψl>U = 0, (4.3)

for all integers p with −mc + 1 ≤ p ≤ 0 and U + 1 ≤ p ≤ U + mc. Similar

equations hold for H†m. These two sets of equations ensure ~ψl≤0 = ~ψl>U = 0. Then

the Hamiltonian has a FB of class U . As an example, consider mc = 1, see Fig. 3.6.

The corresponding condition simplifies to H†1 ~ψU = H1~ψ1 = 0.

With that, we arrive at our core result—a novel systematic local FB generator based

on CLS properties. For convenience, we set mc = 1 which corresponds to nearest

neighbor hopping and is one of the most typical cases considered both experimentally

and theoretically. Then we have to find those ν × ν matrices H0, H1 that solve the

following set of equations for 1 ≤ l ≤ U :

H†1~ψl−1 +H0

~ψl +H1~ψl+1 = EFB

~ψl , (4.4)

H†1~ψ1 = H1

~ψU = 0 , ~ψ0 = ~ψU+1 = 0 . (4.5)

Choosing a set of H0, H1, we need to solve the eigenvalue problem Eq. (4.4) under

the constraint of Eq. (4.5), which makes H1 singular and ~ψ1 and ~ψU the left and right

eigenvectors of the zero mode(s) of H1. We do so by considering increasing values of

ν and U in the following sections.

4.2 Two-band problem 53

4.2 Two-band problem

The lattice and band structure of a cross-stitch lattice with U = 1, ν = 2, mc = 1 was

reported in Ref. [49] and corresponds to

H0 =

(0 0

0 0

), H1 = −

(1 1

1 1

). (4.6)

In Fig. 4.1 (b) the sawtooth lattice (ST1) with U = 2, ν = 2, and mc = 1 is shown [49]

together with its CLS and band structure, with

H0 = −

(0√

2√

2 0

), H1 = −

(0√

2

0 1

). (4.7)

ST1 and its FB were recently experimentally probed in photonic waveguide lattices [189].

-π 0 π-4-3-2-1012

k

E

(a)$ (b)$ (c)$+$ +$

($

+$ +$

1$ 2$3$

($

+$ +$

($

($

Figure 4.1: (a) Top: canonical ν = 2 chain for U = 2. Circles denote latticessites (different sizes correspond to different onsite energies), lines denote hopping con-nections (different lines correspond to different hopping strengths), and filled circlesdenote the locations of a CLS. Bottom: generalized sawtooth chain after basis rotation(see text for details). Signs indicate the signs of the CLS amplitudes. (b) The knownsawtooth ST1 chain. (c) New sawtooth ST2 chain. Top of (b,c): lattice structure,bottom of (b,c): band structure.

Without a loss of generality, we will use a canonical form of H for a generic two-

band network: a unitary transformation on each unit cell will diagonalizeH0 sorting its

diagonal elements (eigenvalues) Hµµ monotonically increasing with µ. A trivial gauge

H → H + ζI (with I the identity matrix) sets H11 = 0, and a subsequent rescaling

of H → κH ensures H22 = 1(the case of a completely degenerate H0 will be treated

separately in Appendix B.2.3). In the simplest yet nontrivial case of two bands ν = 2,


which completely fixes the non-degenerate matrix H0:

H0 =

(0 0

0 1

), H1 =

(a b

c d

). (4.8)

Since H1 is singular (as required by theorem 3.9) and of size 2, it has exactly one zero

mode, and can be parameterized by its spectral decomposition H1 = α|θ, δ〉〈ϕ, γ| as

follows:

H1 = α

(cos θ cosϕ eiγ cos θ sinϕ

e−iδ sin θ cosϕ e−i(δ−γ) sin θ sinϕ

). (4.9)

The prefactor α = |α|eiφα can be complex, and |ϕ, γ〉 and |θ, δ〉 are the left and right

eigenvectors of the non-zero eigenvalue of H1 (see Appendix B.2). The upper plot

in Fig. 4.1 (a) illustrates this canonical network structure. A rotation of the unit cell

basis by an angle ω shifts the angles θ → θ + ω and ϕ → ϕ + ω, and modifies

H0 =

(cos2 ω cosω sinω

cosω sinω sin2 ω

). As we previously discussed, FB networks are

defined up to unitary transformations. Applying θ → θ+ ω and ϕ→ ϕ+ ω, we find a

generalized sawtooth (gST) chain with three different hoppings t1,2,3 per triangle, and

an onsite energy detuning (see bottom of Fig.4.1 (a) and Appendix B.3 for details).

4.2.1 U=1 case

We start with U = 1 (Fig. 4.1 (a) in Ref. [49]). Equations (4.4–4.5) reduce to

H0~ψ1 = EFB

~ψ1, H1~ψ1 = H†1

~ψ1 = 0 . (4.10)

Then the FB energy is EFB = 0 or EFB = 1. For EFB = 0 it follows θ = π/2 or 3π/2

and ϕ = π/2 or 3π/2. Respectively for EFB = 1 we find θ = 0 or π and ϕ = 0 or

π. The canonical form of H1 has exactly one nonzero element on the diagonal, e.g.

for EFB = 0 it is H1 =

(0 0

0 |α|eiφα

). We therefore obtain the detangled structure

of the cross-stitch lattice (see Fig. 2.5 (a)). The dispersive band energy is given by

E(k) = C + 2|α| cos (k + φα) where C = 0 for EFB = 1 and C = 1 for EFB = 0. The

case of degenerate H0 ≡ 0 does not change the structure of H1 and leads to EFB = 0

and C = 0. Interestingly, the cross-stitch lattice family in Ref. [49] was characterized

by three parameters: the location of the FB energy, the width of the dispersive band,

and an overall gauge. Here we obtain a four-dimensional control parameter space. The

first three are the overall gauge ζ , the rescaling κ, and the band width control |α|, which


reproduce the findings from Ref. [49]. The additional fourth control parameter is the

phase φα. It corresponds to a time-reversal symmetry breaking effective magnetic field

in 1D, and completes the class ofmc = 1, ν = 2, U = 1 FB lattices. Remarkably, there

is another hidden U = 1 case with two FBs for which H1 has precisely one nonzero

element on one of the two off-diagonals: θ = 0, ϕ = π/2 or θ = π/2, ϕ = 0. To

observe this, one must also redefine the unit cell (see Appendix B.2 for details).

4.2.2 U=2 case

We proceed to the nontrivial U = 2 case. Here, the Hamiltonian HU=2 is a 2× 2 block

matrix

H2 =

(H0 H1

H†1 H0,

)(4.11)

where H0 is given by Eq. (4.8) and H1 is given by Eq. (4.9).

The equations Eqs. (4.4–4.5) read

H0~ψ1 +H1

~ψ2 = EFB~ψ1 , (4.12)

H†1~ψ1 +H0

~ψ2 = EFB~ψ2 , (4.13)

H1~ψ1 = 0 , (4.14)

H†1~ψ2 = 0 . (4.15)

The details of solving the above equations are given in Appendix B.2. The final result

reads:

δ = γ, |α| =√− sin(2θ) sin(2ϕ)

| sin(2(θ − ϕ))|. (4.16)

The solutions in Eq. (4.16) and the Hamiltonian H are invariant under the transforma-

tion {ϕ→ ϕ+ pπ , θ → θ + qπ , φα → φα + (p+ q)π} with p, q being integers. The

irreducible angle parameter space therefore reduces to 0 ≤ ϕ, θ ≤ π. Since |α| is real,

the solutions only exist for 0 ≤ θ ≤ π2∩ π

2≤ ϕ ≤ π or π

2≤ θ ≤ π ∩ 0 ≤ ϕ ≤ π

2, i.e.

two disjointed regions are shown for the FB energy EFB in Fig. 4.2. The corresponding

band structure is given by (see Appendix B.2 for details)

EFB =cos(θ) cos(ϕ)

cos(θ − ϕ), (4.17)

E(k) =sin(θ) sin(ϕ)

cos(θ − ϕ)+ 2|α| cos(θ − ϕ) cos(k + φα). (4.18)


-3-2-10123

Figure 4.2: Flatband energy EFB (θ, ϕ) for mc = 1, ν = 2, U = 2. The coloredsquares host FB networks, while the white ones do not. The color code shows theenergy of the FB. Dashed-dotted lines denote same onsite energies, dotted lines denotet1 = t2, dashed lines denote t2 = t3, and solid lines denote t1 = t3, all in the gSTchain. Filled circles represent the ST1 chain in Fig. 4.1 (b) and filled squares the ST2chain in Fig.4.1 (c).

The bandwidth ∆w of the dispersive band is given by

∆w = 2

√− sin(2θ) sin(2ϕ)

| sin(θ − ϕ)|, (4.19)

and is always bounded by |∆w| ≤ 2.

The FB energy is always gapped away from the dispersive band by a gap ∆g =

∆E − ∆w

2with ∆E being the distance between the FB energy and the dispersive band

center (except for a few isolated points discussed below). The ratio ∆w/∆E is shown

in Fig. 4.3. This ratio is zero for θ = π/2 + ϕ. There, the FB energy is gapped

infinitely far away from the dispersive band. Using a proper rescaling parameter κ and

a gauge ζ , we can always renormalize the band gap to a finite number, at the expense of

flattening the dispersive band. This special line corresponds to the case of degenerate

H0 and two FBs of class U = 1 (see Appendix B.2.3 for details). On the boundary

lines ϕ, θ = 0, π/2, π the band width ∆w strictly vanishes, reducing the problem to


0

0.5

1.0

1.5

2.0

Figure 4.3: Ratio of the dispersive band width to the distance between the FB andthe dispersive band center ∆w/∆E versus (θ, ϕ) for mc = 1, ν = 2, U = 2 in oneirreducible quadrant. The color code shows the value of the ratio.

a trivial H1 = 0 case with two FBs of class U = 1. One exception is the points

{θ = π − ϕ , ϕ = 0, π/2, π} where the band width ∆w stays finite but the gap ∆g

vanishes. Here, the FB becomes class U = 1 and touches the dispersive band of finite

width (see Fig. 4.3).

The class U = 2 is the largest possible irreducible CLS for ν = 2, mc = 1

networks, as can be straightforwardly checked using the above generator construction

(see Appendix B.1). In particular, we find all one-parameter families of gST chains for

which either the onsite energies are equal (dashed-dotted lines in Fig. 4.2), or a pair

of the hoppings t1,2,3 is equal (solid, dashed, and dotted lines in Fig. 4.2). The known

ST1 chain is given by the intersection of dotted and dashed-dotted lines, in which the

two hoppings are equal t1 = t2 6= t3 and the onsite energies are equal. We discover a

novel intersection point (square symbols in Fig. 4.2) where all three hoppings are equal

t1 = t2 = t3 (but the onsite energies differ). This is a new ST2 chain (Fig. 4.1(c)),

which should be easily realized experimentally: simple geometry allows all hoppings

to be made equal, an external DC bias can fine-tune the onsite energy differences, and

the CLS is easily addressed having identical absolute amplitude values on occupied

sites.

In this section, we introduced a novel FB generator for 1D translational invariant

tight-binding networks with two bands. We construct the whole FB family of two-

band networks with nearest neighbor hoppings, and we found that the largest possible


CLS size is U = 2. To solve the eigenvalue problem, we considered H0 is given and

parameterized H1, and then identified the parameter space that gives FBs. However,

in the case of higher numbers of bands, H0 cannot be considered as a given variable

and the parameterization ofH1 involves more parameters, resulting in an overcomplete

problem that is hard to solve. If we fix the CLS andH0, solving the eigenvalue problem

for H1 becomes easier. In the following section, we use the latter approach to generate

FB Hamiltonians for an arbitrary number bands and arbitrary CLS size U in 1D.

4.3 Arbitrary number of bands

In the case of more than two bands, i.e. ν > 2, we have to find ν × ν matrices H0, H1

and the CLS that satisfy the eigenvalue problem (Eq. (4.4)) and destructive interference

conditions (Eq. (4.5)). It is convenient to write the eigenvalue problem as follows:

H1~ψ2 = (EFB −H0)~ψ1 (4.20)

H†1~ψl−1 +H1

~ψl+1 = (EFB −H0)~ψl, 2 ≤ l ≤ U − 1 (4.21)

H†1~ψU−1 = (EFB −H0)~ψU (4.22)

H1~ψ1 = H†1

~ψU = 0 (4.23)

~ψl = 0 , l < 0, l > U . (4.24)

This set of equations is the starting point of our FB generator. Our goal is to generate

all possible matricesH1 which allow for the existence of a FB, given a particular choice

ofH0. Note thatH0 can be diagonal (canonical form), but any non-diagonal Hermitian

choice of H0 is allowed as well.

One way to look for solutions is to parameterize H1 and to compute FB energy

EFB and the CLS ΨCLS for a given set of U and ν. In order to satisfy Eq. (4.23), we

choose H1 from the space Z of ν × ν matrices with one zero eigenvalue. Then the

directions of the vectors ~ψ1, ~ψU are fixed by the choice of H1, leaving their two norms

as free variables. Together with the remaining unknown CLS components and the FB

energy, we arrive at V = (U − 2)ν + 3 variables. The total number of equations

in Eqs. (4.20–4.22) is E = Uν. Since ν ≥ 2, it follows that the set of equations is

overdetermined. We need 2ν−3 additional constraints which will lead us to the proper

codimension(2ν − 3) manifold in the space Z . For ν = 2, the codimension(1) man-

ifold was computed explicitly and a closed form of the functional dependence of the

CLS and FB energy on H1 was obtained in the previous section. For larger values of

ν (and U ), the constraint computation becomes difficult, and therefore we will simply

4.3 Arbitrary number of bands 59

invert the approach: we will define the CLS (thereby setting U ) and EFB and gener-

ate the proper H1 matrix manifold. This will turn an overcomplete set of equations

into an undercomplete one, which is easier to be analyzed. One of the difficulties of

Eqs. (4.21–4.24) is that EFB is also unknown. We will now show that it can be easily

expressed in terms of the CLS and the Hamiltonian.

Let us assume that ψ1 is not orthogonal to ψU . Multiplying 〈ψU | from the left with

Eq. (4.20), the FB energy EFB follows as 1

EFB =〈ψU |H0|ψ1〉〈ψ1|ψU〉

. (4.25)

For practical purposes, we can choose the CLS normalization condition 〈ψ1|ψU〉 =

1. Note that if ψ1 is orthogonal to ψU , the CLS class is reduced to a U − 1 class

by an appropriate unitary transformation including a redefinition of the unit cell (see

Theorem 3.5).

Knowing EFB, we can treat the problem of FB generation (Eqs. 4.20–4.24) as an

inverse eigenvalue problem: [25] given EFB and ΨCLS—as well as part of the Hamil-

tonian, i.e. H0—we reconstruct the Hamiltonian matrix H in Eq. (3.18). The idea of

finding hopping matrices for a fixed CLS was first introduced by Nishino, Goda, and

Kusakabe [116, 117]. Our results, even if limited to 1D, are much more systematic;

compared to the work of Nishino, Goda, and Kusakabe, we classify a CLS by its size

U , introduce the constraints on ΨCLS ensuring that it is a U -class CLS, and show how

to resolve these constraints.

4.3.1 The generator

We consider a 1D translational invariant tight-binding network with nearest neighbor

hoppings mc = 1, arbitrary number ν of bands, and the CLS class U ≤ ν. We propose

the following algorithm to construct a Hamiltonian of a FB with the above parameters:

1. Fix the number of bands ν and the size of the CLS U .

2. Choose H0 either as a diagonal (canonical form) or as any Hermitian matrix.

3. Choose a real EFB.

4. Choose ~ψ1 (or ~ψU ).

5. Exclude H1 from Eqs. (4.20–4.24), arrive at a set of two linear and further non-

linear constraints, and solve them for the remaining CLS components ~ψl.1For mc > 1, one has to assume Hm,m < mc are also input parameters.


6. Solve the linear system Eqs. (4.20–4.24) to find H1.

The system of equations (4.20–4.24) is linear, and therefore it is easy to solve or to

show that it has no solution. Typically, if this system has a solution, it will be under-

complete and show up with multiple solutions compatible with the input CLS. It is

therefore enough to find a particular solution H1 to Eqs. (4.20–4.24). A generic solu-

tion is H1 = H1 + δH1, where δH1 follows from the following homogeneous system

of equations:

δH1~ψ2 = 0

δH†1~ψl−1 + δH1

~ψl+1 = 0, 2 ≤ l ≤ U − 1

δH†U−1~ψU−1 = 0 (4.26)

δH1~ψ1 = δH†1

~ψU = 0

~ψl = 0 l < 0, l > U.

The perturbation δH1 is a deformation of the Hamiltonian H that preserves the CLS

and the FB energy, and only affects the dispersive part of the spectrum.

It is also possible to further constrain the network connectivity by choosing specific

elements of H0 and/or H1 to be zero. This is easily accounted for in H0, which is an

input parameter. The case of H1 is more involved, as discussed in Section 4.3.2.

4.3.2 Solutions

We proceed to classify FBs in the order of increasing U . The U = 1 case has already

been completed in Ref. [49]; therefore we start our classification with U = 2.

U=2 case

We fix the number of bands to ν, and chooseH0,EFB, and |ψ1〉. The inverse eigenvalue

problem Eq. (4.20–4.24) now reads

H1 |ψ2〉 = (EFB −H0) |ψ1〉

〈ψ1|H1 = 〈ψ2| (EFB −H0)

H1 |ψ1〉 = 0 (4.27)

〈ψ2|H1 = 0.


The eigenfunction ΨCLS = (~ψ1, ~ψ2) cannot be chosen arbitrarily; its second part |ψ2〉has to satisfy the following set of linear and non-linear compatibility constraints:

〈ψ1|ψ2〉 = 1

〈ψ1|H0|ψ2〉 = EFB (4.28)

〈ψ1|EFB −H0|ψ1〉 = 〈ψ2|EFB −H0|ψ2〉 .

The first constraint is simply a choice of ΨCLS normalization. The second constraint

follows from Eq. (4.25) and uses EFB as an input variable. The last identity results

from multiplying the first equation in Eq. (4.27) by 〈ψ2| from the left, and multiplying

the second equation in Eq. (4.27) by |ψ1〉 from the right. It is not possible to solve the

third constraint analytically in general, but we present in Appendix B.5.1 a numerical

algorithm that allows to resolve these constraints and enumerate all the solutions, if

existing. If existing, the solution to |ψ2〉 has ν − 3 free parameters. For the special

case of two bands ν = 2, FB energy EFB can not be chosen arbitrarily and needs to be

included into the procedure as a to-be-defined variable. Note that this particular case

can be solved in closed analytical form following a different solution strategy (see the

previous section).

Once ΨCLS = (~ψ1, ~ψ2) is known, we can solve Eq. (4.27) for H1. First we note

that the last two equations (the destructive interference conditions) can be taken into

account with the following ansatz for H1:

H1 = Q2M Q1, Qi = I− |ψi〉〈ψi|〈ψi| |ψi〉

. (4.29)

Then Eq. (4.27) becomes an inverse eigenvalue problem; the details of the derivation

are presented in Appendix B.4 and the solution is

H1 = G1 + δH1,

G1 =(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ1|EFB −H0|ψ1〉, (4.30)

δH1 = Q12KQ12,

where K is an arbitrary ν × ν matrix and Q12 is a joint transverse projector on

|ψ1〉 , |ψ2〉: Q12 |ψi〉 = 0, i = 1, 2. If the denominator 〈ψ1|EFB −H0|ψ1〉 ≡ 0, the

above solution is replaced with a more complicated expression involving two different

projectors (see Appendix B.4 for details).

It is instructive to count the number F of free parameters in the above solution,


(a)

(b)

(c)

Figure 4.4: Examples of flatband Hamiltonians with CLSs of class U = 2, ν = 3. Thesites occupied by a CLS are indicated by filled black circles. Each subfigure containsa visualization of the lattice (top) and the band structure (bottom). The FB is coloredin orange. (a) Diagonal H0; (b) non-diagonal H0, and (c) non-diagonal and fullyconnected H0. Appendix B.6.1 contains a detailed description of the Hamiltonians.

given a fixed H0, EFB, and |ψ1〉 for ν ≥ 3. It is the sum of two contributions: the

number of free parameters in δH1 and in the particular solution G1, which are (ν− 2)2

and (ν − 3), respectively. The final result is F = ν2 − 3ν + 1. It then follows

that the FB Hamiltonians form a codimension(2ν − 2) subspace, since H0 is arbitrary,

dim(H1) = ν2, and the total number of free parameters at fixedH0 is Ft = F+1+ν =

ν2 − 2(ν + 1). This is a remarkable result, since it shows that FB Hamiltonians are

only weakly fine-tuned, e.g. for ν = 3 we find five free parameters when choosing the

nine elements of H1 for an arbitrarily chosen H0. Note that the above counting does

not apply to the ν = 2 case, which is studied in the previous section and amounts to

two free parameters when choosing the four elements of H1.

Equations (4.28) and (4.30) provide the complete solution to the problem of finding

all the 1D nearest neighbor Hamiltonians with one FB and a CLS of class U = 2.

Figure 4.4 shows some examples of U = 2 and ν = 3 Hamiltonians constructed using

the above scheme.

U ≥ 3 case

Let us consider larger U values. For simplicity, we use U = 3 in the examples. First,

we fix the number of bands to ν, and choose H0, EFB, and |ψ1〉. Then we have the

following inverse eigenvalue problem with U + 2 equations (U for each CLS-occupied


unit cell, and two for the destructive interference conditions):

H1 |ψ2〉 = (EFB −H0) |ψ1〉

H†1 |ψ1〉+H1 |ψ3〉 = (EFB −H0) |ψ2〉

H†1 |ψ2〉 = (EFB −H0) |ψ3〉

H1 |ψ1〉 = 0 (4.31)

H†1 |ψ3〉 = 0.

The set of constraints for ΨCLS reads

〈ψ1 |ψ3〉 = 1

〈ψ1|H0|ψ3〉 = EFB

〈ψ1|EFB −H0|ψ2〉 = 〈ψ2|EFB −H0|ψ3〉 (4.32)

〈ψ1|EFB −H0|ψ1〉+ 〈ψ3|EFB −H0|ψ3〉 =

= 〈ψ2|EFB −H0|ψ2〉 .

Again these identities are derived from Eq. (4.31) by multiplying them with 〈ψ1| and

〈ψU | and rearranging the terms in order to eliminate H1. Notice that the set of com-

patibility constraints for ΨCLS amounts to U + 1 equations. Note also that precisely

two of those U + 1 equations, with 〈ψ1| given, are linear, and the remaining are U − 1

nonlinear equations for the remaining CLS amplitudes. It is not possible to solve the

nonlinear equations analytically in general, but we present in Appendix B.5.2 a numer-

ical algorithm that allows to resolve these constraints and enumerate all the solutions,

if existing, for the case U = 3.

Instead of using the ansatz (4.29) for H1, we take a more suitable approach to

generate FB Hamiltonians (i.e. matrices H1) for U ≥ 3. With a given ΨCLS, which

satisfies the constraints from Eq. (4.32), the set of equations in Eq. (4.31) is a linear

system with respect to H1:

T h1 = Λ. (4.33)

Here, h1 is a ν2-dimensional vector resulting from the vectorization of the matrix H1,

T is a rectangular ν(U + 2)× ν2 matrix whose elements are composed of the elements

of the CLS, such that the product T h1 is the left hand side of Eq. (4.31), and Λ is a


ν(U + 2) vector originating from the right hand side of Eq. (4.31):

Λ = (EFB −H0)

~ψ1

~ψ2

. . .

~ψU~0

~0

. (4.34)

The zero-vector components ~0 result from the destructive interference. The linear sys-

tem of Eq. (4.33) can then be solved, for example by using a least squares solver.

Figure 4.5 shows some examples of U = 3 FBs, which we generated by resolving the

constraints in Eq. (4.32) and solving Eq. (4.33).

4.3.3 Chiral symmetry

An important subclass of FB networks is those with chiral symmetry (see Section

2.3.3), and our generator simplifies in the case of chiral symmetry. Chiral lattices

are bipartite networks with minority and majority sublattices. This imposes a specific

structure of the hopping integrals and the CLS amplitudes ~ψl. For that, we split the

lattice sites from each unit cell into two subsets, each belonging to one of the two

sublattices. This leads to a splitting of each ~ψl into two sublattice vectors, as well as

to a corresponding block structure of the matrices H0, H1. As a result, the CLS of a

chiral FB will always reside exclusively on the majority sublattice [133]:

H0 =

(0 A†

A 0

), H1 =

(0 T †

S 0

)

~ψl =

(~ϕl

0

), l = 1, . . . , U . (4.35)

Here A, S, and T are (ν − µ) × µ matrices, µ is the number of sites on the majority

sublattice in the unit cell, and ~ϕl is a µ-component vector residing on the majority sub-

lattice sites in a unit cell. By definition ν − µ ≤ µ < ν. The spectrum of the system

enjoys particle-hole symmetry around E = 0. A chiral flatband has energy EFB = 0

and is symmetry protected. For ν < 2µ, there are µ−bν/2c FBs at EFB = 0 [133]. In-

creasing the range of hopping mc > 1 while preserving chiral symmetry will keep the

chiral FBs in place. Moreover, one can preserve the chiral FBs by partially destroying

the chiral and sublattice symmetry; this is achieved by adding hopping terms on the


(a)

(b)

(c)

(d)

Figure 4.5: Examples of three-band FB lattices with a CLS of class U = 3. The sitesoccupied by a CLS are indicated by black filled circles. (a) Diagonal choice forH0. (b)Chain-like structure for H0. (c) Generic choice for H0. (d) EFB is set negative enoughto become the ground state. Details of these examples are presented in Appendix B.6.2.

minority sublattice only, since the chiral FB CLS is occupying majority sublattice sites

only:

H0 =

(0 A†

A B

), H1 =

(0 T †

S W

)

~ψl =

(~ϕl

0

), l = 1, . . . , U , (4.36)


where B and W are (ν − µ) × (ν − µ) matrices. Note that the overall particle-hole

symmetry of the system is lost, but the original chiral FBs are still present at EFB = 0.

For a bipartite network, the hopping matrix H1 has a specific structure given by

Eq. (4.36), that simplifies Eq. (4.27) to

S |ϕ2〉 = −A |ϕ1〉 (4.37)

S |ϕ1〉 = 0 (4.38)

T |ϕ1〉 = −A |ϕ2〉 (4.39)

T |ϕ2〉 = 0, (4.40)

where EFB = 0. The minority sublattice hopping matrices B,W dropped out as

expected. The above equations are considerably simpler than the generic U = 2

Eq. (4.27): the above system splits into two independent inverse eigenvalue problems

for S and T . Details of the solution are presented in Appendix B.4.3, the final answer

of which is

S = −A |ϕ1〉〈ϕ2|Q1

〈ϕ2|Q1|ϕ2〉+KSQ12

T = −A |ϕ2〉〈ϕ1|Q2

〈ϕ1|Q2|ϕ1〉+KTQ12, (4.41)

whereKT andKS are arbitrary matrices of size (ν−µ)×µ, andQ12 is a joint transverse

projector on |ϕ1,2〉. There are no restrictions on the entries ofA,B,W and |ϕ1,2〉—they

are all free parameters—in contrast to the generic U = 2 FB construction. Therefore,

the number of free parameters is (ν − µ)(2ν + µ − 2) − 1 (see Appendix B.4.3 for

details). The above solution fails for 〈ϕ2|Q1|ϕ2〉 = 〈ϕ1|Q2|ϕ1〉 ≡ 0; as a result,

|ϕ2〉 ∝ |ϕ1〉, and the CLS and the FB are of class U = 1.

Figure 4.6 shows an example of a bipartite lattice with ν = 4. There are two

sites in the unit cell of each sublattice, and B 6= 0, W 6= 0. In this example, the

parameters ~ϕ2, ~ϕ2, A,B,W are arbitrarily chosen, and KT = 0, KS = 0 (see details

in Appendix B.6.1).

4.3.4 Network constraints

For practical purposes,FB fine-tuning of a Hamiltonian network can involve additional

network constraints, e.g. the strict vanishing of certain hopping terms between specific

sites of the network [127]. This typically happens when arranging network sites in a

plane. Let us consider the typical problem of finding a nearest neighbor FB Hamilto-

4.4 Summary 67

Figure 4.6: Example of a bipartite FB Hamiltonian with U = 2, ν = 4. The sitesof the CLS are indicated with filled black squares. Links are colored differently forthe convenience of visualisation of the chain. In this example, chiral symmetry is bro-ken on the minority sublattice due to the presence of B 6= 0, W 6= 0 in Eq. (4.36).Nevertheless, the chiral FB is preserved. Details are given in Appendix B.6.1.

nian with specific network constraints. These network constraints dictate the locations

of the zero entries in H0 and H1. They can be incorporated into the matrix T of

Eq. (4.33) as a mask M : T → TM that ensures the zero entries in H1 are in the right

positions. The solution of the resulting system is then searched for similarities to the

non-constrained case.

Especially when H0 and H1 are sparse, e.g. the number of variables in H1 is equal

to or greater than the number of equations, it is possible to solve Eqs. (4.20–4.24)

analytically (see Appendix B.7). Figure 4.7 shows examples of networks with FBs

generated for a 1D kagome chain and for chains with hoppings allowed only inside

network plaquettes.

4.4 Summary

In this chapter, we introduced a novel flatband generator for 1D translational invariant

tight-binding networks. First, we solved a two-band problem to obtain a family of

FB Hamiltonians parameterized by a two-parameter family, where we found that the

largest possible size of the irreducible CLS for this case is U = 2. Particularly, we


(a)

(b)

(c)

Figure 4.7: Examples of FB Hamiltonians constructed in specific networks. The sitesoccupied by a CLS are marked by black filled circles. (a) 1D kagome with ν = 5and U = 2 CLSs. The crossing of the three bands indicates that the Hamiltonian canbe detangled into two independent sub-Hamiltonians. (b) and (c) Examples of ν = 3Hamiltonians with U = 2 and U = 3 CLSs, respectively. The details of all theseHamiltonians are provided in Appendix B.7.

showed that by using unitary transformations, all FB networks with two bands can

be mapped into generalized sawtooth chains, and we found a highly symmetric ST2

chain.

Extending the above methods to an arbitrary number of bands, we presented a sys-

tematic construction of 1D Hamiltonians with ν > 2 bands including one FB for an

arbitrary size CLS U ≤ ν, and illustrated the method with several examples. The task

of finding FB Hamiltonians is reduced to solving specific inverse eigenvalue problems

subject to certain non-linear constraints. Flatband energy enters as a parameter and

can be tuned. For the U = 2 case, we derived analytical solutions to the inverse eigen-

value problem supplemented with a numerical algorithm to resolve the constraints. For

U ≥ 3, analytical solutions are not accessible, yet numerical algorithms can be applied

to generate FB Hamiltonians. We illustrated the method by generating several U = 3

FB Hamiltonians. The same construction allows us to incorporate various network

geometry constraints into the search algorithm. Our results show that FB Hamiltoni-

ans, while being the result of fine-tuning in the space of all tight-binding Hamiltonian

networks, allow for a surprisingly large number of free parameters, that change the

network but leave the flatness of the FB untouched. In the next chapter, we extend our

1D FB generator to two dimensions.

Chapter 5

Flatband generator in two dimensions

In the previous chapter, we introduced a flatband (FB) generator in 1D. There are more

interesting phenomena though in 2D FB lattices that have been reported theoretically

and observed experimentally. Therefore, it is highly desirable to extend the 1D FB

generator to 2D. The methods for 1D can be implemented in 2D, but with some added

complexity. First, identifying the CLS shape is more difficult, and second, depending

on CLS size and shape, the destructive interference conditions are more complicated.

Last but not least, the expanded hopping range also adds more complexity to the eigen-

value problem as well as the destructive interference. Regarding these complications,

we mostly restrict our attention to identifying all possible FB Hamiltonians with near-

est and next nearest neighbor unit cell hoppings, with CLSs occupying a maximum of

four unit cells in a 2× 2 plaquette.

The outline of this chapter is as follows. We start in Section 5.1 by introducing the

eigenvalue problem before discussing the classification of CLSs in 2D and destructive

interference conditions in Section 5.2. We present the FB generator scheme for this

case in Section 5.3. Results for nearest neighbor hoppings and next nearest neighbor

hoppings are given in Sections 5.4 and 5.5, respectively.

5.1 The eigenvalue problem

We consider a 2D translational invariant tight-binding network with nearest and next

nearest neighbor hoppings and ν sites per unit cell. We use block matrix representation

with the same notations and conventions from Section 3.3. The eigenvalue problem

here reads

H0~ψn +

∑χ

H†χ~ψl′χ +

∑χ

Hχ~ψlχ = E~ψn, n ∈ Z , (5.1)

69

70 Flatband generator in two dimensions

where ν-component vector ~ψn is the wave function of the nth unit cell, Hχ is the near-

est neighbor hopping matrix for the χth direction, and lχ and l′χ are the indices of the

nearest neighboring unit cells along the χth direction. Note that we refer to neighbor-

ing unit cells as neighbors, which is different from conventional notation. In this nota-

tion, the nearest neighbors are the nearest neighboring unit cells along primitive lattice

translation vectors, and the next nearest neighbors are the nearest neighbors along the

diagonal (non-primitive) directions. More precisely, when we consider only nearest

neighbors, χ = 1, 2 in Eq. (5.1) to represent two directions along the two primitive

lattice translation vectors. In the case of next nearest neighbor hoppings, χ = 1, 2, 3

in Eq. (5.1), where χ = 3 is the third hopping direction, and the neighbors along this

direction become the next nearest neighbors (see the example in Fig. 5.1).

Figure 5.1: Example of nearest neighbors and next nearest neighbors in simple 2Dlattices with one site per unit cell, where ~a1, ~a2 are primitive translation vectors and~a3 = ~a1 + ~a2. (a) Square lattice highlighting only nearest neighbors along ~a1, ~a2. (b)Hexagonal lattice highlighting two next nearest neighbors along ~a3.

5.2 Classification of CLSs in 2D

As discussed in Section 3.1, we classify a 2D CLS by shape vector U = (U1, U2),

where U1, U2 are the range of a CLS along two primitive translation vectors ~a1, ~a2.

However, as shown in Fig. 5.2, not all of the U1U2 unit cells in the U1×U2 plaquette are

occupied. Especially, when U1, U2 are large, the CLS can take complicated shapes.

In such cases, we need more parameters to specify which sites are occupied in the

U1×U2 plaquette. Then, U could be a U1×U2 matrix composed of 1 or 0, representing

occupied or empty unit cells in such a plaquette. However, when U1, U2 are small, all

possible shapes can be identified by just introducing extra parameter s to count the

number of unoccupied unit cells. In this case, the shape vector reads U = (U1, U2, s).

Then the CLS size U , i.e. the number of unit cells occupied by the irreducible CLS, is

given by U = U1U2− s. To the best of our knowledge, most known examples fall into

5.2 Classification of CLSs in 2D 71

the U = (U1 = 2, 1 ≤ U2 ≤ 2, 1 ≤ s ≤ 2) classes (see Fig. 5.2), and thus we limit

our focus to these classes in this thesis.

Figure 5.2: The U-classification of four 2D FB lattices. Shaded areas give the rangeof the CLS (i.e. the plaquette), and the darker shaded regions show the occupied unitcells (black circles). Red boxes denote the unit cell, and~a1, ~a2 are primitive translationvectors, and ~a3 = ~a1 ± ~a2. (a) Lieb: U = (2, 2, 1), (b) checkerboard: U = (2, 2, 1),(c) kagome: U = (2, 2, 1), and (d) dice: U = (2, 2, 0).

Under this limitation, the wave functions of a maximum four unit cells are relevant

in Eq. (5.1) (i.e. all other wave functions are zero), and we label the CLS components

as ~ψi=1,...,4. We also use bra-ket notations |ψi〉 and ~ψi interchangeably throughout

this chapter. Then, with these notations, we can write the eigenvalue problem and

destructive interference conditions for this limited case of U = (U1 = 2, 1 ≤ U2 ≤2, 1 ≤ s ≤ 2).

In the case of nearest neighbor hoppings, hopping matrices H1, H2 describe the

hoppings along primitive lattice translation vectors ~a1, ~a2, respectively. There are

several possible shapes along with hoppings, as illustrated in Fig. 5.3. We can write

the eigenvalue problem and destructive interference conditions as follows:


H1~ψ2 +H2

~ψ3δU2,2 = (EFB −H0)~ψ1,

H†1~ψ1 +H2

~ψ4δU2,2δs,0 = (EFB −H0)~ψ2,(H1

~ψ4δs,0 +H†2~ψ1

)δU2,2 = (EFB −H0)~ψ3δU2,2,(

H†1~ψ3δs,0 +H†2δs,0

)δU2,2 = (EFB −H0)~ψ4δU2,2δs,0,

H1~ψ1 = H†1

~ψ2 = 0,

H2~ψ1 = H2

~ψ2 = 0,

H1~ψ3δU2,2 = H†2

~ψ3δU2,2 = 0,

H1~ψ3δU2,2δs,1 +H†2

~ψ2 = 0,

H†1~ψ4δU2,2δs,0 = H†2

~ψ4δU2,2δs,0 = 0 .

(5.2)

Putting the corresponding values of U1, U2, s, we get the eigenvalue problem and

destructive interference conditions corresponding to CLS class U = (U1, U2, s).

Figure 5.3: Classification of U = (2, U2, s) CLSs and destructive interference. Eachsquare represents a unit cell. (a) Single unit cell and hopping directions representedby arrows. (b) U=(2,2,1) case. (c) U=(2,1,0) case. (d) U=(2,2,2) case. (e) U=(2,2,0)case. In (b)–(e), the arrows represent destructive interference. When two arrows cross,waves from two different unit cells interfere destructively.

Many known FB lattices, such as checkerboard, kagome, and dice lattices, involve

diagonal hoppings, i.e. next nearest neighbor hoppings, which are important to con-

sider. In this case, there are two different situations: one when hoppings along both

diagonal directions are non-zero, as shown in Fig. 5.4 (a), and one when hoppings

along only one diagonal direction are non-zero, as shown in Fig. 5.4 (b). For simplic-

ity, we only consider the latter case, as the most of the known examples fall into this

5.2 Classification of CLSs in 2D 73

category.

Figure 5.4: Different configurations of U = (2, U2, s) CLS classes for different s ina 2D lattice with next nearest neighbor hoppings. (a) Single unit cell with hoppingsalong both diagonal directions. (b) Single unit cell with hoppings along one diagonaldirection. In (a) and (b), arrows indicate the the hopping directions. (c)–(f) Differentcases with hoppings along one diagonal direction, where the arrows represent de-structive interference, i.e. the waves hopping along the arrow direction become zero.When two arrows cross, waves from two different unit cells interfere destructively. (c)U=(2,2,1), (d) U=(2,1,0), (e) U=(2,2,2), and (f) U=(2,2,0).

We still consider U = (U1 = 2, 1 ≤ U2 ≤ 2, 0 ≤ s ≤ 1), but now introduce a

new hopping matrix H3 to describe next nearest hoppings, i.e. the hoppings between

nearest neighbors along the diagonal direction, see Fig. 5.4 (b). Then the eigenvalue

problem and destruction interference conditions become:

H1~ψ2 +H2

~ψ3δU2,2 = (EFB −H0)~ψ1,

H†1~ψ1 +H2

~ψ4δU2,2δs,0 +H†3~ψ3δU2,2 = (EFB −H0)~ψ2,(

H1~ψ4δs,0 +H†2

~ψ1 +H3~ψ2

)δU2,2 = (EFB −H0)~ψ3δU2,2,(

H†1~ψ3 +H†2

~ψ2

)δU2,2δs,0 = (EFB −H0)~ψ4δU2,2δs,0,

H1~ψ1 = H2

~ψ1 = 0,

H†1~ψ4δU2,2δs,0 = H2

~ψ4δU2,2δs,0 = 0,

H†3~ψ2 = H3

~ψ3δU2,2 = 0,

H2~ψ2 +H†3

~ψ1 = 0,

H1~ψ3δU2,2 +H3

~ψ1δU2,2 = 0,

H†2~ψ3δU2,2 +H3

~ψ4δU2,2δs,0 = 0,

H†1~ψ2 +H3

~ψ4δU2,2δs,0 = 0 .

(5.3)


Putting corresponding values of U1, U2, s into the above equation, we get the eigen-

value problem and destructive interference conditions corresponding to CLS class

U = (U1, U2, s) with next nearest neighbor hoppings.

5.3 The flatband generator

According to Definition 3.13 in Section 3.3.2, a 2D FB generator for a class U FB is a

scheme to generate the set of all possible ~ψi=1,...,U and matricesH0, H1, H2 that satisfy

the eigenvalue problem and destructive interference conditions as from Eq. (5.2) or Eq.

(5.3).

As we did in 1D, here we want to generate all possible H1, H2 (and H3 in case

of next nearest neighbor hoppings) that satisfy Eq. (5.2) (Eq. (5.3)) for a given H0

(canonical form or any Hermitian matrix) and particular choice of CLS class U. For a

given H0 and CLS, these equations become an inverse eigenvalue problem of finding

H1, H2 (and H3 for next nearest neighbor hoppings). Since these equations depend on

the shape of the CLS, i.e. U2, s, we solve them case by case. In general, our generator

works as follows.

1. Choose a shape vector U = (2, 1 ≤ U2 ≤ 2, 0 ≤ s ≤ 1).

2. Choose a hopping range, either nearest neighbor or next nearest neighbor.

3. Write down corresponding form of Eq. (5.2) or Eq. (5.3) for the chosen U.

4. Choose an arbitrary (or a particular) H0 that is either canonical or any Hermitian

matrix.

5. Choose a real EFB.

6. Choose an arbitrary ~ψ1 or ~ψU .

7. Exclude H1, H2 from the equations obtained in step 3 to get non-linear con-

straints on remaining CLS components ~ψi, and solve these constraints to find the

remaining CLS components.

8. With the chosen H0 and CLS obtained from the previous step, solve the equa-

tions obtained in step 3.

Below we present our results case by case. First, we note the U = (2, 2, 2) (see Fig. 5.3

(d)). The eigenvalue problem for this case reduces to H0~ψi=1,2 = EFB ~ψi=1,2, because

the nearest neighbors of ~ψi=1,2 are all zero. This leads to one of two possibilities: either

5.4 Nearest neighbor hoppings 75

a U = 1 case or two degenerate FB with the same EFB. A generic way to construct

U = 1 FBs is presented in Section 2.3.2, and the case of a degenerate FB is not our

focus in this thesis. Therefore, we do not tackle this U = (2, 2, 2) case further in this

chapter.

5.4 Nearest neighbor hoppings

Consider the two-band case, where H0, H1, H2 are 2× 2 matrices, and at least one of

them has to have two zero modes. This results in one of three possibilities: a U = 1

solution, isolated 1D chains, or isolated unit cells (see Appendix C.1). This leads to

the following theorem.

Theorem 5.1. In a 2D two-band translational invariant tight-binding network with

nearest neighbor hoppings, the only possible CLS class is U = 1.

Therefore, from now on we consider more than two bands, i.e. ν ≥ 3 cases.

U=(2,1,0) case

A schematic of the CLS and destructive interference conditions for this case is shown

in Fig. 5.3 (c). Here, U2 = 1, s = 0, and Eq. (5.2) reduces to a 1D form that yields

the following solution (see Appendix C.2.1):

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉,

H2 = Q12MQ12,

(5.4)

where Q1 and Q2 are transverse projectors of |ψ1〉, |ψ2〉, i.e. Q1,2|ψi=1,2〉 = 0 (see

Section 4.3, and Appendix B.4.2). For given H0, EFB, the CLS components ~ψ1, ~ψ2

have to be chosen respecting the following constraints,

〈ψ2|(EFB −H0)|ψ1〉 = 0,

〈ψ1|(EFB −H0)|ψ1〉 = 〈ψ2|(EFB −H0)|ψ2〉 .(5.5)

For givenH0, EFB, |ψ1〉 and ν ≥ 3 there are ν−3 free variables inH1 (see Section

4.3), and (ν− 2)2 free variables in H2. Therefore, there are in total (ν− 2)2 + ν− 3 =

ν2 − 3ν + 1 variables in the solution to Eq. (5.4). If we only fix H0, then EFB, |ψ1〉contribute other ν+1 free parameters, which makes the total number of free parameters

ν2 − 3ν + 1 + ν + 1 = ν2 − 2(ν − 1).


U=(2,2,1) case

A schematic of the CLS and destructive interference conditions for this case is shown

in Fig. 5.3 (b). In this case U2 = 2, s = 1, and our generator gives the following

solution for the ν > 3 case (see details in Appendix C.2.2):

H1 =1

〈ψ3|Q1,2|a〉Q1,2|a〉〈z|Q1,2,3 +

(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ2| (EFB −H0) |ψ2〉

− 〈ψ3| (EFB −H0) |ψ3〉〈ψ2| (EFB −H0) |ψ2〉〈ψ1|Q2,3|y〉

Q2,3|y〉〈ψ2| (EFB −H0)

H2 =1

〈ψ1|Q2,3|y〉Q2,3|y〉〈ψ3|(EFB −H0)

+1

〈ψ2|Q3|u〉

(〈ψ1|Q3|u〉〈ψ1|Q2,3|y〉

Q2,3|y〉 −Q3|u〉)〈z|Q1,2,3,

(5.6)

where Qi are transverse projectors on |ψi〉, Q2,3 is a transverse projector on |ψi=2,3〉,and Q1,2,3 is a transverse projector on |ψi=1,2,3〉 (see Section 4.3 and Appendix B.4.2).

The ν component vectors |y〉, |z〉, |a〉, |u〉 are free parameters, which are defined as

Q2,3|y〉 = H2|ψ3〉 = (EFB −H0)|ψ1〉 −H1|ψ2〉, 〈z|Q1,2,3 = 〈ψ3|H1 = −〈ψ2|H2.

(5.7)

The CLS components |ψi=1,2,3〉 must satisfy the following constraints:

〈~ψ2|H0|~ψ1〉 = EFB〈~ψ2|~ψ1〉

〈~ψ3|H0|~ψ1〉 = EFB〈~ψ3|~ψ1〉

〈~ψ3|H0|~ψ2〉 = EFB〈~ψ3|~ψ2〉

〈~ψ2| (EFB −H0) |~ψ2〉+ 〈~ψ3| (EFB −H0) |~ψ3〉 = 〈~ψ1| (EFB −H0) |~ψ1〉.

(5.8)

When Q1,2,3|z〉 = 0, it corresponds to the case that the hoppings from ~ψ2 and ~ψ3

are zero individually, instead of the sum of the hoppings from ~ψ2 and ~ψ3 being zero.

The solution of this case is given in Appendix C.2.3. Note that, when Q1,2,3|z〉 = 0

and Q2,3|y〉 = (EFB −H0)|ψ1〉, the solution in Eq. (5.6) also reduces to the solution

in Appendix C.2.3.

ν = 3 case: In this case, transverse projector Q1,2,3 on |ψi=1,2,3〉 is a 3 × 3 matrix.

If |ψi=1,2,3〉 are linearly independent, then Q1,2,3 becomes zero, and so Q1,2,3|z〉 = 0.

The solution for this case is given in Appendix C.2.3, as discussed above. If we require

5.5 Next nearest neighbor hoppings 77

Q1,2,3 6= 0, and so Q1,2,3|z〉 6= 0, then |ψi=1,2,3〉 must be linearly dependent as

c1|ψ1〉+ c2|ψ2〉+ c3|ψ3〉 = 0, (5.9)

which gives the following solution (see details in Appendix C.2.2):

H1 =Q2|a〉〈ψ2| (EFB −H0)

〈ψ3|Q2|a〉

H2 =Q3|c〉〈ψ3| (EFB −H0)

〈ψ2|Q3|c〉,

(5.10)

where |a〉, |c〉 are free parameters, and EFB, H0, |ψ2〉, |ψ3〉 are chosen respecting the

constraints〈ψ2| (EFB −H0) |ψ2〉 = 0

〈ψ3| (EFB −H0) |ψ3〉 = 0

〈ψ3| (EFB −H0) |ψ2〉 = 0

(EFB −H0) (α|ψ2〉+ β|ψ3〉) = 0,

(5.11)

where α, β are proportionality factors such that

|ψ1〉 = α|ψ2〉+ β|ψ3〉. (5.12)

According to Conjecture 3.5, the CLS given by Eq. (5.12) always yields band touching.

U=(2,2,0) case

In this case, we only solve the three-band problem using a different method than above.

The CLS and destructive interference conditions are illustrated in Fig. 5.3 (e). Here, we

use a direct parameterization to solve Eq. (5.2) and obtain a lengthy analytic solution,

which is given in Appendix C.2.4. There are three free parameters in this solution.

5.5 Next nearest neighbor hoppings

Following the same procedure as for nearest neighbor hoppings except for steps 3 and

7, here we put corresponding values of U2, s into Eq. (5.3) to obtain solutions for

U = (2, U2, s) cases.


U=(2,1,0) case

The CLS and destructive interference for this case are illustrated in Fig. 5.4 (d). Here,

U2 = 1, s = 0, and Eq. (5.3) gives the following solution (see Appendix C.3.2):

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉,

H2 = Q12|c〉〈d|Q12,

H3 = Q12|e〉〈f |Q12 ,

(5.13)

whereQ12 is a transverse projector of ~ψ1, ~ψ2, andH0, EFB, ~ψ1, ~ψ2 are chosen respect-

ing the following constraints

〈ψ2|(EFB −H0)|ψ1〉 = 0,

〈ψ1|(EFB −H0)|ψ1〉 = 〈ψ2|(EFB −H0)|ψ2〉 .(5.14)

From Eq. (5.13), we can see that, for the two-band case, the solution gives de-

coupled 1D chains. Because, for the two-band case, the transverse projector becomes

zero, which makes H2 = H3 = 0. Therefore, no two-band solution exists for the

U = (2, 1, 0) case.

U=(2,2,1) case

The CLS and hoppings for this case are illustrated in Fig. 5.4 (c). Here, U2 = 2, s = 1,and Eq. (5.3) yields the following solution (see details in Appendix C.3.1):

H1 =Q2|z〉〈x|Q1

〈ψ3|Q2|z〉+

(〈ψ3|Q2|z〉(EFB −H0)|ψ1〉 − 〈ψ3|Q2|z〉Q3|u〉 − 〈x|Q1|ψ2〉Q2|z〉)〈ψ3|Q2|z〉 ((〈ψ1|(EFB −H0)|ψ1〉 − 〈ψ1|Q3|u〉) 〈ψ3|Q2|z〉 − 〈ψ1|Q2|z〉〈x|Q1|ψ2〉)

× (〈ψ3|Q2|z〉〈ψ2|(EFB −H0)− 〈ψ3|Q2|z〉〈w|Q3 − 〈ψ1|Q2|z〉〈x|Q1) ,

H2 = −Q3|y〉〈x|Q1

〈ψ2|Q3|y〉+

(〈ψ2|Q3|y〉Q3|u〉+ 〈x|Q1|ψ3〉Q3|y〉)〈ψ2|Q3|y〉 (〈ψ2|Q3|y〉〈ψ1|Q3|u〉+ 〈x|Q1|ψ3〉〈ψ1|Q3|y〉)

× (〈ψ2|Q3|y〉〈ψ3|(EFB −H0)− 〈ψ2|Q3|y〉〈v|Q2 + 〈ψ1|Q3|y〉〈x|Q1) ,

H3 = −Q2|z〉〈y|Q3

〈ψ1|Q2|z〉+

(〈ψ1|Q2|z〉Q2|v〉+ 〈y|Q3|ψ2〉Q2|z〉) (〈ψ1|Q2|z〉〈w|Q3 + 〈ψ3|Q2|z〉〈y|Q3)

〈ψ1|Q2|z〉 (〈ψ1|Q2|z〉〈ψ3|Q2|v〉+ 〈y|Q3|ψ2〉〈ψ3|Q2|z〉).

(5.15)

5.6 Summary 79

In this solution, |x〉, |y〉, |z〉, |u〉, |v〉, |w〉 are introduced to decouple the equations for

H1, H2, H3 (see Appendix C.3.1), and they are defined as

H†1|ψ3〉 = −H†2|ψ2〉 = Q1|x〉,

H2|ψ2〉 = −H†3|ψ1〉 = Q3|y〉,

H1|ψ3〉 = −H3|ψ1〉 = Q2|z〉,

H2|ψ3〉 = Q3|u〉,

H3|ψ2〉 = Q2|v〉,

H†3|ψ3〉 = Q3|w〉 .

(5.16)

These vectors (|x〉, |y〉, |z〉, |u〉, |v〉, |w〉) and |ψi=1,2,3〉 must satisfy the following con-

straints (see details in Appendix C.3.1):

〈ψ1|(EFB −H0)|ψ1〉 − 〈ψ1|Q3|u〉 = 〈ψ2|(EFB −H0)|ψ2〉 − 〈w|Q3|ψ2〉 = 〈ψ1|Hx|ψ2〉,

〈ψ2|(EFB −H0)|ψ3〉 = 〈ψ1|Q2|z〉 = 〈ψ1|Hx|ψ3〉,

〈ψ2|(EFB −H0)|ψ1〉 = 〈w|Q3|ψ1〉,

〈ψ2|(EFB −H0)|ψ1〉 = 〈ψ2|Q3|u〉,

〈ψ3|(EFB −H0)|ψ1〉 = 〈x|Q1|ψ2〉 = 〈ψ3|Hx|ψ2〉,

〈x|Q1|ψ3〉 = 〈ψ3|Q2|z〉 = 〈ψ3|Hx|ψ3〉,

〈ψ3|(EFB −H0)|ψ2〉 = 〈ψ1|Q3|y〉 = 〈ψ1|Hy|ψ2〉,

〈ψ3|(EFB −H0)|ψ3〉 − 〈v|Q2|ψ3〉 = 〈ψ1|Q3|u〉 = 〈ψ1|Hy|ψ3〉,

〈ψ3|(EFB −H0)|ψ1〉 = 〈v|Q2|ψ1〉,

〈ψ2|Q3|y〉 = −〈x|Q1|ψ2〉 = 〈ψ2|Hy|ψ2〉,

〈ψ2|Q3|u〉 = −〈x|Q1|ψ3〉 = 〈ψ2|Hy|ψ3〉,

〈y|Q3|ψ1〉 = 〈ψ1|Q2|z〉 = −〈ψ1|Hyx|ψ1〉,

〈ψ1|Q2|v〉 = −〈y|Q3|ψ2〉 = 〈ψ1|Hyx|ψ2〉,

〈w|Q3|ψ1〉 = −〈ψ3|Q2|z〉 = 〈ψ3|Hyx|ψ1〉,

〈ψ3|Q2|v〉 = 〈w|Q3|ψ2〉 = 〈ψ3|Hyx|ψ2〉 .(5.17)

5.6 Summary

In this chapter, we extended the flatband generator in one dimension to two dimensions.

We classified all CLSs that occupy a maximum of 4 unit cells in a 2 × 2 plaquette.


Then we setup eigenvalue problems, and identified destructive interference conditions

for different CLS classes. By solving these sets of equations, we obtained analytic

solutions for the Hamiltonians of given flatband classes.

Our method not only covers all known examples so far, it also gives a large number

of extra free parameters. These results can be extended to larger CLSs in 2D, and

higher lattice dimensions as well. In the next chapter, we extend our flatband generator

to the non-Hermitian regime.

Chapter 6

Non-Hermitian flatband generator

Recently, systems described by non-Hermitian Hamiltonians have been drawing more

attention, with an increasing number of studies being carried out to understand the fate

of flatbands (FBs) under non-Hermitian perturbations or non-Hermitian settings. Vari-

ous schemes have been proposed to construct FB lattices in the non-Hermitian regime.

In this chapter, we introduce a generator—a systematic classification and construction

scheme for non-Hermitian FB Hamiltonians—for 1D a non-Hermitian tight-binding

network with two bands.

We start this chapter by introducing background and our motivation for the non-

Hermitian FB generator in Section 6.1, before moving on to main definitions in Section

6.2. Then we introduce our FB generator scheme in Section 6.3, present our results in

Section 6.4, and conclude in Section 6.5.

6.1 Overview of non-Hermitian physics and flatbands

in non-Hermitian systems

Non-Hermitian systems [11, 108] exhibit extraordinary properties such as complex

spectra, non-orthogonal eigenstates [26, 33], exceptional points [20, 54, 68, 69, 70,

193], etc. Moreover, in open quantum systems, non-Hermitian Hamiltonians can de-

scribe coupling with the environment, which simplifies analysis and reduces the large

number of degrees of freedom [45, 144, 145]. After the discovery that parity-time

(PT )-symmetric non-Hermitian Hamiltonians can have real eigenvalues [18, 19, 201],

PT symmetry was demonstrated in optical systems with gain and loss [100], which in

turn led to an explosion of research outcomes in PT -symmetric non-Hermitian pho-

tonics [43, 44, 48, 64, 85, 88, 112, 146, 177, 191, 200]. Non-Hermitian topology has

also been drawing more interest [62, 93, 198, 199].

81

82 Non-Hermitian flatband generator

In terms of macroscopically degenerate FBs, i.e. dispersionless energy bands of

translational invariant tight-binding networks, most have have been studied in Hermi-

tian systems [97, 98, 105, 106, 107, 173, 174], and experimentally realized in photonic

systems[65, 111, 183], ultra-cold atoms [74], and exciton-polariton condensates [103].

Such achievements though cannot be applied to the above-mentioned non-Hermitian

systems, such as coupled laser cavities, where gain and loss is unavoidable. There-

fore, designing models that support FBs in the presence of non-Hermiticity, or in other

words gain and loss, is highly desirable.

Accordingly, research concerning FBs in non-Hermitian systems has been ini-

tiated [32, 56, 135]. Indeed, FBs have been found in various non-Hermitian sys-

tems [57, 94, 129, 202]. Most of these studies have been carried out in the context of

symmetry, such as PT symmetry, chiral symmetry, and non-Hermitian particle-hole

symmetry [57, 129]. While FBs can exist in non-Hermitian systems without any sym-

metry, little is known about FBs in the non-symmetric regime, and further the eigen-

states of non-Hermitian FBs have been largely untouched. It has become important

then to propose a systematic classification and construction scheme for non-Hermitian

FB Hamiltonians, as has been done for 1D Hermitian systems [97, 98].

We provide here such a scheme, which we term a generator, for non-Hermitian FB

Hamiltonians in 1D two-band networks. Non-Hermiticity in our model comes from

the asymmetry in the hoppings to the right and left, as well as onsite gain and loss.

We obtain FB Hamiltonians simply by requiring eigenvalues of the non-Hermitian k-

space Hamiltonian to be k-independent (i.e. flat). Therefore, symmetry is not required

in our model. Since non-Hermitian Hamiltonians give complex band structures, non-

Hermitian FBs naturally fall into three different categories: one in which both real and

imaginary parts are flat (the completely flat case), one in which either real or imaginary

parts are flat (the partially flat case), and one in which the modulus is flat (the flat

modulus case). Using a simple band calculation approach along with some approaches

from Refs. [97, 98], we identify all possible non-Hermitian FB Hamiltonians in the

above-mentioned three categories.

6.2 Non-Hermitian Hamiltonian

We consider a one-dimensional (d = 1) translational invariant non-Hermitian lattice

with ν > 1 sites per unit cell. As we are extending the work in Chapter 4 ([97]) to

the non-Hermitian regime, we adopt the same definitions and conventions. However,

in this case H0 is no longer Hermitian and hoppings to the right and left directions

6.3 The generator 83

might be different. Therefore, we define right and left hopping matrices Hr, Hl, and

Hr 6= H†l . Then the eigenvalue problem reads

H0~ψn +Hr

~ψn−1 +Hl~ψn+1 = E~ψn, n ∈ Z. (6.1)

Discrete translational invariance allows us to use the Floquet–Bloch theorem to write

the eigenvector of this equation as ~ψn =∑

k e−iknuk, where the lattice constant is

taken to be 1. Using the Bloch eigenfunction we can then write Eq. (6.1) as

H(k)uk = Ekuk, (6.2)

where the k-space Hamiltonian is

H(k) = H0 +Hle−ik +Hre

ik. (6.3)

The FB generator for the Hermitian version of the Hamiltonian in Eq. (6.3), where

Hl = H†r , was introduced in Chapter 4 ([97]). In this work, we identify the H(k) that

gives a FB.

6.3 The generator

There are two possible ways to construct non-Hermitian FB Hamiltonians. One way

is the direct extension of the FB generator in Chapter 4 ([97]), which is based on

compact localized states (CLSs), to the non-Hermitian regime (see Appendix D.1).

However the CLS based generator only gives completely flat FBs. Here though we

are mainly concerned with the second approach: band calculation, which can generate

non-Hermitian FBs of all three above mentioned categories.

In this method, we require some eigenvalues ofH(k) in Eq. (6.3) to be completely

or partially flat, and identify the Hamiltonians, i.e. the matrices H0, Hr, Hl, that

satisfy such requirement.

We take the simplest case with ν = 2 sites per unit cell. In this case, hoppings

within the unit cell are described by intracell hopping matrix H0 (which can be trans-

formed to more generic form by unitary transformation similar to the one described in

Section 4.2):

H0(ν, µ) =

(0 ν

0 µ

), (6.4)

where µ = 0, 1, ν = 0, 1, and they cannot be 1 at the same time. Therefore, H0 has


three different forms:

H0 =

(0 0

0 0

), degenerate,

H0 =

(0 0

0 1

)non− degenerate,

H0 =

(0 1

0 0

), abnormal.

(6.5)

The intercell hoppings are described by left and right hopping matrices

Hl =

(f g

h l

), Hr =

(a b

c d

). (6.6)

Notice that the form of H0 is completely fixed by the symmetries in this case. Then,

the eigenvalues xk, yk of the k-space Hamiltonian in Eq. (6.3) satisfy

xk + yk = µ+ eik(a+ d) + e−ik(f + l)

xkyk = e2ik detHr + e−2ik detHl

+ (νf − µh)eik + (νa− µc)e−ik

+ df − cg − bh+ al.

(6.7)

By solving Eq. (6.7) under the condition that at least one of xk, yk has to be fully or

partially (real, imaginary parts, or modulus) k independent, we can get the hopping

matrices Hr, Hl having FBs.

6.4 Results

We present the results for each category of non-Hermitian FB: completely flat, partially

flat, and modulus flat.

6.4.1 Completely flat case

Flatbands that are completely flat have k independent real and imaginary parts. This

leads to two different cases: one in which both bands are completely flat, and one in

which only one band is completely flat.

6.4 Results 85

Both bands are completely flat

In this case, assuming both xk = x, yk = y are k independent in Eq. (6.7) and

requiring all k-dependent terms to vanish, we get

a+ d = 0,

f + l = 0,

det Hr = ad− bc = 0,

det Hl = fl − hg = 0,

νf − µh = 0,

νa− µc = 0,

xy = df − cg − bh+ al,

x+ y = µ.

(6.8)

From these it follows d = −a, l = −f , and either f = a = 0, or h = c = 0, or none

(for details see Appendix D.2.1).

Solving Eq. (6.8) for different forms of H0 in Eq. (6.5), we can get Hl, Hr that

gives two FBs at FB energies EFB = ±x. Here, we only show the solution for the

degenerate µ = ν = 0 case (see Appendix D.2.1 for other cases):

Hr =

(a b

−a2

b−a

),

Hl =

(f g

−f2

g−f

),

x2 = 2af − a2g

b− bf 2

g,

(6.9)

which has two FBs at EFB = ±x. Note that the parameters in Eq. (6.9) can be

complex.

Interestingly, the solution to Eq. (6.9) can give real FBs in the absence of Hermitic-

ity and PT symmetry. For example, suppose a, b, f ∈ R, g = −b in Eq. (6.9), then

the Hamiltonian in Eq. (6.3) becomes

H(k) =

e−ik(e2ika+ f

)be−ik

(−1 + e2ik

)e−ik(f2−a2e2ik)

b−e−ik

(e2ika+ f

),

(6.10)

which gives two real FBs at EFB = ±(a + f); however, the Hamiltonian in this

equation is neither Hermitian nor PT symmetric.


One band is completely flat

Requiring either x or y to be k independent in Eq. (6.7) yields detHr = 0, detHl = 0.

Therefore, with one band that is completely flat, we can parameterize Hr, Hl as

Hr =

(a b

c bca

), Hr =

(f g

h ghf

). (6.11)

Putting d = bca

into Eq. (6.7) and requiring xk = x to be k independent gives the

following solution (see Appendix D.2.2):

b =

(−x±

√x2 − 4af

)(fν + gx) + 2afg

2f 2,

c =(x− µ)

((x2 ± x

√x2 − 4af

)− 2af

)2(fν + gx)

,

h =f 2(µ− x)

fν + gx.

(6.12)

This solution gives the following band structure:

EFB = x,

Ek =2(fν + gµ)

(−(x− µ) +

(aeik + fe−ik

))2(fν + gx)

,

∓eikν(x− µ)

(√x2 − 4af − x

)2(fν + gx)

.

(6.13)

Putting corresponding values of µ, ν into Eqs. (6.12) and (6.13), we can get solu-

tions for the degenerate, non-degenerate, and abnormal cases with corresponding band

structures. From Eq. (6.12), we can see that the parameters a, f, g can be complex.

Using CLS method we can get the same solution as Eq. (6.12), which gives aU = 3

CLS. This is interesting compared to the Hermitian case, in which the maximum CLS

size is U = 2 (see Chapter 4 or Ref. [97]). It can be shown that, when f = −a − x,

the solution in Eq. (6.13) and the CLS reduce to U = 2 (see Appendices D.1.2 and

D.1.3 for details). It can also be shown that the CLS size of U = 3 is the maximum

attainable.

We can alternatively use the inverse eigenvalue method; in Appendix D.1.4 we

demonstrate this for the U = 2 case.

6.4 Results 87

6.4.2 Partially flat case

The case of FBs that are partially flat is a special case that only available in non-

Hermitian lattices. As previously mentioned, partially flat means that either the real

or the imaginary part of a band is k independent. The eigenstates in this case are not

compactly localized, and therefore the CLS approach does not apply. We stick to the

band calculation approach.

We assume that xk = x1 + ix2, y = y1 + iy2, where xi, yi ∈ R, i = 1, 2. By

separating the real and imaginary parts of Eq. (6.7), we have

x1 + y1 = µ+ (a+ d+ f + l) cos(k)

x2 + y2 = −(a+ d− f − l) sin(k)

x1y1 − x2y2 =al − bh− cg + df

+ (aµ− cν + fµ− hν) cos(k)

+ (detHl + detHr) cos(2k)

x2y1 + x1y2 = (−aµ+ cν + fµ− hν) sin(k)

+ (detHl − detHr) sin(2k).

(6.14)

Requiring some of x1, x2, y1, y2 to be k independent and solving Eq. (6.14), we can

get Hr, Hr that gives partially flat FBs. There are many possible cases for such FBs,

namely where the real (imaginary) parts of both bands are flat, the real (imaginary) part

of one band is flat, and the real (imaginary) part of one band and the imaginary (real)

part of the other band is flat. Here we show the results for the case in which the real

parts of both bands are flat; details of this case and other cases are given in Appendix

D.3.

In this example, x1, y1 in Eq. (6.14) are k independent, and we assume an abnormal

H0, i.e. ν = 1, µ = 0. Then Eq. (6.14) gives

H0 =

(0 1

0 0

),

Hl =

(a (a−d)2

4h− h

4x21

−h d

),

Hr =

(−a− x1

h2−x21(a−d+2x1)2

4hx21

h x1 − d

) (6.15)


which yields the following band structure

xk = i(a+ d) sin(k)− ih sin(k)

x1

+ x1,

yk = i(a+ d) sin(k) +ih sin(k)

x1

− x1 .

(6.16)

Using the same method, we can solve the case in which the imaginary parts of both

bands are flat.

6.4.3 Modulus-flat case

In addition to the two previous cases, the third category of non-Hermitian FBs is the

special flat-modulus case, where the band has a k-dependent phase with a non-trivial

winding number. In this case, we assume that x = reiθk , r ∈ R in Eq. (6.7). If

we assume the phase is an integer multiple of k, i.e. θk = mk, then m can take two

possible values m = ±1. For each value of m, we solve the Eq. (6.7) to get the

hopping matrices Hr, Hl that gives a FB at EFB = reiθk , r ∈ R. As an example,

we present the results for m = 1 and abnormal H0, i.e. µ = 0, ν = 1 (see details in

Appendix D.3.5).

For m = 1 and abnormal H0 we have

Hr =

(r b

0 d

),

Hl =

(0 g

0 l

),

(6.17)

which gives the following band structure

E1 = eikr,

E2 = deik + e−ikl + 1.(6.18)

Using a similar method, other cases of H0 and the m = −1 case can be solved. No-

tably, the FB at E1 in Eq. (6.18) has a non-trivial winding number.

6.5 Summary

We considered a 1D translational invariant two-band network with non-Hermitian hop-

pings, and introduced a systematic classification of non-Hermitian FB Hamiltonians.

6.5 Summary 89

This classification provides a systematic way to construct non-Hermitian FB lattices

with two bands. The non-Hermitian FB in this case were constructed by fine-tuning the

hoppings, and the construction does not require PT or chiral symmetry. Completely

flat FBs host CLSs, on which the classification of the Hamiltonian is based. We found

that the maximum class of CLSs for the non-Hermitian two-band network is U = 3,

which differs from the Hermitian case where the maximum is U = 2. For partially flat

FBs, the CLSs are not necessary, so we used band calculation methods to identify the

partially flat FB Hamiltonians. Interestingly, for the flat-modulus case, we found that

the band has a non-trivial winding number.


Chapter 7

Flatband in a microwave photoniccrystal

Previous chapters introduced our flatband (FB) generators in 1D, 2D, and non-Hermitian

systems. As an application, in this chapter we present a tight-binding model for a

microwave photonic crystal. The spectrum of the photonic crystal consists of Dirac

points at two different energies; while conventional tight-binding models can explain

the spectrum around the lower Dirac points, they cannot explain the spectrum around

the upper ones. Our tight-binding model, comprising honeycomb and kagome sublat-

tices, fits the density of states (DOS) of the microwave photonic crystal nicely, includ-

ing the spectrum around the upper Dirac points. The methods and ideas behind our

FB generator, such as matrix representation, play an important role in setting up this

model and simplifying the calculations.

7.1 Photonic crystals and Dirac billiards

The peculiar band structure of graphene and the linear dispersion relation around the

touch points of its conduction and valence bands originate from the symmetry prop-

erties of its honeycomb structure [14], which is formed by two interpenetrating tri-

angular lattices with threefold symmetry. Consequently, photons (bosons) or waves

propagating in a spatially periodic potential with a honeycomb structure may com-

prise energy spectra regions where they are effectively described by the Dirac equation

for spin-1/2 fermions. Actually, there exist numerous realizations [128] of artificial

graphene using 2D electron gases exposed to a honeycomb potential lattice [113, 158],

molecular assemblies arranged on a copper surface [61], ultracold atoms in optical lat-

tices [172, 181], and photonic crystals [15, 16, 22, 23, 75, 81, 89, 123, 136, 137, 147].

91

92 Flatband in a microwave photonic crystal

It was shown in [132] that under certain conditions, photonic crystals with a tri-

angular lattice geometry exhibit a linear Dirac dispersion relation. In Ref. [39] a

microwave Dirac billiard was constructed from a photonic crystal, consisting of a

rectangular basin containing about 900 metallic cylinders (as illustrated in Fig. 7.1)

arranged into a triangular lattice with a top plate in order to obtain a microwave res-

onator [38, 39, 40]. Below a certain frequency of microwaves sent into the resonator,

which is inversely proportional to its height, the electric field is perpendicular to the top

and bottom plates, that is, the associated Helmholtz equation is two-dimensional and

mathematically equivalent to the Schrödinger equation of the corresponding quantum

billiard [63, 162, 163]. Due to the periodic lattice structure formed by the cylinders,

Figure 7.1: Photograph of the basin plate of a microwave Dirac billiard containing888 metal cylinders. It was constructed from brass and coated with lead to achievesuperconductivity at liquid helium temperature. The red crosses mark the positions ofthe antennas. The inset shows a magnification of the lattice structure from a long side.Taken from [39].

the frequencies of wave propagation as a function of the two quasimomentum com-

ponents exhibit a band structure similar to that of graphene. Indeed, the resonance

spectra of microwave Dirac billiards exhibit Dirac points where they are governed by

the relativistic Dirac equation [22]. The honeycomb structure arises from the voids

formed by three neighboring cylinders yielding a triangular cell [55], which can be

considered as an open resonator. There, the electric field intensity of the propagating

modes is localized, as marked by red and blue circles in Fig. 7.2. Thus, the cells host

quasibound states which, depending on the eigenfrequency, might partially overlap

with neighboring ones. The size of the electric field intensity at the voids provides the

on-site excitations. The void structure terminates with a zigzag edge along the longer

edges of the rectangle and with an armchair edge along the shorter ones.

Figure 7.3 shows a comparison of the DOS, ρ(f) = π2

N

∑n δ(f − fn) with N de-

noting the number of sites of the honeycomb lattice, i.e., voids formed by the metal

7.2 Tight-binding model for a microwave photonic crystal 93

Figure 7.2: Schematic view of the triangular lattice structure of the billiard. The redand blue circles mark the voids between the cylinders, corresponding to the atoms ingraphene arranged on the two independent triangular lattices that form the hexagonallattice. It is terminated by armchair and zigzag edges along the short and long sides,respectively, and translationally invariant with respect to all sides. Taken from Ref.[39].

cylinders and fn the eigenfrequencies (left panel), with the calculated band-structure

function f(~q) (right panel) along the path ΓMKΓ inside the first Brillouin zone (inset).

Here, K denotes the positions of the Dirac points located at the corners of the Brillouin

zone, Γ which is the center of the BZ where the band terminates, and M the saddle

points [28]. The positions of the experimental band gaps and Dirac points agree well

with those in the calculated band structure. The latter are bordered by sharp peaks, the

van Hove singularities [182], which correspond to the saddle points in the band struc-

ture. Generally, ρ(f) exhibits maxima at frequencies where the band barely changes

with the quasimomentum vector ~q, i.e., regions of low group velocity, |~∇f(~q)| ' 0.

The narrow region of exceptionally high resonance density at the upper edge of the

first band gap is associated with a FB separating the two Dirac points, with the bands

framing them.

7.2 Tight-binding model for a microwave photonic crys-

tal

In Ref. [39], a tight-binding model (TBM) was used to describe the experimental DOS.

In distinction to the first TBM description of graphene [186], where only nearest- and

second-nearest neighbor interactions of the pz orbitals were considered, also third-

nearest-neighbour couplings and overlaps between the wavefunctions centered at the

associated atoms had to be included in order to attain good agreement between the

experimental and computed DOS. The latter takes into account the partial overlap of


Quasimomentum q

Dirac point 1

Dirac point 2

band gap

'

300500 100

20

25

30

35

40

45

50

Figure 7.3: Comparison of the experimental DOS (left panel) with the computed bandstructure of an infinitely extended photonic crystal with the same lattice constant (rightpanel). The frequencies of the Dirac points, the band gaps, the van Hove singularities,and the other peaks of ρ(f) agree well with the computed ones as well as the locationsof the saddle points and the FB. Taken from [39].

the quasibound states, i.e., of the electric field mode components localized at the voids

formed by the metal cylinders with neighboring ones [138]. This indicates that the

quasibound states localized at the voids extend into the regions of neighboring ones.

Accordingly, the band-structure function f(~q) was obtained by solving the following

generalized eigenvalue problem

HTB|Ψ~q(~r)〉 = f(~q)SWO|Ψ~q(~r), 〉 (7.1)

with the TBM Hamiltonian

HTB =

(γ0 + γ2h2(~q) γ1h1(~q) + γ3h3(~q)

γ1h1(~q) + γ3h3(~q) h0 + γ2h2(~q)

)(7.2)

and the wavefunction overlap matrix

SWO =

(1 + s2h2(~q) s1h1(~q) + s3h3(~q)

s1h1(~q) + s3h3(~q) 1 + s2h2(~q)

). , (7.3)

7.3 The honome lattice 95

These incorporate the nearest-neighbor coupling γ1 with the second- and third-nearest

neighbor couplings γ2 and γ3, and the corresponding overlap parameters s1, s2, and s3.

The functions hn(~q), n = 1, 2, 3 associated with the different couplings were obtained

from Ref. [138].

The parameters were determined by fitting the DOS deduced from the band-structure

function f(~q) to the experimental one. In order to achieve a good agreement of the

DOSs deduced from the TBM and the experimental eigenfrequencies, respectively, the

nearest neighbor, second and third nearest neighbor couplings between the quasibound

states were taken into account, i.e., the electric field mode components localized at the

voids between the metallic cylinders and also the corresponding overlaps. The spec-

trum around the lower Dirac point was well described by the TBM. However, for the

upper Dirac point, the TBM reproduced only the positions of the peaks exhibited by

the DOS, the band edges, and the Dirac point, whereas the overall shape exhibited

clear deviations. That is, the TBM does not seem to be suitable to describe the band

structure including both Dirac points.

The fact that the TBM yielded a good description only after including wavefunc-

tion overlaps, along with the occurrence of a FB together with two Dirac points and the

bands framing them, led us to the idea that the microwave Dirac billiard may actually

provide for the experimental realization of a honeycomb-kagome lattice. In this pic-

ture, the non-vanishing electric field intensity between two neighboring metal cylinders

corresponds to atoms of the kagome lattice. The fact that the lower Dirac point and the

bands framing it are well described by the tight-binding model for a finite-sized hon-

eycomb lattice implies that the coupling between the honeycomb and kagome lattice

is weak.

7.3 The honome lattice

The DOS of the tight-binding model in the previous section fits well with the exper-

imental spectrum around the lower Dirac point. However, the spectrum around the

upper Dirac point, including the FB, was not explained theoretically. We want to pro-

pose a tight-binding model that fits the experimental spectrum around both Dirac points

and the FB as well.

By observing the nodes of the wave functions obtained from the Helmholtz equa-

tion, we came up with a tight-binding model with two sublattices, honeycomb and

kagome. We dubbed this new lattice as honome lattice, see Fig. 7.4.


Figure 7.4: The honome lattice consists of two sublattices: honeycomb (red) andkagome (blue). (a) Structure of the honome lattice. The shaded area shows the unitcell. All hoppings within nearest neighboring unit cells are shown. The hopping in-dices are ordered according to increasing distance, with each representing all hop-pings of the same distance. (b) Honome lattice up to third nearest neighbor hoppings.(c) Honome lattice showing 4th to 6th nearest neighbor hoppings.

We write the wave function of a unit cell as

~ψm,n = (am,n, bm,n, cm,n, dm,n, em,n)T , (7.4)

where am,n, bm,n are wave functions of honeycomb sublattice sites, cm,n, dm,n, em,nare wave functions of kagome sublattice sites, and m,n are unit cell indices giving the

position of the unit cell with respect to the origin, i.e. the lattice vector of the (m,n)

unit cell is ~Rm,n = m~e1 + n~e2.

Within nearest neighboring unit cells, the maximum hopping range is 15th neigh-boring hopping in conventional terms. Then we can express the hoppings in matrix

7.3 The honome lattice 97

form:

H0 =

ε t3 t1 t4 t4

t3 ε t1 t1 t1

t1 t1 0 t2 t2

t4 t1 t2 0 t2

t4 t1 t2 t2 0

, H1 =

t6 t8 t7 t7 t11

t8 t6 t4 t4 t9

t7 t7 t6 t5 t10

t11 t9 t10 t6 t12

t7 t4 t5 t2 t6

H2 =

t6 t3 t4 t4 t1

t13 t6 t9 t7 t4

t9 t4 t6 t5 t2

t14 t7 t10 t6 t5

t15 t9 t12 t10 t6

, H3 =

t6 t13 t9 t15 t14

t3 t6 t4 t9 t7

t4 t9 t6 t12 t10

t1 t4 t2 t6 t5

t4 t7 t5 t10 t6

,

(7.5)

where tj is the jth neighbouring hopping. H0 is the intra unit cell hopping matrix, Hi

describes hoppings between nearest neighboring unit cells along ~ei, where ~e1, ~e2 are

primitive lattice translation vectors, and ~e3 = −~e1 − ~e2. Then the eigenvalue problem

for this lattice is

H0~ψm,n +H1

~ψm+1,n +H†1~ψm−1,n +H2

~ψm,n+1

+H†2~ψm,n−1 +H3

~ψm+1,n+1 +H†3~ψm−1,n−1

= E~ψm,n.

(7.6)

According to Bloch theorem, one can write ~ψm,n = ~φ~ke−i~k·~Rm,n , and by putting

this into Eq. (7.6) we get the k-space Hamiltonian,

H(k) = H0 +3∑j=1

(Hje

−ikj +H†j eikj), (7.7)

where ki = ~k · ~ei, ~k = (kx, ky) , ~e1 = (1, 0) , ~e2 =(−1

2,√

32

), and ~e3 =

(−1

2,−√

32

).

Eigenvalues of H(k) give the band structure.

For given hoppings, here we consider a maximum 5th neighboring hoppings in

conventional terms. The DOS can be computed in two different ways. One is to

compute the eigenvalues of the direct (real) space Hamiltonian, and the other one is

to take reciprocal lattice sites corresponding to the direct lattice sites and, using band

structure, calculate the energy corresponding to each reciprocal lattice site.

In order to achieve the best fit with experimental results, we have to find the best

parameters, i.e. hopping values and onsite energies. In the next section we introduce

an algorithm to find these optimal parameters.


7.4 Methods

The experimental data contains 4000 resonant frequencies, and fitting the DOS of the

experiment and the honome for all these 4000 points would be very resource consum-

ing and inefficient. Therefore, we only fit some special points; one option is to fit the

relative positions of the van Hove singularities (peaks and Dirac points of the DOS)

only.

7.4.1 Positioning van Hove singularities

The position of a van Hove singularity is its horizontal coordinate, which is resonant

frequency (energy), see Fig. 7.5. Then the relative position of van Hove singularity p

is defined as

p =ωvh − ω0

w, (7.8)

where ωvh is the resonant frequency (energy) corresponding to this van Hove singular-

ity, ω0 is the lower band edge of the lowest band, and w is the width of the spectrum.

Figure 7.5 shows the relative positions of all van Hove singularities.

We extract from the experimental data the relative positions of ith peaks pei and

relative positions of ith Dirac point dei of the experiment. For example, the band edge

of the experiment is given in Ref. [39], which is 19.64. We can extract the position of

the 5th peak, which is pe5 = 43.421. Then the width of the experimental spectrum is

wexp = 43.421− 19.64 = 23.781. (7.9)

Similarly, we extract the positions (resonant frequencies) for all van Hove singularities

and calculate the relative positions using Eq. (7.8); see Table 7.1.

VHS pe1 pe2 pe3 FBe de1 de2RP 0.098524 0.218872 0.809259 0.573672 0.15586 0.902927

Table 7.1: Relative positions (RP) of van Hove singularities (VHS) extracted fromexperimental data.

When we calculate the relative positions of the van Hove singularities of the hon-

ome lattice, we assume that the peaks of the energy (frequency) spectrum are located at

theM point, the Dirac points are located at theK point, and the band edges are located

at the Γ point. This allows us to obtain the positions of the van Hove singularities by

calculating the eigenvalues of H(k) at these M, K, Γ points. Their positions in first

Brillouin zone are shown in Fig. 7.5. In some cases, we have analytical solutions, e.g.

7.4 Methods 99

in the case of 3rd neighboring hopping range, while in other cases we have no analytic

solution and thus calculate the van Hove singularities numerically. The relative posi-

tions are then calculated according to Eq. (7.8), where we take the difference between

the 5th peak and the lowest band edge as the width of the spectrum w

w = Max (m1,m2,m3,m4,m5)− γ, (7.10)

where γ is the lowest band edge, which is the lowest eigenvalue of H(k) at the Γ

point, i.e. at kx = ky = 0. The reason for choosing the spectral width in this way is

that the highest band, which is beyond the Helmholtz regime, in experiment cannot be

described by the honome model.

7.4.2 Deviations from experiment

For a given set of parameters (hopping values and onsite energy ε), we can calculate

the position of the van Hove singularities.

With these positions, we can find the difference between the calculated values and

the experimental results by∆di = di − dei ,

∆pi = pi − pei ,

∆fb = Efb − Eefb ,

(7.11)

where di, dei are the ith Dirac point from theory and experiment, respectively, pi, pei are

the ith peak from theory and experiment, respectively, and Efb, Eefb are the FB from

theory and experiment, respectively.

Afterwards, we look for the parameters (hoppings, onsite energies) which minimize

∆di, ∆pi, and δfb. The next section details this process.

Alternatively, we can also calculate the standard deviations

∆2 =∑i

(∆di)2 +

∑j

(∆pj)2 , (7.12)

and try to minimize them.

7.4.3 The algorithm

After computing the relative positions of all van Hove singularities, we look for the

parameters which minimize the differences (and standard deviations) between the rel-

ative positions of calculated and experimental van Hove singularities. To do so, we use


Figure 7.5: Relative positions of peaks and Dirac points.

two different algorithms: plain Monte Carlo and importance sampling. Note that both

are Monte Carlo methods.

Plain Monte Carlo algorithm: The plain Monte Carlo algorithm operates as fol-

lows:

1. Choose a hopping range, i.e. the number of hoppings.

2. Start from an initial hopping value, and calculate the eigenvalues of H(k) at

M, K, Γ points. From these values, calculate the relative positions of the peaks

and Dirac points, and their deviations from experimental results.

3. Randomly generate hopping values.

4. Repeat the calculations in step 2 for the randomly generated hoppings.

5. Compare the calculation results from steps 2 and 4.

6. If the new set of hoppings decreases the deviations, accept the new set. If the

new set of hoppings increases or does not change the deviations, reject the new

set (i.e. keep the previous set of hoppings).

7. Repeat steps 3 to 6 until the pre-determined number of cycles is complete.

7.5 Results 101

Importance sampling: In this method, instead of randomly generating all hoppings

in each cycle, we modify one of the hoppings by a small random step. The algorithm

operates as follows:

1. Choose a hopping range, i.e. the number of hoppings.

2. Start from an initial hopping value; one option is to initialize the hoppings with

the values obtained from the plain Monte Carlo method. Then calculate the

eigenvalues ofH(k) atM, K, Γ points. From these values, calculate the relative

positions of the peaks and Dirac points, and their deviations from experimental

results.

3. Randomly choose a hopping and add a small random number (random shift) to

it.

4. Repeat the calculations in step 2 for the modified hoppings.

5. Compare the calculation results from steps 2 and 5.

6. If the random step decreases the deviations, accept the change. If the random

step increases or does not change the deviations, reject the change (i.e. keep the

previous set of hoppings).

7. Repeat steps 3 to 5 until the pre-determined number of cycles is complete.

We start by considering the hoppings t1, t2, t3 and onsite energy ε, and then add

more hoppings to improve the fitting. Note that we can always perform a rescaling

such that one of the hoppings is 1; therefore, in our later calculations, we set t1 = 1.

7.5 Results

We consider two different hoppings ranges: up to 3rd and up to 5th nearest neighbor

hopping. The results show good agreement with the experiment. The parameters giv-

ing the best fitting results are not unique, which reflects the fine-tuned nature of the

system. Note that as the parameters we obtained gives a spectrum that is rescaled and

shifted from the experimental spectrum, we rescaled and shifted this spectrum to fit the

experimental spectrum.


7.5.1 Fitting with three hoppings

In the case of up to 3rd nearest neighbor hoppings, only t1, t2, t3 are non-zero in

Eqs. (7.5), and (7.7), which gives an analytic solution for the eigenvalues of H(k),

at M, K, Γ points.

Then using Eq. (7.8), we can find the relative positions of the peaks and Dirac

points. We minimize the deviations in Eq. (7.11) using the plain Monte Carlo algo-

rithm, as introduced in Section 7.4.3. We obtained a perfect FB in this case and good

fitting with the experiment, as shown in Fig. 7.6.

Figure 7.6: Fitting with three hoppings and minimizing the deviations by plain theMonte Carlo method. (Left) Density of states, where the red line is the spectrumof the honome lattice and the blue line is the experimental spectrum. (Right) Bandstructure of the honome lattice along the path Γ,M,K,Γ inside the first Brillouinzone (see Fig. 7.3). The values of the parameters are t1 = 1 and (ε, t2, t3) =(−4.79586, 0.803977,−0.431564).

Using the parameters obtained from the plain Monte Carlo method, we then tried

the importance sampling method in order to further minimize the deviations in (7.11).

Results are shown in Fig. 7.7. We also tried to minimize the standard deviation in

(7.12), which gives similar results.

7.5.2 Fitting with five hoppings

In the case of up to 5th nearest neighbor hoppings, only ti, i = 1, . . . , 5 are non-zero

in Eqs. (7.5) and (7.7), and one can compute analytically the eigenvalues of H(k) at

Γ, M points, but not at the K point, where the eigenvalues are computed numerically.

Then using Eqs. (7.8) and (7.10), we minimize the deviations in Eq. (7.11) via the

plain Monte Carlo method to get the hopping parameters that best fit the experimental

data. The result shows a little broadening of the FB, which is closer to the experimental

7.5 Results 103

Figure 7.7: Fitting with three hoppings and minimizing the deviations by the im-portance sampling method. (Left) Density of states, where the red line is the spec-trum of the honome lattice and the blue line is the experimental spectrum. (Right)Band structure of the honome lattice along the path Γ,M,K,Γ inside the first Bril-louin zone (see Fig. 7.3). The values of the parameters are t1 = 1 and (ε, t2, t3) =(−4.88694, 0.812533,−0.43105).

Figure 7.8: Fitting with five hoppings and minimizing the deviations by the plainMonte Carlo method. (Left) Density of states, where the red line is the spectrumof the honome lattice and the blue line is the experimental spectrum. (Right) Bandstructure of the honome lattice along the path Γ,M,K,Γ inside the first Brillouinzone (see Fig. 7.3). The values of the parameters are t1 = 1 and (ε, t2, t3, t4, t5) =(−4.98748, 0.847096,−0.486464,−0.0425766, 0.0225378).

result, as shown in Fig. 7.8. Minimizing the standard deviation in Eq. (7.12) produces

a similar result.

Using the parameters obtained from the plain Monte Carlo method, we tried the

importance sampling method in order to further minimize the deviations in Eq. (7.11).

Results are shown in Fig, 7.9. We again tried to minimize the standard deviation in Eq.

(7.12), and we got a result similar to the one achieved from minimizing the deviations.

Both methods produce the fits that show good agreement with the experiment, as


Figure 7.9: Fitting with five hoppings and minimizing the deviations by the impor-tance sampling method. (Left) Density of states, where the red line is the spectrumof the honome lattice and the blue line is the experimental spectrum. (Right) Bandstructure of the honome lattice along the path Γ,M,K,Γ inside the first Brillouinzone (see Fig. 7.3). The values of the parameters are t1 = 1 and (ε, t2, t3, t4, t5) =(−5.02261, 0.852182,−0.486464,−0.04, 0.0256164).

can be seen in Figs. 7.6–7.9. In particular, the spectrum of our model, the honome

lattice, also displays the Dirac points at two different energies, in agreement with the

experiment. The lower Dirac point arises from the honeycomb sublattice, and the upper

Dirac point arises from the kagome sublattice. The singularity in the experimental

spectrum comes from the FB of the honome lattice. This singularity (i.e. the FB) has

a little broadening, as we show in the case of five hoppings, which is resulted from

the slight distortion of the FB by long-range hoppings, i.e. t4, t5 in our examples (see

Figs. 7.8, 7.9).

7.6 Summary

In this chapter, we introduced a tight-binding model for a 2D microwave photonic

crystal. Patterns of eigenfuctions obtained from numerical diagonalization indicates

bipartite structure of the nodes of the eigenfunctions. This suggested to search for

effective tight-binding description based on chiral lattices, that were discussed before,

and lead us to study the tight-binding model on the combination of the honeycomb and

kagome lattices, that was dubbed honome lattice. Then the honome lattice was used

to describe the spectral properties of the microwave photonic crystal. We developed

a numerical algorithm to fit an experimental spectrum with the density of states of

the honome lattice, and our fitting result shows good agreement with the experimental

observation. Particularly, the singularity in the experimental spectrum is well explained

7.6 Summary 105

by the singularity of the density of states of the honome lattice around the FB energy.

In this work, we used block matrix formalism to help simplify the fitting algorithm.

Moreover, we implemented the fine-tuning property of FB Hamiltonians to achieve

the best fit. This work proved that the concepts and methods used in our FB generator

are useful in practical applications.


Chapter 8

Conclusions and outlook

Tight-binding models with dispersionless energy bands in their single particle spec-

trum have proven their importance in various applications and diverse settings ranging

from the Hubbard model to Bose–Einstein condensates and photonics. Macroscopic

degeneracy of such systems allows physicists to tune flatbands (FBs) to various ex-

otic phases of matter, with the help of different perturbations. The FB models studied

thus far are limited to some specific classes that have special lattice geometry and

symmetry. However, there are a vast amount of FB Hamiltonians that have not been

uncovered so far. The absence of systematic classification and construction methods

for FB Hamiltonians led us to work on this thesis.

We introduced in this thesis a systematic classification of FB lattices and devel-

oped novel methods to generate FB Hamiltonians with specific compact localized state

(CLS) properties. Our ultimate aim was to identify all possible FB Hamiltonians of

different classes in 1D and 2D systems.

After reviewing the basic concepts of the tight-binding model, FBs, and macro-

scopic degeneracy, we reviewed existing FB construction methods, and applications

and realizations of FBs. Next we proposed a classification approach for FBs according

to the size and shape of their irreducible CLSs. Furthermore, we studied the properties

of CLSs, and with the help of block matrix representation, we formalized destructive

interference conditions and CLS existence conditions in a mathematically rigorous

way. Based on these conditions, we developed a test procedure to identify the CLS

class of a given FB Hamiltonian, and the inversion of this test procedure led us to the

core idea of a FB generator.

Having the essential tools and the idea of FB generation, we realized the com-

plete classification and systematic generation of FB Hamiltonians in 1D that cover all

possible Hamiltonians possessing at least one FB. Particularly, we have achieved the

107

108 Conclusions and outlook

complete parameterization of all FB networks in the two-band case. Remarkably, we

found that all two-band FB networks can be mapped into generalized sawtooth lattices

using local unitary transformations. Extension of the two-band case to higher num-

bers of bands in 1D was more challenging, for which we developed a new method that

we called the inverse eigenvalue method. This method yielded analytic solutions for

U = 2 FBs and numerical solutions for U > 2 cases. With this method, in principle,

all possible FBs in 1D can be generated.

Moving on to higher dimensions, we extended our approach to 2D. We started by

introducing a classification of 2D FB lattices whose CLSs occupy a maximum of four

unit cells in a 2 × 2 plaquette. Implementing the methods used for 1D, we obtained

analytic solutions for various CLS classes in the plaquette. We were able to cover most

known examples with our analytic solutions.

Extending the idea of our FB generator to the non-Hermitian regime, we consid-

ered a 1D two-band network with non-Hermitian couplings. Using k-space represen-

tation, we identified the conditions required to have a FB. Solving these conditions, we

acquired analytic solutions for non-Hermitian FB Hamiltonians. We were able to gen-

erate completely flat FBs, partially flat FBs, and the special case of modulus-flat FBs.

Unlike conventional methods, our generator does not rely on a particular symmetry,

such as PT symmetry.

Finally, turning to applications, we applied our methodologies to explain the spec-

tral properties of a microwave photonic crystal. We designed a novel tight-binding

model, which we term a honome lattice, to fit the experimental data obtained from

spectral measurements. In particular, we were able to explain the singularity in the ex-

perimental spectrum in terms of the singularity in the density of states of the honome

coming from the FB. We developed two algorithms that both fit the experimental data

with nice accuracy.

The topics explored in this thesis will impact FB-related research in a number of

aspects. First of all, our FB generator provides more flexibility in the experimental

design of FB lattices, because this method does not require special lattice geometry

or symmetry, and allows for the fine-tuning of parameters. By tuning the control pa-

rameters in our generator, it is possible to design FB modes that suit the experimental

conditions.

Moreover, our approaches are based on real-space analysis. The analytic solutions

provided here for FB Hamiltonians in 1D and 2D lattices will enable researchers to

further understand the properties of FB lattices. In particular, from our 2D analytic

solution, band touching properties can be studied more systematically using real-space

109

analysis instead of k-space analysis [139]. This will give more insight into the band

topology of 2D FB lattices from the real-space perspective.

In addition to these effects on ongoing research, our work will open new topics

for future studies. First, some of the conjectures introduced here require strict proof.

Should these conjectures be proved wrong, finding counter-examples will be interest-

ing and will lead to new questions. Along these lines, following from the theorems

proven here for the 1D case, rigorous proof for 2D and higher dimensions is desir-

able; to do so, extending the FB generator to more generic cases, like arbitrary CLS

sizes, can help to carry over these theorems to even higher dimensions. This is also

appealing given that in higher dimensions, more interesting phenomena are expected.

Further, our approach is based on real-space analysis, and extending the current ap-

proach to 3D will provide a theoretical platform for designing real materials with FBs

as well as searching for natural compounds having FBs.

Our FB generator in non-Hermitian systems can be extended to higher numbers

of bands in 1D and higher lattice dimensions. Interestingly, we have found a non-

trivial winding number of a modulus-flat FB. The consequence of such a non-trivial

winding number is an open question that demands investigation. Moreover, extending

the current approach is encouraging as there is plenty of room to connect our results in

non-Hermitian systems to more generic cases. In this way, there are myriad things to

be investigated such as system dynamics, topology, perturbation effects, and so on.

Recently, nearly FBs found in bilayer graphene systems have been drawing atten-

tion [30, 102, 192]. Such systems host a FB when two layers are tilted with respect to

each other and a particular tilting angle is reached. The origin of such a FB is not clear

and leads to a number of questions: What is the fate of CLSs? If CLSs exist in such

systems, what is the CLS class? What are the CLS properties? All of these questions

are open for further studies, and our methods in this work may provide a tool to answer

them.

This thesis focused on the single particle (non-interacting) systems. Exploring the

effects interactions is an interesting and promising topic. One example is superconduc-

tivity in FB, that has been drawing attention recently [80, 185, 195]. Our results rise

a number of interesting questions: How does the Bardeen-Cooper-Schrieffer (BCS)

interaction interplays with flatbands in general? Does it always lead to superconduc-

tivity? Is the superconductivity always enhanced? How does the enhancement depend

on the type of the flatband?

To sum up, our methods enhance our understanding of the properties of FB sys-

tems, provide a flexible tool to design FB lattices suiting experimental conditions, and

110 Conclusions and outlook

open new interesting topics for future research.

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Appendix A

Supplementary materials for CLSproperties

A.1 Linear dependence and reducibility of CLSs in 1D

(U=3 case)

Consider themc = 1, U = 3 case. Suppose ~ψ =(~ψ1, ~ψ2, ~ψ3

)is the compact localized

eigenstate, where ~ψi is a ν-component vector for the ith unit cell with ν sites. Now we

have the following eigenvalue equations with CLS conditions:

H0~ψ1 +H1

~ψ2 = EFB~ψ1,

H†1~ψ1 +H0

~ψ2 +H1~ψ3 = EFB

~ψ2,

H†1~ψ2 +H0

~ψ3 = EFB~ψ3,

H1~ψ1 = 0,

H†1~ψ3 = 0.

(A.1)

Suppose the components of the CLS ~ψ are linearly dependent as

~ψ2 = α~ψ1 + β ~ψ3, (A.2)

129

130 Supplementary materials for CLS properties

and then Eq. (A.1) becomes

H0~ψ1 + βH1

~ψ3 = EFB~ψ1,

H†1~ψ1 +H0

(α~ψ1 + β ~ψ3

)+H1

~ψ3 = EFB

(α~ψ1 + β ~ψ3

),

αH†1~ψ1 +H0

~ψ3 = EFB~ψ3,

H1~ψ1 = 0,

H†1~ψ3 = 0.

(A.3)

Plugging Eq. (A.2) into the second equation in (A.3) above and using the first and third

equations gives

H†1~ψ1 +H0

(α~ψ1 + β ~ψ3

)+H1

~ψ3 = EFB

(α~ψ1 + β ~ψ3

),

H†1~ψ1 + αH0

~ψ1 + βH0~ψ3 +H1

~ψ3 = αEFB~ψ1 + βEFB

~ψ3,

H†1~ψ1 + αH0

~ψ1 + βH0~ψ3 +H1

~ψ3 = αH0~ψ1 + αβH1

~ψ3 + βαH†1~ψ1 + βH0

~ψ3.(A.4)

We can see that when αβ = 1, the above equation is satisfied. Therefore, any linear

dependence of the CLS components cannot lead to a CLS of U=3, and we should have

an extra condition αβ = 1 or α = 1/β. Now we show that under this constraint (αβ = 1

or α = 1/β), the state ~ψ =(~ψ1, ~ψ2, ~ψ3

), with linear dependence as in (A.2), is actually

a U = 2 class.

Suppose we have a CLS ~ψ =(~ψ1, α~ψ1 + β ~ψ3, ~ψ3

), where α = 1/β, and then the

second equation in (A.3) is always satisfied and equation (A.3) can be rewritten as

αH0~ψ1 +H1

~ψ3 = αEFB~ψ1

αH†1~ψ1 +H0

~ψ3 = EFB~ψ3

H1~ψ1 = 0

H†1~ψ3 = 0,

or

H0~ψ1 + βH1

~ψ3 = EFB~ψ1

H†1~ψ1 + βH0

~ψ3 = βEFB~ψ3

H1~ψ1 = 0

H†1~ψ3 = 0.

(A.5)

From (A.5) we can see that(~ψ1, β ~ψ3

)and

(α~ψ1, ~ψ3

)can be a CLS of classU = 2.

Now if we write

~ψ =(~ψ1, α~ψ1 + β ~ψ3, ~ψ3

)=(~ψ1, β ~ψ3, 0

)+(

0, α~ψ1, ~ψ3

),

then we can see that ~ψ is actually a linear combination of two U = 2 CLSs.

This indicates that there is no U = 2 class for the ν = 2 case. Suppose we have a

U = 3 CLS in a ν = 2 lattice, giving H0, H1 as 2 × 2 matrices; therefore, being the

A.1 Linear dependence and reducibility of CLSs in 1D (U=3 case) 131

eigenvectors of the 2 × 2 matrix, the three components of the CLS should be linearly

dependent. According to the analysis above, it reduces to the U = 2 class.

132 Supplementary materials for CLS properties

Appendix B

Supplementary materials for theflatband generator in 1D

B.1 On the linear independence of CLSs

Linear dependence for mc = 1 and U = 2

Consider a CLS of U = 2 class ~ψ =(~ψ1, ~ψ2

)with mc = 1. Assume that the two

components ~ψ1, ~ψ2 are linearly dependent such that ~ψ1 = a~ψ2. Since(~ψ1, ~ψ2

)is a

CLS, it follows

H1~ψ1 = 0 ,

H†1~ψ2 = 0 .

(B.1)

This yields

aH1~ψ2 = 0 , H†1

~ψ2 = 0 or H1~ψ1 = 0 ,

1

aH†1

~ψ1 = 0 . (B.2)

Thus, ~ψ1, ~ψ2 are left and right eigenvectors of H1 at the same time, and therefore either~ψ1 or ~ψ2 serves as the only component of a CLS of class U = 1.

Interestingly, a similar (but more lengthy) proof can be obtained for the ν =

2, mc = 1, U = 3 case. Given a CLS ~ψ =(~ψ1, ~ψ2, ~ψ3

), it can be shown that

the linear dependence of {~ψ1, ~ψ2, ~ψ3} implies that the FB is of class U ≤ 2. Then

ν > 2 linear dependence is only a necessary but not a sufficient condition.

133

134 Supplementary materials for the flatband generator in 1D

Orthogonality for ν = 2 and U = 2

Consider U = 2 with ~ψ1 ⊥ ~ψ2. Then a suitable rotation of the basis in each unit

cell will result in ~ψ1 = (1, 0) and ~ψ2 = (0, 1). This seeming U = 2 case can be

reduced to U = 1 by redefining the unit cell. Indeed, after the above rotation we may

denote each site in the unit cell by al and bl. Then a CLS is given by al = δl,l0 and

bl = δl,l0+1 (up to prefactors and renormalization factors). Redefining the unit cell

using al = al and bl = bl+1 turns the above CLS into class U = 1. We can conclude

that for mc = 1, ν = 2, and U = 2, ~ψ1 and ~ψ2 must be neither parallel (linearly

dependent) nor orthogonal in order for the FB to not be reducible to U = 1.

B.2 Generator and band structure for two-band U = 2

FB networks

For the ν = 2, mc = 1, U = 2 case we solve the following equations:

H0~ψ1 +H1

~ψ2 = EFB~ψ1 , (B.3)

H†1~ψ1 +H0

~ψ2 = EFB~ψ2 , (B.4)

H1~ψ1 = 0 , (B.5)

H†1~ψ2 = 0 . (B.6)

We can always diagonalize H0 and gauge and rescale the full Hamiltonian to obtain

H0 =

(0 0

0 1

). (B.7)

For non-singular Λ = EFB −H0, we find ~ψ2 from (B.4) and insert it into (B.3) to get

Λ−1H1Λ−1H†1~ψ1 = ~ψ1 , (B.8)

Λ−1H†1~ψ1 = ~ψ2 , (B.9)

where Λ−1 = 1EFB−H0

. Similarly, we have

Λ−1H†1Λ−1H1~ψ2 = ~ψ2 , (B.10)

Λ−1H1~ψ2 = ~ψ1 . (B.11)

B.2 Generator and band structure for two-band U = 2 FB networks 135

B.2.1 Real H1

Consider all elements ofH1 to be real values. Equations (B.5) and (B.6) allow us to re-

define H1 in terms of unit vectors |ϕ〉 and |θ〉, which are the left and right eigenvectors

of the non-zero eigenvalue of H1, and which are orthogonal to ~ψ1 and ~ψ2:

H1 = α|θ〉〈ϕ| ,

|ϕ〉 =

(cosϕ

sinϕ

),

|θ〉 =

(cos θ

sin θ

),

(B.12)

where the scalar products are

〈~ψ1|ϕ〉 = 0 , (B.13)

〈~ψ2|θ〉 = 0 . (B.14)

Using these definitions and solving Eqs. (B.8) and (B.5), we obtain


cos(θ − ϕ), (B.15)

|α| =

√−tan(θ) tan(ϕ) csc2(θ − ϕ)

(tan(θ) tan(ϕ) + 1)2=


| sin(2(θ − ϕ))|. (B.16)

B.2.2 Complex H1

A complex H1 can be parameterized as

H1 = α|θ, δ〉〈ϕ, γ| = α


e−iδ sin θ cosϕ e−i(δ−γ) sin θ sinϕ

),

|ϕ, γ〉 =

(cosϕ

eiγ sinϕ

),

|θ, δ〉 =

(cos θ

eiδ sin θ

),

(B.17)

where α is a complex number.


Following the same procedure as for real H1 we obtain

EFB =eiδ cot(θ) cos(ϕ)

eiγ sin(ϕ) + eiδ cot(θ) cos(ϕ),

|α| = 2e2i(γ+δ) sin(2θ) sin(2ϕ)((eiγ − eiδ)2 (− cos(2(θ + ϕ))) + (eiγ + eiδ)2 cos(2(θ − ϕ))− 4ei(γ+δ)

)× 1

(eiγ cos(θ) cos(ϕ) + eiδ sin(θ) sin(ϕ))2 .

(B.18)

Because |α| is real, which imposes δ = γ, consequently


cos(θ − ϕ), (B.19)

|α| =


| sin(2(θ − ϕ))|. (B.20)

Solutions (B.15), (B.16), (B.19), and (B.20) are identical. Since |α| is real, solu-

tions only exist in the parameter regions 0 ≤ θ ≤ π2∩ π

2≤ ϕ ≤ π or π

2≤ θ ≤ π ∩ 0 ≤

ϕ ≤ π2.

In Bloch representation, the Hamiltonian reads

H(k) = H†1eik +H0 +H1e

−ik . (B.21)

With the above parameterization ( (B.17) and (B.7)), the band structure follows as

EFB =cos θ cosϕ

cos(θ − ϕ),

Ek =cos θ cosϕ

cos(θ − ϕ)+ 2|α| cos(θ − ϕ) cos(k + φα) ,

(B.22)

where φα is the phase of α = |α|eiφα .

B.2.3 Degenerate H0

The solutions α and EFB in (B.19) and (B.20) diverge for θ − ϕ = ±π2

and θ = φ,

and so does H1. We renormalize the Hamiltonian by multiplying it with 1α

. Then H0

vanishes, and H1 turns finite.

However, when θ − ϕ = ±π2, the dispersion bandwidth in (B.22) is finite, and

after normalization the dispersive band becomes flat as well. Therefore, we have two

coexisting FBs on the lines θ − ϕ = ±π2. According to (B.13) and (B.14), ~ψ1 ⊥ ~ψ2.

B.3 Generalized sawtooth chain 137

In such a case we can always perform a rotation in each unit cell such that ψ1 = (1, 0),

ψ2 = (0, 1). A subsequent redefinition of the unit cell turns the CLS into class U = 1

(see the above subsection on orthogonality for ν = 2 and U = 2).

When θ = ϕ, according to (B.13) and (B.14) ~ψ1 ‖ ~ψ2, and are therefore linearly

dependent, turning the FB into the U = 1 class. In this case

H1 = |θ, γ〉〈θ, γ| =

(2 cos2 θ 1

2eiγ sin(2θ)

12e−iγ sin(2θ) 2 cos2 θ

). (B.23)

The corresponding Bloch Hamiltonian in momentum space reads

H(k) = H†1eik +H1e

−ik = cos k

(2 cos2 θ eiγ sin(2θ)

e−iγ sin(2θ) 2 cos2 θ

), (B.24)

which yields one flat and one dispersive band

EFB = 0 , Ek = 2 cos k . (B.25)

B.2.4 FB energy equals one of the eigenvalues of H0: Reduction toU = 1

When FB energy EFB equals one of the eigenvalues 0, 1 of H0, we have to solve the

original equations (B.3–B.6). A simple calculation shows that the only remaining FB

solutions are again of class U = 1.

B.3 Generalized sawtooth chain

In our FB generator, we have considered the canonical form of H0, which was diago-

nal. We can perform unitary transformations (rotations) of the unit cell basis that will

modify H1 and make H0 non-diagonal, turning the whole model into a non-canonical

one. Using the rotation matrix

R (ω) =

(cosω − sinω

sinω cosω

), (B.26)


we define ψi = R(ω)~ψi, Hm = R(ω)HmR†(ω) where m = −1, 0, 1 and H−m = H†m,

and

H0 =

(0 0

0 1

), H1 = α


e−iγ sin θ cosϕ sin θ sinϕ

), (B.27)

with |α| =√− sin(2θ) sin(2ϕ)

| sin(2(θ−ϕ))| . We consider the case γ = δ = 0, and thus H0 and H1 read

H0 = R†(ω)H0R(ω) =

(sin2 ω cosω sinω

cosω sinω cos2 ω

),

H1 = α

(cos(θ + ω) cos(ϕ+ ω) cos(θ + ω) sin(ϕ+ ω)

cos(ϕ+ ω) sin(θ + ω) sin(θ + ω) sin(ϕ+ ω)

).

(B.28)

We can always find a value of ω that will zero one row or one column of H1. This

simplifies the non-canonical lattice into a generalized sawtooth chain. It has in general

three different hopping strengths and an onsite energy detuning between the two sites in

a unit cell. As an example, we consider the case when the first column of H1 vanishes:

ϕ+ ω = ±π2. (B.29)

It follows that

H0 =

(cos2(ϕ) − cos(ϕ) sin(ϕ)

− cos(ϕ) sin(ϕ) sin2(ϕ)

)=

(ε1 t1

t1 ε2

),

H1 = α

(0 − sin(θ − ϕ)

0 cos(θ − ϕ)

)=

(0 t2

0 t3

).

(B.30)

For the particular case of the ST1 chain (sawtooth chain with two equal hoppings and

zero onsite energy detuning, see Section 4.2 in the main text), where ε1 = ε2 and

t1 = t2, we find

ϕ =3π

4, θ = arctan

(3 + 2

√2),

ϕ =π

4, θ = π − arctan

(3− 2

√2).

(B.31)

This leads to the following tight-binding equations:

Ean = −√

2bn −√

2bn+1 , Ebn = −√

2an −√

2an−1 − bn+1 − bn−1 . (B.32)

The FB is located at EFB = 2 (Fig. 4.1 (b) in Section 4.2 in the main text). The CLS

has the form a0 = a1 = 1 and b1 = −√

2 (up to a normalization factor) with all other

B.3 Generalized sawtooth chain 139

amplitudes vanishing.

Detuning the angles θ, ϕ away from this point, we deform the ST1 model while

maintaining one FB.

Let us require t1 = t2 = t3. Then it follows

θ =π

2− 1

2tan−1

(1

2

), ϕ =

3π

4− 1

2tan−1

(1

2

),

θ =π

2− 1

2tan−1

(1

2

), ϕ =

3π

4− 1

2tan−1

(1

2

).

(B.33)

This is a novel, high-symmetry sawtooth chain (ST2 chain, see Section 4.2 in main

text). It can be obtained for example with the following matrices

H0 = −

(0 1

1 1

), H1 = −

(0 1

0 1

). (B.34)

This leads to the following tight-binding equations:

Ean = −bn − bn+1 , Ebn = −bn − an − an−1 − bn+1 − bn−1 . (B.35)

The FB is located at EFB = 1 (Fig. 4.1 (c) in Section 4.2 of the main text). The

CLS has the simple form a0 = a1 = −b1 = 1 (up to a normalization factor) with all

other amplitudes vanishing. Note that we can also set other columns and rows of H1

in (B.28) to zero and obtain all points in the θ, ϕ diagram corresponding to ST1 and

ST2 chains. All these points are shown by filled squares and filled circles in Fig. 4.3

in Section 4.2 in the main text.


B.4 Inverse eigenvalue problem: A toy example and

the solution of the U = 2 CLS

This appendix explains the solution of the inverse eigenvalue problems (4.27). As

discussed in the main text, 1D flatband lattices with CLS class U satisfy

H1~ψ2 = (EFB −H0) ~ψ1, (B.36)

H†1~ψl−1 +H1

~ψl+1 = (EFB −H0) ~ψl l = 2, . . . , U − 1, (B.37)

H†1~ψU−1 = (EFB −H0) ~ψU , (B.38)

H1~ψ1 = 0, (B.39)

H†1~ψU = 0. (B.40)

Assuming that EFB, H0, ~ψl=1,...,U are given, Eqs. (B.36)–(B.40) constitute an inverse

eigenvalue problem for a block-tridiagonal matrix, where diagonal blocks are H0 and

off-diagonal ones are H1.

B.4.1 Toy example

As a warmup, we solve a toy inverse eigenvalue problem: reconstruct ν × ν matrix T

given its action |y〉 on some vector |x〉:

T |x〉 = |y〉 . (B.41)

The solution is not unique, as a generic solution can be represented as T = T∗ + δT ,

where T∗ is any particular solution of (B.41) and δT |x〉 = 0. One possible solution is

easily found to be

T∗ =|y〉〈x|〈x|x〉

, δT = QxK, (B.42)

where Qx is a transverse projector on x. This construction is straightforward to gen-

eralize to the case of many vectors (we assume here implicitly that the equations are

consistent):

T |xk〉 = |yk〉 , k = 1..m. (B.43)

B.4 Inverse eigenvalue problem: A toy example and the solution of the U = 2CLS 141

The generic solution to this problem is given by

T∗ =∑ij

Aij |yi〉〈xj| , A−1ij = 〈xi|xj〉 , (B.44)

δT = QK, (B.45)

where Q is the orthogonal projector on the subspace spanned by {xk} and K is an

arbitrary ν × ν matrix. For later convenience we refer to T∗ as a particular solution

and δT as a free part.

B.4.2 U=2 case

In this case,Eqs. (B.36)–(B.40) read

H1 |ψ2〉 = (EFB −H0) |ψ1〉 ,

H†1 |ψ1〉 = (EFB −H0) |ψ2〉 ,

H1 |ψ1〉 = 0, (B.46)

H†1|ψ2〉 = 0.

We know H0, |ψ1〉 , |ψ2〉 and EFB = 〈ψ1|H0|ψ2〉, and we need to determine H1. As

discussed above for the toy case, a generic solution to this problem can be decomposed

into a particular solution and a free part. The last two equations in the above set are

satisfied by the following ansatz:

H1 = Q2MQ1, Qi = I− |ψi〉〈ψi|〈ψi|ψi〉

. (B.47)

Plugging this ansatz back into the system, we find

Q2M Q1 |ψ2〉 = (EFB −H0) |ψ1〉 , (B.48)

〈ψ1|Q2M Q1 = 〈ψ2| (EFB −H0) .

Note the identity

〈ψ1|H1|ψ2〉 = 〈ψ1|EFB −H0|ψ1〉 = 〈ψ2|EFB −H0|ψ2〉 , (B.49)


that follows straightforwardly from the first two equations of (B.46). Defining the

projectors

R12 = I− Q1 |ψ2〉〈ψ2|Q1

〈ψ2|Q1|ψ2〉, (B.50)

R21 = I− Q2 |ψ1〉〈ψ1|Q2

〈ψ1|Q2|ψ1〉,

we can write

M = T +R21K R12, (B.51)

where T is a particular solution of (B.48). The second term, whereK is an arbitrary ν×ν matrix, satisfies (B.48) by construction and is the free part of the solution. Therefore,

we only need to find a particular solution to the system to get the generic solution. This

is achieved by the same ansatz T = |x〉〈y| as in the toy case discussed above. The

ansatz yields the following equations:

Q2 T Q1 |ψ2〉 = 〈y|Q1|ψ2〉Q2 |x〉 = (EFB −H0) |ψ1〉 , (B.52)

〈ψ1|Q2 T Q1 = 〈ψ1|Q2|x〉〈y|Q1 = 〈ψ2| (EFB −H0). (B.53)

From these, vectors x and y are fixed (up to unimportant normalization):

〈y|Q1 =1

〈ψ1|Q2|x〉〈ψ2| (EFB −H0),

Q2 |x〉 =1

〈y|Q1|ψ2〉(EFB −H0) |ψ1〉 ,

=〈ψ1|Q2|x〉

〈ψ2|EFB −H0|ψ2〉(EFB −H0) |ψ1〉 ,

=〈ψ1|Q2|x〉

〈ψ1|EFB −H0|ψ1〉(EFB −H0) |ψ1〉 .

We used the condition (B.49) to replace the denominator in the fourth line. Also note

that the expression for y from the first line was used to simplify the second line (elim-

inate y). The particular solution is then

Q2TQ1 =(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ1|EFB −H0|ψ1〉. (B.54)

B.4 Inverse eigenvalue problem: A toy example and the solution of the U = 2CLS 143

Thanks to (B.49), it is symmetric with respect to |ψ1〉 , |ψ2〉. This and the above men-

tioned free part Q21KQ12 give the full family of solutions (4.30):

H1 =(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ1|EFB −H0|ψ1〉+Q2R21KR12Q1.

This expression is further simplified by noticing that R12Q1 and Q2R21 are the same

projector on the subspace spanned by |ψ1〉 , |ψ2〉 that we denote Q12: (R12Q1)2 =

R12Q1, idem for Q2R21, and both vanish when acting on |ψ1,2〉 as can be straightfor-

wardly verified. We can therefore replace these combinations by Q12:

H1 =(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ1|EFB −H0|ψ1〉+Q12KQ12. (B.55)

This solution is supplemented by the following non-linear constraints

〈ψ2|ψ1〉 = 1, (B.56)

〈ψ2|H0|ψ1〉 = EFB,

〈ψ1|EFB −H0|ψ1〉 = 〈ψ2|EFB −H0|ψ2〉 ,

that are obtained by eliminating H1 from (B.46) using destructive interference condi-

tions, i.e. the last two equations in (B.46).

In case the denominator in (B.55) is zero, the single-projector ansatz fails, and a

two-projector ansatz has to be used:

H1 =(EFB −H0) |ψ1〉〈ψ2|Q1

〈ψ2|Q1|ψ2〉+

+Q2 |ψ1〉〈ψ2| (EFB −H0)

〈ψ1|Q2|ψ1〉+Q12KQ12,

(B.57)

as can be verified by a direct substitution. In this special solution, the denominators

only vanish when Ψ1 ∝ Ψ2, i.e. in the U = 1 case.

B.4.3 Bipartite lattices and chiral symmetry

In this section, we solve the inverse eigenvalue problem for U = 2 for the special

case of bipartite lattices. We consider a bipartite lattice with ν sites per unit cell that

split into majority and minority sublattices with µ and ν − µ sites, respectively. Since

the lattice is bipartite, the sites on one sublattice only have neighbors belonging to the

other sublattice. This enforces the following structure on the hopping matrices and the


wave functions of the CLS (see Eq. (4.36)):

H0 =

(0 A†

A B

), H1 =

(0 T †

S W

),

~ψ1 =

(ϕ1

0

), ~ψ2 =

(ϕ2

0

). (B.58)

Here, ϕ1,2 are µ-component vectors describing the wave amplitudes of the majority

sublattice sites. A, S, T are (ν − µ) × µ matrices, while B,W are (ν − µ) × (ν − µ)

matrices. B,W formally break the bipartiteness of the lattice, but do not affect the

EFB = 0 FB(s). This special structure simplifies Eq. (B.46):

S |ϕ2〉 = −A |ϕ1〉 , (B.59)

T |ϕ1〉 = −A |ϕ2〉 , (B.60)

S |ϕ1〉 = 0, (B.61)

T |ϕ2〉 = 0. (B.62)

These equations need to be resolved with respect to S and T . The last two equations

are satisfied by the anzätse S = S ′Q1, T = T ′Q2, where Qi is a transverse projector

on ϕi. The remaining two equations are identical to the toy problem discussed above

(Appendix B.4.1), and their solution is precisely Eq. (4.41):

S = −A |ϕ1〉〈ϕ2|Q1

〈ϕ2|Q1|ϕ2〉+KSQ12, (B.63)

T = −A |ϕ2〉〈ϕ1|Q2

〈ϕ1|Q2|ϕ1〉+KTQ12,

(B.64)

where Q12 is a joint transverse projector on |ϕ1,2〉.

Now let’s count the number of free parameters. |ϕ1〉, |ϕ2〉 all are free parameters

with each containing µ free parameters. A contains (ν−µ)µ free variables. B,W each

contains (ν−µ)2 free parameters. KSQ12 and Q21KT are (ν−µ)×µ and µ× (ν−µ)

matrices, and, because of the transverse projectors, they contain (ν − µ)(µ − 2) and

µ(ν−µ− 2) free parameters, respectively. Therefore, (B.63) contains a total 2µ− 1 +

(ν−µ)µ+ (ν−µ)(µ− 2) +µ(ν−µ− 2) + 2(ν−µ)2 = (ν−µ)(2ν+µ− 2)− 1 free

parameters. The extra −1 corresponds to the overall normalization of the CLS that is

not fixed.

B.5 Resolving the non-linear constraints 145

B.5 Resolving the non-linear constraints

Let us discuss how one can efficiently resolve the set of non-linear constraints that

appear in the inverse eigenvalue problem, for example (B.56). Since these are a non-

linear system of equations, one can always try a numerical solver. However, our ex-

perience was not particularly successful: the solver was not converging, and found no

solution more often than not. Instead, it is possible to design a numerical algorithm

that eliminates the constraints one by one to either find and enumerate all the solutions,

or to prove that there are none.

B.5.1 U=2 case

The non-linear equations that we need to solve are:

〈ψ1|ψ2〉 = 1, (B.65)

〈ψ1|H0|ψ2〉 = EFB, (B.66)

〈ψ1|EFB −H0|ψ1〉 = 〈ψ2|EFB −H0|ψ2〉 . (B.67)

We assume that EFB, H0, and ψ1 (or ψ2) are given input parameters.

Then we need to solve the above equations for ψ2. The first two equations (B.65–

B.66) are linear and are easily satisfied with the following expansion for ψ2 by the

choice of the basis vectors e1 and e2:

|ψ2〉 =ν∑k=1

xk |ek〉 , (B.68)

|e1〉 =1√〈ψ1|ψ1〉

|ψ1〉 , (B.69)

|e2〉 =1√

〈ψ1|H0Q1H0|ψ1〉Q1H0 |ψ1〉 , (B.70)

〈el|em〉 = δlm, l,m = 1, 2, . . . ν. (B.71)

Here, Q1 is a transverse projector on |ψ1〉. With this choice of basis vectors, the equa-

tions (B.65–B.66) imply:

x1 =1√〈ψ1|ψ1〉

,

x2 =1√

〈ψ1|H0Q1H0|ψ1〉

[EFB −

〈ψ1|H0|ψ1〉〈ψ1|ψ1〉

].


The remaining basis vectors are fixed by requiring their orthonormality, for exam-

ple, by using Gram–Schmidt orthogonalization. Next we plug the expansion (B.68)

into (B.67) and separate out the terms with e1, e2:

〈ψ1|EFB −H0|ψ1〉 =ν∑

ij=1

x∗ixj 〈ei|EFB −H0|ej〉

=2∑

ij=1

x∗ixj 〈ei|EFB −H0|ej〉

+2∑i=1

ν∑j=3

[x∗ixj 〈ei|EFB −H0|ej〉+ x∗jxi 〈ej|EFB −H0|ei〉

]+

ν∑ij=3

x∗ixj 〈ei|EFB −H0|ej〉 .

This expression can be rewritten as follows:

ν−2∑ij=1

y∗iMijyj +ν−2∑i=1

[u∗i yi + uiy∗i ] = w, (B.72)

Mij = 〈ei+2|EFB −H0|ej+2〉 , (B.73)

ui =2∑j=1

xj 〈ei+2|EFB −H0|ej〉 , (B.74)

w =2∑

ij=1

x∗ixj 〈ei|EFB −H0|ej〉 − 〈ψ1|EFB −H0|ψ1〉 , (B.75)

where yi = xi+2. The equations with yi are further simplified by the shift zi = yi +

M−1ij uj that eliminates the linear term. This gives the following equation in quadratic

form

ν−2∑ij=1

z∗iMijzj = w +ν−2∑ij=1

u∗iMijuj. (B.76)

Notice that the right hand side of the above equation is real. The matrix M is Hermi-

tian, and can be diagonalized: Mij =∑

αEα |rα〉〈rα|. The above equation is solved

B.5 Resolving the non-linear constraints 147

with the help of this spectral decomposition:

ν−2∑α=1

Eα|tα|2 = w, (B.77)

w = w +ν−2∑α=1

Eα|sα|2, (B.78)

tα = 〈rα|zα〉 sα = 〈rα|u〉 . (B.79)

The presence or absence of a solution is decided by the mutual signs of w and Eα: if

w > 0 and Eα < 0 ∀α, then there is no solution. If one Eα > 0, there is a single

solution, and for two or more Eα > 0 there is a multiparametric family of solutions.

Knowing tα, it is straightforward to reconstruct the original ~ψ2.

Above, M was assumed non-singular. If it is singular, then M−1ij is the Moore–

Penrose pseudoinverse [17], and we have yi = zi + gi−M−1ij uj where g ∈ kerM . For

gi the quadratic terms in (B.72) vanish (by definition of gi) and only gi linearly enter

the equation, while zi can be treated as in the non-singular case (for convenience we

assume that the first k eigenvalues of M are zero):

ν−2∑α=k+1

Eαt2α = w −

k∑α=1

[〈u|rα〉+ 〈rα|u〉] . (B.80)

The presence of zero modes renormalizes w.

The more refined version of counting relies on the above solution and the counting

of Eα with the “right" sign. This tells us that for ν = 2, 3, there is a single solution for

fixed ~ψ1, EFB, H0. For larger ν, there could be a single solution or multiparametric

families of solutions, from 0 to ν − 3.

B.5.2 U=3 case

In this case, the nonlinear constraints (4.32) read

〈ψ1|ψ3〉 = 1,

〈ψ1|H0|ψ3〉 = EFB,

〈ψ1|EFB −H0|ψ2〉 = 〈ψ2|EFB −H0|ψ3〉 , (B.81)

〈ψ3|EFB −H0|ψ3〉 = 〈ψ2|EFB −H0|ψ2〉 ,

− 〈ψ1|EFB −H0|ψ1〉 .


Resolution of this set of constraints is very similar to the U = 2 case, so here we

only outline the main steps. We search to resolve the above equations with respect

to Ψ3, taking Ψ1,Ψ2 as inputs. The first three lines are linear, and we solve them by

expanding Ψ3 over a suitable orthonormal basis:

|ψ3〉 =∑k

xk |ek〉 ,

|e1〉 =1√〈ψ1|ψ1〉

|ψ1〉 ,

|e2〉 =1√

〈ψ1|H0Q1H0|ψ1〉Q1H0 |ψ1〉 ,

|e3〉 =Q∗(EFB −H0) |ψ2〉√

〈ψ2|(EFB −H0)Q∗(EFB −H0)|ψ2〉,

...

〈el|em〉 = δl,m, l,m = 1, 2, . . . , ν

Q1 = I− |ψ1〉〈ψ1|〈ψ1| |ψ1〉

,

and Q∗ is a joint transverse projector on |ψ1〉 and Q1H0 |ψ1〉. Then x1 and x2 are the

same as in the U = 2 case, and x3 is directly expressed through the third equation in

Eqs. (B.81). The last (fourth) equation in (B.81) is solved in the same way as that in

the U = 2 case: it is reduced to solving a quadratic form.

B.6 Examples for FB generators

In this section, we present details of the example FB Hamiltonians generated using the

algorithm discussed in the main text. In all of these examples, we pick some H0, EFB

and part of ψl as inputs. Next, following the algorithm outlined in Appendix B.5,

we construct a set of {ψl} consistent with the CLS structure. Then we find the hop-

ping matrix H1 using the algorithm from Section 4.3.1 in the main text (detailed in

Appendix B.4). For simplicity we drop the free part K in all the examples below.

B.6.1 ν = 3, U = 2 case

Example shown in Figure 4.4a: We start with a three-band case ν = 3 and no additional

constraints on the form of H1. We assume a canonical (diagonal) form of H0 and

B.6 Examples for FB generators 149

choose ~ψ1 as

H0 =

0 0 0

0 1 0

0 0 2

, ~ψ1 = (1,−1, 1) . (B.82)

Using the FB algorithm, we find the particular solution:

EFB = 0.5, ~ψ2 = (1.5, 1.5, 1) , (B.83)

H1 =

−0.25 0.25 0.5

−0.25 0.25 0.5

0.75 −0.75 −1.5

. (B.84)

Example shown in Figure 4.4b: Taking non-diagonal H0 and ~ψ1 as

H0 =

0 1 0

1 0 1

0 1 0

, ~ψ1 = (1,−1, 1) , (B.85)

we construct the following FB Hamiltonian:

H1 =

0.19926929 −0.47727273 −0.67654202

−0.33211549 0.79545455 1.12757003

0.19926929 −0.47727273 −0.67654202

, (B.86)

~ψ2 = (4.46130814, 1.5, −1.96130814) , EFB = 0.5. (B.87)

Example shown in Figure 4.4c: Taking all the sites in the unit cell connected to

each other and the same ~ψ1 and EFB as in the above example gives

H0 =

0 1 1

1 0 1

1 1 0

, ~ψ1 = (1,−1, 1) , (B.88)


and we land at the following Hamiltonian:

H1 =

0.18163216 −0.16071429 −0.34234644

−0.90816078 0.80357143 1.71173221

0.18163216 −0.16071429 −0.34234644

, (B.89)

~ψ2 = (1.84761673, 0.25, −0.59761673) , EFB = 0.5. (B.90)

Bipartite lattice U = 2, Figure 4.6: We consider the ν = 4, µ = 2 case and pick

the following input variables:

A =1

4

( √3 1

3√

3

), B =

(1 −2

−2 1

),

~ϕ1 =1√2

(1, 1) , ~ϕ2 =1

2

(1,√

3),

W =

(2 −1

−1 2

).

Solving Eq. (B.63), i.e. (4.41), yields the following solution:

S =

(√3 + 2

)( −1 1

−√

3√

3

)2√

2,

T =

(√3 + 1

)( 3 3√

3

−√

3 −3

)4√

2.

The corresponding hopping matrices H0, H1 read

H0 =

0 0

√3

434

0 0 14

√3

4√3

414

1 −234

√3

4−2 1

,

H1 =

(0 T

S W

),

~ψ1 =

(1√2,

1√2, 0, 0

),

~ψ2 =

(1

2,

√3

2, 0, 0

).

B.6 Examples for FB generators 151

B.6.2 ν = 3, U = 3 case

Example shown in Figure 4.5a: We pickH0 in canonical form and choose ~ψ1 as follows

H0 =

0 0 0

0 1 0

0 0 2

, ~ψ1 = (1,−1, 1)

Solving the non-linear constraints (B.81), i.e., Eq. (4.32), we get ~ψ2, ~ψ3. Then solving

the Eq. (4.33), which is equivalent to Eqs. (4.20–4.24), we get

H1 =

−0.06548573 −0.27210532 −0.2066196

−0.15130619 −0.28682832 −0.13552213

−0.14682469 0.75742396 0.90424865

,~ψ2 = (−0.05144152,−1.53640189,−0.38025523) ,

~ψ3 = (0.58333333,−0.33333333, 0.08333333) ,

EFB = 0.5.

Example shown in Figure 4.5b: We choose H0 and ~ψ1 as

H0 =

0 −1 0

−1 0 1

0 1 0

, ~ψ1 = (1,−1, 1) .

The corresponding FB H1 is

H1 =

0.23624218 0.15535892 −0.08088326

−0.87350793 −0.69073091 0.18277702

1.31303601 0.95651792 −0.35651809

,~ψ2 = (3.14189192,−2.05220768,−0.94681365) ,

~ψ3 = (1.08333333,−0.33333333,−0.41666667) ,

EFB = 0.5.

Example shown in Figure 4.5c: For the following input

H0 =

0 −1 2

−1 0 1

2 1 0

, ~ψ1 = (1,−1, 1) ,


we find the following FB nearest neighbor hopping matrix H1:

H1 =

0.06915801 −0.66620419 −0.7353622

−0.31644957 −0.3029663 0.01348327

−0.46657738 −0.38011423 0.08646314

,~ψ2 = (0.77717503, 2.50899893, 1.05355773) ,

~ψ3 = (0.03571429,−0.57142857, 0.39285714) ,

EFB = 0.5.

Example shown in Figure 4.5d: The following input data

H0 =

0 1 0

1 0 1

0 1 0

, ~ψ1 = (1,−1, 1) ,

provides an example FB Hamiltonian in which the FB is the ground state:

H1 =

−0.52279625 0.17024672 0.69304298

−0.62702148 −0.11461122 0.51241027

−0.73124671 −0.39946915 0.33177756

,~ψ2 = (0.25537008, 0.28652804,−0.59920373) ,

~ψ3 = (0.25,−0.5, 0.25) ,

EFB = −1.5.

B.7 Network constraints

We present here details of the examples where network connectivity was provided as an

input to the FB generator. In all cases, one can find particular solutions to the resulting

non-linear system of equations.

Often network connectivity implies very sparse H0 and H1. Therefore, inserting

sparse H0 and H1 into Eqs. (4.20–4.24) gives a set of equations that can be solved

analytically. More precisely, as seen in the examples below, when H0 and H1 are so

sparse that the number of unknowns (non-zero elements ofH1, H0 and part of the CLS)

is less than or equal to the number of equations, we can solve the Eqs. (4.20–4.24)

analytically. Note that, instead of inserting H1 and H0 into Eqs. (4.20-4.24), we can

get the same set of equations from (4.33) by zeroing the elements of h1 corresponding

B.7 Network constraints 153

to the zero elements of H1.

B.7.1 U=2 case

1D kagome

We consider a d = 1 version of the 2D kagome lattice. The nearest neighbor Hamilto-

nian is restricted by lattice connectivity to

H0 =

0 t2 0 0 0

t2 0 t1 0 0

0 t1 0 t1 0

0 0 t1 0 t2

0 0 0 t2 0

, H1 =

0 t1 t1 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 t1 t1 0

.

Destructive interference, i.e. the last two equations in (4.27), implies that

~ψ1 = (x1,−x2, x2,−x2, x3) , ~ψ2 = (0, a, b, c, 0) .

If we insert ~ψ1, ~ψ2 above into the equations in (4.27), we find

−x2t2 + (y2 + y3) t1

x2t1 + x1t2

−2x2t1

x2t1 + x3t2

−x2t2 + (y3 + y4) t1

= EFB

x1

−x2

x2

−x2

x3

,

at2

(b+ y1) t1

(a+ c+ y1 + y5)

(b+ y5) t1

ct2

= EFB

0

a

b

c

0

.


One possible solution of this is

a = c = 0,

t1 = −EFB2

,

t2 =EFB

2,

x1 = −x,

x2 = x,

x3 = −x,

a = 0,

b = x,

c = 0.

This solution gives a FB with energy EFB. Thus the final solution is

~ψ1 = (−x,−x, x,−x,−x) ,

~ψ2 = (0, 0, x, 0, 0) ,

H1 =

0 −EFB2−EFB

20 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 −EFB2−EFB

20

,

H0 =

0 EFB2

0 0 0EFB

20 −EFB

20 0

0 −EFB2

0 −EFB2

0

0 0 −EFB2

0 EFB2

0 0 0 EFB2

0

.

This lattice has a FB with flatband energy EFB.


U = 2, ν = 3 example

The connectivity of the network shown in Figure 4.7b implies the following hopping

matrices:

H0 =

0 t1 0

t1 0 t2

0 t2 0

, H1 =

s1 s2 0

s4 s5 s6

0 s7 s8

.

We parameterize the CLS amplitudes as follows: ~ψ1 = (x, y, z), ~ψ2 = (a, b, c). Then

Eq. (4.27) gives:as1 + bs2

as4 + bs5 + cs6

bs7 + cs8

=

xEFB − t1y

−t1x− t2z + yEFB

zEFB − t2y

,

s1x+ s4y

s2x+ s5y + s7z

s6y + s8z

=

aEFB − bt1

−at1 + bEFB − ct2cEFB − bt2

,

s1x+ s2y

s4x+ s5y + s6z

s7y + s8z

=

0

0

0

,

as1 + bs4

as2 + bs5 + cs7

bs6 + cs8

=

0

0

0

.

Here, H0, EFB, and ~ψ1 are free parameters. If we fix x = 1, y = 2, z = 1, t1 =

1, t2 = 2, EFB = 3, then we find one particular solution of above the equations

s1 =2√

2

3, s2 = −

√2

3,

s4 =2√

2

3, s5 =

√2

3,

s6 = −1

3

(4√

2), s7 = −

√2

3,

s8 =2√

2

3, a =

1√2,

b = − 1√2, c = −

√2,


from which follow the hopping matrices and CLS amplitudes

H0 =

0 1 0

1 0 2

0 2 0

, H1 =

2√

23−√

23

02√

23

√2

3−1

3

(4√

2)

0 −√

23

2√

23

,

~ψ1 = (1, 2, 1), ~ψ2 =

(1√2,− 1√

2,−√

2

).

B.7.2 U=3 case

U = 3, ν = 3 example

We consider the network shown in Fig. 4.7c. Its connectivity requires the following

hopping matrices:

H0 =

0 t1 0

t1 0 t2

0 t2 0

, H1 =

s1 s1 0

− s12− s1

2− s6 s6

0 2s6 −2s6

.

According to destructive interference conditions (B.39) and (B.40), we paramterize~ψ1, ~ψ2, ~ψ3 as

~ψ1 = (−y, y, y), ~ψ2 = (a, b, c), ~ψ3 = (d, 2d, d).

Then the main equations (4.31) become:(a+ b)s1

(c− b)s6 − 12(a+ b)s1

2(b− c)s6

=

−y (EFB + t1)

y (EFB + t1 − t2)

y (EFB − t2) ,

32(2d− y)s1

(y − d)s6 − 32(d+ y)s1

(2d− y)s6

=

aEFB − bt1

bEFB − at1 − ct2cEFB − bt2

,

12(2a− b)s1(

a− b2

)s1 − (b− 2c)s6

(b− 2c)s6

=

d (EFB − 2t1)

d (2EFB − t1 − t2)

d (EFB − 2t2) .


Again, the above system admits many solutions. We pick one with t1 = 1, t2 = 2, b =12

and get

a =1

80

(3√

21 + 23), c =

1

80

(√21 + 41

),

d =1

40

(−7

√3

2−√

7

2

), y =

1

40

(√3

2+ 3

√7

2

),

EFB =5

2, s1 = −

√72

3, s6 = −

√32

2.

Therefore, the CLS amplitudes and hopping matrices are:

~ψ1 =

140

(−√

32− 3√

72

)140

(√32

+ 3√

72

)140

(√32

+ 3√

72

) ,

~ψ2 =

180

(3√

21 + 23)

12

180

(√21 + 41

) ,

~ψ3 =

140

(−7√

32−√

72

)120

(−7√

32−√

72

)140

(−7√

32−√

72

) ,

H1 =

−√

72

3−√

72

30√

72

6

√32

2+

√72

6−√

32

2,

0 −√

32

√32

H0 =

0 1 0

1 0 2

0 2 0

,

which give a FB with energy EFB = 5/2. Schematics and the band structure of this

lattice are shown in Figure 4.7c.


Appendix C

Supplementary materials for theflatband generator in 2D

C.1 Two-band problem

First we consider the U = (2, 1, 0) case in Fig. 5.3 (c) in the main text. In this case,

the eigenvalue problem and destructive interference conditions in Eqs. (5.2) read

H0~ψ1 +H1

~ψ2 = EFB ~ψ1,

H0~ψ2 +H†1

~ψ1 = EFB ~ψ3,

H1~ψ1 = 0, H†1

~ψ2 = 0,

H2ψs = 0, H†2ψs = 0, 1 ≤ s ≤ 2.

(C.1)

We can see that in the y direction U = 1, therefore we have CLS conditions in the y

direction as

H2~ψ1 = H†2

~ψ1 = 0, H2~ψ2 = H†2

~ψ2 = 0. (C.2)

The above equation implies that H2 has two zero eigenvalues, which in turn implies

that H2 = 0 or

H2 =

(t t

−t −t

), (C.3)

having only one non-zero eigenvector (−x, x). Thus, the problem reduces to the U = 1

class.

Using the same analysis, the U = (2, 2, 2) case in Fig. 5.3 (d) in the main text also

reduces to U = 1. Redefining the x, y directions by tilting them 45 degrees, we can

transform the problem to a U = (2, 1, 0) problem.

Now if we consider the U = (2, 2, 0) case in Fig. 5.3 (e), the eigenvalue problem

159


and destructive interference conditions in Eqs. (5.2) are:

H0~ψ1 +H1

~ψ2 +H2~ψ3 = EFB ~ψ1,

H0~ψ2 +H†1

~ψ1 +H2~ψ4 = EFB ~ψ2,

H0~ψ3 +H1

~ψ4 +H†2~ψ1 = EFB ~ψ3,

H0~ψ4 +H†1

~ψ3 +H†2~ψ2 = EFB ~ψ4,

H1~ψ1 = 0, H1

~ψ3 = 0,

H†1~ψ2 = 0, H†1

~ψ4 = 0,

H2~ψ1 = 0, H2

~ψ2 = 0,

H†2~ψ3 = 0, H†2

~ψ4 = 0.

(C.4)

As we see from the above equation, the 2 × 2 matrices H1, H2 have two zero modes,

either implying H1 = H2 = 0 or reducing to the U = 1 case, according to the previous

subsection.

Similarly, it can also be shown that for the U = (2, 2, 1) in Fig. 5.3 (b), the only

possibility is either H1 = H2 = 0 or a reduction to U = 1.

C.2 More than two bands with nearest neighbor hop-

pings

C.2.1 U=(2,1,0) case

Putting the values U2 = 1, s = 0 to Eq. (5.2), we get the eigenvalue problem and

destructive interference conditions in this case as follows

H1|ψ2〉 = (EFB −H0)|ψ1〉,

〈ψ1|H1 = 〈ψ1|(EFB −H0),

H1|ψ1〉 = 0,

〈ψ2|H1 = 0,

H2|ψ1〉 = 0,

H2|ψ2〉 = 0,

〈ψ1|H2 = 0,

〈ψ2|H2 = 0.

(C.5)

C.2 More than two bands with nearest neighbor hoppings 161

Using destructing interference (C.5), we eliminateH1, H2 from the eigenvalue problem

to get the CLS conditions as below

〈ψ2|(EFB −H0)|ψ1〉 = 0,

〈ψ1|(EFB −H0)|ψ1〉 = 〈ψ2|(EFB −H0)|ψ2〉.(C.6)

The destructive interference conditions suggests following form of H1, H2:

H1 = Q2|u〉〈v|Q1, H2 = Q12MQ12 , (C.7)

where |u〉, |v〉 are arbitrary vectors, M is an arbitrary matrix, Qi is a transverse pro-

jector on |ψi〉, and Q12 is transverse projector on |ψi=1,2〉, i.e. Q12|ψi=1,2〉 = 0 (see

Appendix B.4.2). Then the eigenvalue problem (C.5) becomes

Q2|u〉〈v|Q1|ψ2〉 = (EFB −H0)|ψ1〉,

〈ψ1|Q2|u〉〈v|Q1 = 〈ψ2|(EFB −H0) ,

which gives

Q2|u〉 =1

〈v|Q1|ψ2〉(EFB−H0)|ψ1〉, 〈v|Q1 =

〈v|Q1|ψ2〉〈ψ1|(EFB −H0)|ψ1〉

〈ψ2|(EFB−H0) .

Putting this into (C.7) we get the solution

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉,

H2 = Q12MQ12.

(C.8)

C.2.2 U=(2,2,1) case



H1~ψ2 +H2

~ψ3 = (EFB −H0)~ψ2,

H†1~ψ1 = (EFB −H0)~ψ2,

H†2~ψ1 = (EFB −H0)~ψ3,

H1~ψ1 = 0, H†1

~ψ2 = 0,

H1~ψ3 = 0, H2

~ψ1 = 0,

H2~ψ2 = 0, H†2

~ψ3 = 0,

H†1~ψ3 +H†2

~ψ2 = 0.

(C.9)


Using the destructive interference conditions in (C.9), we can eliminateH1, H2 and

obtain the CLS constraints:

〈~ψ2|H0|~ψ1〉 = EFB〈~ψ2|~ψ1〉,

〈~ψ3|H0|~ψ1〉 = EFB〈~ψ3|~ψ1〉,

〈~ψ3|H0|~ψ2〉 = EFB〈~ψ3|~ψ2〉,

〈~ψ2| (EFB −H0) |~ψ2〉+ 〈~ψ3| (EFB −H0) |~ψ3〉 = 〈~ψ1| (EFB −H0) |~ψ1〉.

(C.10)

Here, bra/ket notations are equivalent to the corresponding vector, i.e. ~ψi = |ψi〉.

We can now define

H2|ψ3〉 = |y〉 = (EFB −H0)|ψ1〉 −H1|ψ2〉, 〈ψ3|H1 = −〈ψ2|H2 = 〈z|. (C.11)

Using destructive interference conditions (and by multiplying |ψi〉, i = 1, 2, 3 to the

above equations from left or right) we can get

|y〉 → Q2,3|y〉, 〈z| → 〈z|Q1,2,3. (C.12)

For ν = 3, the only possibility to satisfy the second equation above for 〈z|Q1,2,3 6=0 is that |ψ1〉, |ψ2〉, |ψ3〉 are linearly dependent,

c1|ψ1〉+ c2|ψ2〉+ c3|ψ3〉 = 0. (C.13)

As long as all the vectors are not proportional, the CLS is not reducible to the U=1

class. For example, the Lieb lattice has ψ1 = ψ2 + ψ3 and it is a U = (2, 2, 1) class.

ν > 3 case

Resolving H1: Suppose H1 can be written as the sum of two projectors:

H1 = |a〉〈b|+ |c〉〈d|. (C.14)

Then using the destructive interference conditions in (C.9), we have

H1 = Q2(|a〉〈b|+ |c〉〈d|)Q1,3. (C.15)


Putting (C.15) and (C.12) into (C.9), we have

Q2|a〉〈b|Q1,3|ψ2〉+Q2|c〉〈d|Q1,3|ψ2〉 = (EFB −H0)|ψ1〉 −Q2,3|y〉, (C.16)

〈ψ1|Q2|a〉〈b|Q1,3 + 〈ψ1|Q2|c〉〈d|Q1,3 = 〈ψ2|(EFB −H0), (C.17)

〈ψ3|Q2|a〉〈b|Q1,3 + 〈ψ3|Q2|c〉〈d|Q1,3 = 〈z|Q1,2,3. (C.18)

In Eqs. (C.16–C.18), there are three equations and four vectors. We can therefore

freely choose one of the vectors. Suppose |a〉 is free a variable, and then

Q2|c〉 =1

〈d|Q1,3|ψ2〉((EFB −H0) |ψ1〉 −Q2,3|y〉 −Q2|a〉〈b|Q1,3|ψ2〉) . (C.19)

Inserting this into (C.17) and (C.18) we have

〈b|Q1,3 =〈b|Q1,3|ψ2〉

〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉〈ψ2| (EFB −H0)

+〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉 − 〈ψ1|Q2|a〉〈b|Q1,3|ψ2〉

(〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉) 〈ψ3|Q2|a〉〈z|Q1,2,3,

〈d|Q1,3 =〈d|Q1,3|ψ2〉

〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉〈ψ2| (EFB −H0)

− 〈ψ1|Q2|a〉〈d|Q1,3|ψ2〉(〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉) 〈ψ3|Q2|a〉

〈z|Q1,2,3.

(C.20)

Putting these solutions into (C.15) gives

H1 =1

〈ψ3|Q2|a〉Q2|a〉〈z|Q1,2,3

+1

〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

− 〈ψ3|Q2|a〉 − 〈ψ1|Q2|a〉(〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉) 〈ψ3|Q2|a〉

Q2,3|y〉〈ψ2| (EFB −H0)

− 〈ψ1|Q2|a〉(〈ψ1| (EFB −H0) |ψ1〉 − 〈ψ1|Q2,3|y〉) 〈ψ3|Q2|a〉

(EFB −H0) |ψ1〉〈z|Q1,2,3.

(C.21)

This solution must satisfy (C.16), which requires

Q2,3|y〉 →〈ψ3| (EFB −H0) |ψ3〉

〈ψ1|Q2,3|y〉Q2,3|y〉,

Q2|a〉 → Q1,2|a〉,(C.22)


and putting into (C.21) yields

H1 =1

〈ψ3|Q1,2|a〉Q1,2|a〉〈z|Q1,2,3 +

(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ2| (EFB −H0) |ψ2〉


Q2,3|y〉〈ψ2| (EFB −H0) .

(C.23)

Resolving H2: Suppose H2 has the following form:

H2 = Q3 (|u〉〈v|+ |w〉〈x|)Q1,2. (C.24)

The inverse eigenvalue problem for this H2 is

Q3|u〉〈v|Q1,2|ψ3〉+Q3|w〉〈x|Q1,2|ψ3〉 =〈ψ3| (EFB −H0) |ψ3〉

〈ψ1|Q2,3|y〉Q2,3|y〉,(C.25)

〈ψ1|Q3|u〉〈v|Q1,2 + 〈ψ1|Q3|w〉〈x|Q1,2 = 〈ψ3|(EFB −H0), (C.26)

〈ψ2|Q3|u〉〈v|Q1,2 + 〈ψ2|Q3|w〉〈x|Q1,2 = −〈z|Q1,2,3. (C.27)

Suppose then that |u〉 is a free parameter, so from (C.25) we have

Q3|w〉 =〈ψ3| (EFB −H0) |ψ3〉〈ψ1|Q2,3|y〉〈x|Q1,2|ψ3〉

Q2,3|y〉 −〈v|Q1,2|ψ3〉〈x|Q1,2|ψ3〉

Q3|u〉. (C.28)

Putting this into (C.26) and (C.27) gives

〈x|Q1,2 =〈x|Q1,2|ψ3〉

〈ψ3| (EFB −H0) |ψ3〉〈ψ3|(EFB −H0)

+〈x|Q1,2|ψ3〉〈ψ1|Q3|u〉

〈ψ3| (EFB −H0) |ψ3〉〈ψ2|Q3|u〉〈z|Q1,2,3,

〈v|Q1,2 =〈v|Q1,2|ψ3〉

〈ψ3| (EFB −H0) |ψ3〉〈ψ3|(EFB −H0)

+〈v|Q1,2|ψ3〉〈ψ1|Q3|u〉 − 〈ψ3| (EFB −H0) |ψ3〉

〈ψ3| (EFB −H0) |ψ3〉〈ψ2|Q3|u〉〈z|Q1,2,3.

(C.29)

Putting the above solutions for |v〉, |w〉, |x〉 into (C.24) we have

H2 =1

〈ψ1|Q2,3|y〉Q2,3|y〉〈ψ3|(EFB −H0)

+1

〈ψ2|Q3|u〉

(〈ψ1|Q3|u〉〈ψ1|Q2,3|y〉

Q2,3|y〉 −Q3|u〉)〈z|Q1,2,3.

(C.30)


Therefore, the solution for the U=(2,2,1), ν > 3 case is

H1 =1

〈ψ3|Q1,2|a〉Q1,2|a〉〈z|Q1,2,3 +

(EFB −H0) |ψ1〉〈ψ2| (EFB −H0)

〈ψ2| (EFB −H0) |ψ2〉


Q2,3|y〉〈ψ2| (EFB −H0) ,

H2 =1

〈ψ1|Q2,3|y〉Q2,3|y〉〈ψ3|(EFB −H0)

+1

〈ψ2|Q3|u〉

(〈ψ1|Q3|u〉〈ψ1|Q2,3|y〉

Q2,3|y〉 −Q3|u〉)〈z|Q1,2,3,

(C.31)

where EFB, H0, |ψi〉, i = 1, 2, 3 are chosen respecting the constraints (5.8), and

|y〉, |z〉, |a〉, |u〉 are free parameters.

When |z〉 = 0 and Q2,3|y〉 = (EFB −H0)|ψ1〉, the solution reduces to that of the

special case in Appendix C.2.3.

Three-band case

In the case of three bands,Q123 = 0, the only possibility thatQ1,2,3 6= 0 thusQ1,2,3|z〉 6=0 is

c1|ψ1〉+ c2|ψ2〉+ c3|ψ3〉 = 0.

Also, since

Q12|ψ3〉 = Q13|ψ2〉 = Q23|ψ1〉 = 0,

any Q12, Q13, Q23 can be used as Q123. We use Q23 for Q123, and then, for given the

inverse eigenvalue problem, suppose

|ψ1〉 = α|ψ2〉+ β|ψ3〉.

The CLS constraints become

〈ψ2| (EFB −H0) |ψ2〉 = 0,

〈ψ3| (EFB −H0) |ψ3〉 = 0,

〈ψ3| (EFB −H0) |ψ2〉 = 0,

(EFB −H0) (α|ψ2〉+ β|ψ3〉) = 0.

(C.32)


Then, for the CLS, EFB, andH0, satisfying the above constraints, the eigenvalue prob-

lem and destructive interference conditions become

β〈ψ3|H1 = 〈ψ2| (EFB −H0) ,

α〈ψ2|H2 = 〈ψ3| (EFB −H0) ,

H1|ψ2〉 = H1|ψ3〉 = 0,

H2|ψ2〉 = H2|ψ3〉 = 0,

〈ψ2|H1 = 0,

〈ψ3|H2 = 0.

(C.33)

Resolving H1: The inverse eigenvalue problem for H1 is

〈ψ3|H1 = 〈ψ2| (EFB −H0) .

Suppose H1 is a projector

H1 = Q2|a〉〈b|Q23,

and then the inverse eigenvalue problem for H1 becomes

〈b|Q23 =1

〈ψ3|Q2|a〉〈ψ2| (EFB −H0) ,

which gives

H1 =Q2|a〉〈ψ2| (EFB −H0)

〈ψ3|Q2|a〉,

where |a〉 is fixed freely (i.e. a free parameter) and not proportional to |ψ2〉.

Similarly, we can resolve H2 as

H2 =Q3|c〉〈ψ3| (EFB −H0)

〈ψ2|Q3|c〉,

where as above |c〉 is fixed freely and not proportional to |ψ2〉.

Therefore, the final solution of the U=(2,2,1) three-band problem is

H1 =Q2|a〉〈ψ2| (EFB −H0)

〈ψ3|Q2|a〉,

H2 =Q3|c〉〈ψ3| (EFB −H0)

〈ψ2|Q3|c〉,

(C.34)

where EFB, H0, |ψ1〉, |ψ2〉 are chosen respecting the constraints (C.32) and |a〉, |c〉are free parameters.


C.2.3 Special case for U=(2,2,1) with nearest neighbor hoppings

In this section we consider the case when |z〉 = 0. Then the eigenvalue problem (C.9)

readsH1

~ψ2 +H2~ψ3 = (EFB −H0)~ψ1,

H†1~ψ1 = (EFB −H0)~ψ2,

H†2~ψ1 = (EFB −H0)~ψ3,

H1~ψ1 = 0, H†1

~ψ2 = 0,

H1~ψ3 = H†1

~ψ3 = 0,

H2~ψ1 = 0, H†2

~ψ3 = 0,

H2~ψ2 = H†2

~ψ2 = 0.

(C.35)

The CLS constraints here are the same with (5.8). Note the bra/ket notations are equiv-

alent to the corresponding vector, i.e. ~ψi,j = |ψi,j〉.

Assume that a CLS satisfying (5.8) is given, and then we solve (C.35). We write

the H1, H2 as single projector

H1 = |x〉〈y|, H2 = |v〉〈w|. (C.36)

In order to satisfy the destructive interference conditions we introduce the following

operators:

Q23 = I − Q3|ψ2〉〈ψ2|〈ψ2|Q3|ψ2〉

− Q2|ψ3〉〈ψ3|〈ψ3|Q2|ψ3〉

,

Q13 = I − |ψ1〉〈ψ1|Q3

〈ψ1|Q3|ψ1〉− |ψ3〉〈ψ3|Q1

〈ψ3|Q1|ψ3〉,

Q12 = I − |ψ1〉〈ψ1|Q2

〈ψ1|Q2|ψ1〉− |ψ2〉〈ψ2|Q1

〈ψ2|Q1|ψ2〉,

Qi = I − |ψi〉〈ψi|〈ψi|ψi〉

.

(C.37)

Then we can write H1, H2 as

H1 = Q23|x〉〈y|Q13, H2 = Q23|v〉〈w|Q12, (C.38)

and putting (C.38) into (C.35) we get

Q23|x〉〈y|Q13|ψ2〉+Q23|v〉〈w|Q12|ψ3〉 = (EFB −H0)|ψ1〉,

Q†13|y〉〈x|Q†23|ψ1〉 = (EFB −H0)|ψ2〉,

Q†12|w〉〈v|Q†23|ψ1〉 = (EFB −H0)|ψ3〉.

(C.39)


In (C.39), there are three equations and four vectors, |x〉, |y〉, |v〉, |w〉, so the vectors

can be chosen freely. We choose |v〉 = |x〉, and then

Q23|x〉(〈y|Q13|ψ2〉+ 〈w|Q12|ψ3〉) = (EFB −H0)|ψ1〉, (C.40)

Q†13|y〉 =(EFB −H0)|ψ2〉〈x|Q†23|ψ1〉

, (C.41)

Q†12|w〉 =(EFB −H0)|ψ3〉〈v|Q†23|ψ1〉

. (C.42)

Now putting (C.41) and (C.42) into (C.40), and using the last identity in the compati-

bility constraints (5.8), we get

Q23|x〉 =〈ψ1|Q23|x〉

〈ψ1|(EFB −H0)|ψ1〉(EFB −H0)|ψ1〉. (C.43)

Putting (C.41), (C.42), and (C.43) into (C.38), we get

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉,

H2 =(EFB −H0)|ψ1〉〈ψ3|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉.

(C.44)

Note that this solution is not limited to three bands. If we define a transverse operator

Q123 = I − |ψ1〉〈ψ1|Q†23

〈ψ1|Q†23|ψ1〉− |ψ2〉〈ψ2|Q13

〈ψ2|Q13|ψ2〉− |ψ3〉〈ψ3|Q12

〈ψ3|Q12|ψ3〉, (C.45)

then we can write

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉+Q23KQ123,

H2 =(EFB −H0)|ψ1〉〈ψ3|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉+Q23MQ123,

(C.46)

where K,M are arbitrary ν × ν matrices. Because of operators Q23, Q123, the extra

K,M terms do not effect the eigenvalue problem (C.35). Note that when ν = 3, the

operator Q123 = 0; therefore, the extra term vanishes for ν = 3.


C.2.4 U=(2,2,0) case with three bands



H1ψ2,1 +H2ψ1,2 = (EFB −H0)ψ1,1

H†1ψ1,1 +H2ψ2,2 = (EFB −H0)ψ2,1,

H1ψ2,2 +H†2ψ1,1 = (EFB −H0)ψ1,2,

H†1ψ1,2 +H†2ψ2,1 = (EFB −H0)ψ2,2,

(C.47)

with destructive interference conditions

H1ψ1,1 = H1ψ1,2 = 0,

H†1ψ2,1 = H†1ψ2,2 = 0,

H2ψ1,1 = H2ψ2,1 = 0,

H†2ψ1,2 = H†2ψ2,2 = 0.

(C.48)

Since H1, H2 have two zero modes, we can parameterize the hopping matrices

H1, H2 in the following way:

H1 = |x〉〈y| =

ad ae af

bd be bf

cd ce cf

, H2 = |u〉〈v| =

gr gs gt

hr hs ht

lr ls lt

, (C.49)

where

|x〉 =

a

b

c

, |y〉 =

d

e

f

, |u〉 =

g

h

l

, |v〉 =

r

s

t

. (C.50)

Matrices H1, H2 have two zero modes.


H1 :

−f0

d

,

−ed

0

,

H†1 :

−c0

a

,

−ba

0

,

H2 :

−t0

r

,

−sr

0

,

H†2 :

−l0

g

,

−hg

0

.

(C.51)

Destructive interference conditions impose numerous constraints on H1, H2 and

the CLS. Here we construct a parameterization of H1, H2 and the CLS such that, with

the parameterization, they satisfy the destructive interference conditions by default.

Consider that ψ1,1 is the right zero mode of both H1, H2, and therefore it is parallel

to the cross product of y and ν and also parallel to one of the right zero eigenvectors

of H1, H2 as

ψ1,1 = α(y × v) ‖

−f0

d

‖−t0

r

. (C.52)

Then ψ2,1 is the left zero mode of H1 and the right zero mode of H2, and therefore

parallel to the cross product of x and µ and also parallel to one of the left (right) zero

eigenvectors of H1 (H2) as

ψ2 = β(x× v) ‖

−ba

0

‖−sr

0

. (C.53)

Next, ψ1,2 is the right zero mode of H1 and the left zero mode of H2, and therefore

parallel to the cross product of y and µ and also parallel to one of the right (left) zero

eigenvectors of H1 (H2) as


ψ1,2 = γ(y × u) ‖

−ed

0

‖−hg

0

. (C.54)

Finally, ψ2,2 is the left zero mode of both H1 and H2, and therefore parallel to the

cross product of x and µ and also parallel to one of the left zero eigenvectors of H1, H2

as

ψ2,2 = η(x× u) ‖

−c0

a

‖−l0

g

. (C.55)

From all the above, we can see that

t = f,

d = r = a = g,

b = s,

e = h,

c = l.

(C.56)

For simplicity, we choose all proportionality factors to be 1. Then the expressions

for all ψ reduce to the following equations:

ψ1 =

(−bf + ef)α

0

(ab− ae)α}

=

−fα1

0

aα1

,

ψ2 =

(−bc+ bf)β

(ac− af)β

0

=

−bβ1

aβ1

0

,

ψ3 =

(ce− ef)γ

(−ac+ af)γ

0

=

−eγ1

aγ1

0

,

ψ4 =

(bc− ce)η

0

(−ab+ ae)η

=

−cη1

0

aη1

.

(C.57)

We solve the above equations to get


α1 = (b− e)α, β1 = (c− f)β, γ1 = −cγ + fγ, η1 = −bη + eη. (C.58)

We can set one of the pre-factors to be 1; here, η = 1. Then (C.57) becomes

ψ1,1 =

−(b− e)fα

0

a(b− e)α

,

ψ2 =

−b(c− f)β

a(c− f)β

0

,

ψ3 =

−e(−cγ + fγ)

a(−cγ + fγ)

0

ψ4 =

−c(−b+ e)

0

a(−b+ e)

.

(C.59)

We define H0 as

H0 =

0 0 0

0 1 0

0 0 ε

. (C.60)

Putting (C.56) and (C.58) into (C.49), we get

H1 =

a2 ae af

ab be bf

ac ce cf

, y =

a2 ab af

ae be ef

ac bc cf

. (C.61)

Now the eigenvalue problem (C.47) becomes


(b− e) (−a2(c− f)(β + γ) + fαEFB)

−a(b− e)(c− f)(bβ + eγ)

a(b− e)(−c(c− f)(β + γ) + α(ε− EFB))

==

0

0

0

,

(c− f) (a2(b− e)(1 + α) + bβEFB)

a(c− f)((b− e)e(1 + α) + β − βEFB)

a(b− e)(c− f)(c+ fα)

=

0

0

0

,

(c− f) (a2(b− e)(1 + α)− eγEFB)

a(c− f)(b(b− e)(1 + α) + γ(−1 + EFB))

a(b− e)(c− f)(c+ fα)

=

0

0

0

,

(b− e) (−a2(c− f)(β + γ)− cEFB)

−a(b− e)(c− f)(bβ + eγ)

a(b− e)(−(c− f)f(β + γ)− ε+ EFB)

=

0

0

0

.

(C.62)

Assuming that a 6= 0, c 6= f, b 6= e, we solve the above equations. Also in the

eigenvalue problem we can set a = 1. (We have set η = 1 already. This can still be

done by dividing all equations by a, then dividing all equations by ηa; thus, we can set

η=1 as well.) They read:

b =

√2(−1 + EFB)

EFB

√−√−α(−1+EFB)2E4

FB(4(1+α)2−αE2FB)+(−1+EFB)EFB(−2(1+α)2+αE2

FB)(1+α)2E2

FB

,

c =

√α√−ε+ EFB√EFB

,

e = −

√−√−α(−1+EFB)2E4

FB(4(1+α)2−αE2FB)+(−1+EFB)EFB(−2(1+α)2+αE2

FB)(1+α)2E2

FB√2

,

f = −√−ε+ EFB√α√EFB

,

β =−α(−1 + EFB)E3

FB +√−α(−1 + EFB)2E4

FB (4(1 + α)2 − αE2FB)

2(1 + α)(−1 + EFB)E2FB

,

γ = −α(−1 + EFB)E3

FB +√−α(−1 + EFB)2E4

FB (4(1 + α)2 − αE2FB)

2(1 + α)(−1 + EFB)E2FB

.

(C.63)

There are many possible solutions, with the above solution representing one of


them.

C.3 Next nearest neighbor hoppings

C.3.1 U=(2,2,1) case

The configuation of this case is shown in Fig. 5.4 (c) in the main text. Putting the

values U2 = 2, s = 1 to Eq. (5.3), we get the eigenvalue problem and destructive

interference conditions in this case as follows

H1|ψ2〉+H2|ψ3〉 = (EFB −H0)|ψ1〉,

H†1|ψ1〉+H†3|ψ3〉 = (EFB −H0)|ψ2〉,

H†2|ψ1〉+H3|ψ2〉 = (EFB −H0)|ψ3〉,

H1|ψ1〉 = 0,

〈ψ2|H1 = 0,

H2|ψ1〉 = 0,

〈ψ3|H2 = 0,

H3|ψ3〉 = 0,

〈ψ2|H3 = 0,

H1|ψ3〉+H3|ψ1〉 = 0,

H2|ψ2〉+H†3|ψ1〉 = 0,

〈ψ3|H1 + 〈ψ2|H2 = 0 .

(C.64)

We defineH†1|ψ3〉 = −H†2|ψ2〉 = Q1|x〉,

H2|ψ2〉 = −H†3|ψ1〉 = Q3|y〉,

H1|ψ3〉 = −H3|ψ1〉 = Q2|z〉,

H2|ψ3〉 = Q3|u〉,

H3|ψ2〉 = Q2|v〉,

H†3|ψ3〉 = Q3|w〉.

Then the eigenvalue problem (C.64) decouples into inverse eigenvalue problems of

C.3 Next nearest neighbor hoppings 175

H1, H2, H3 respectively. The inverse eigenvalue problem for H1 is

H1|ψ2〉 = (EFB −H0)|ψ1〉 −Q3|u〉,

H1|ψ3〉 = Q2|z〉,

〈ψ1|H1 = 〈ψ2|(EFB −H0)− 〈w|Q3,

〈ψ3|H1 = 〈x|Q1,

H1|ψ1〉 = 0,

〈ψ2|H1 = 0,

(C.65)

and the inverse eigenvalue problem for H2 is

H2|ψ2〉 = Q3|y〉,

H2|ψ3〉 = Q3|u〉,

〈ψ1|H2 = 〈ψ3|(EFB −H0)− 〈v|Q2,

〈ψ2|H2 = −〈x|Q1,

H2|ψ1〉 = 0,

〈ψ3|H2 = 0,

(C.66)

while the inverse eigenvalue problem for H3 is

H3|ψ1〉 = −Q2|z〉,

H3|ψ2〉 = Q2|v〉,

〈ψ1|H3 = −〈y|Q3,

〈ψ3|H3 = 〈w|Q3,

H3|ψ3〉 = 0,

〈ψ2|H3 = 0.

(C.67)

By multiplying |ψ1〉, |ψ2〉, |ψ3〉 from the left and right to the above inverse eigen-

value problems and comparing them, we can show that |x〉, |y〉, |z〉, |u〉, |v〉, |z〉, |ψ1〉, |ψ2〉, |ψ3〉


should satisfy the following constraints

〈ψ1|(λ−H0)|ψ1〉 − 〈ψ1|Q3|u〉 = 〈ψ2|(λ−H0)|ψ2〉 − 〈w|Q3|ψ2〉 = 〈ψ1|Hx|ψ2〉,

〈ψ2|(λ−H0)|ψ3〉 = 〈ψ1|Q2|z〉 = 〈ψ1|Hx|ψ3〉,

〈ψ2|(λ−H0)|ψ1〉 = 〈w|Q3|ψ1〉,

〈ψ2|(λ−H0)|ψ1〉 = 〈ψ2|Q3|u〉,

〈ψ3|(λ−H0)|ψ1〉 = 〈x|Q1|ψ2〉 = 〈ψ3|Hx|ψ2〉,

〈x|Q1|ψ3〉 = 〈ψ3|Q2|z〉 = 〈ψ3|Hx|ψ3〉,

〈ψ3|(λ−H0)|ψ2〉 = 〈ψ1|Q3|y〉 = 〈ψ1|Hy|ψ2〉,

〈ψ3|(λ−H0)|ψ3〉 − 〈v|Q2|ψ3〉 = 〈ψ1|Q3|u〉 = 〈ψ1|Hy|ψ3〉,

〈ψ3|(λ−H0)|ψ1〉 = 〈v|Q2|ψ1〉,

〈ψ2|Q3|y〉 = −〈x|Q1|ψ2〉 = 〈ψ2|Hy|ψ2〉,

〈ψ2|Q3|u〉 = −〈x|Q1|ψ3〉 = 〈ψ2|Hy|ψ3〉,

〈y|Q3|ψ1〉 = 〈ψ1|Q2|z〉 = −〈ψ1|Hyx|ψ1〉,

〈ψ1|Q2|v〉 = −〈y|Q3|ψ2〉 = 〈ψ1|Hyx|ψ2〉,

〈w|Q3|ψ1〉 = −〈ψ3|Q2|z〉 = 〈ψ3|Hyx|ψ1〉,

〈ψ3|Q2|v〉 = 〈w|Q3|ψ2〉 = 〈ψ3|Hyx|ψ2〉.

If we assume H1 has the following form

H1 =Q2|z〉〈x|Q1

〈ψ3|Q2|z〉+Q23|a〉〈b|Q13,

where 〈ψ3|Q2|z〉 = 〈x|Q1|ψ3〉 = 〈ψ3|H1|ψ3〉 (this is achieved by multiplying |ψ3〉 to

the second and fourth equation of (C.65)), then all equations except for the first and

third are satisfied in the inverse eigenvalue problem (C.65) for H1. The first and third

equations become

Q2|z〉〈x|Q1|ψ2〉〈ψ3|Q2|z〉

+Q23|a〉〈b|Q13|ψ2〉 = Q2|U〉,

〈ψ1|Q2|z〉〈x|Q1

〈ψ3|Q2|z〉+ 〈ψ1|Q23|a〉〈b|Q13 = 〈W |Q1,

(C.68)

whereQ2|U〉 = H1|ψ2〉 = (EFB −H0)|ψ1〉 −Q3|u〉,

〈W |Q1 = 〈ψ1|H1 = 〈ψ2|(EFB −H0)− 〈w|Q3.(C.69)


Solving (C.68) for Q23|a〉, 〈b|Q13 we get

Q23|a〉 =1

〈b|Q13|ψ2〉

(〈ψ3|Q2|z〉Q2|U〉 − 〈x|Q1|ψ2〉Q2|z〉

〈ψ3|Q2|z〉

),

〈b|Q13 =〈b|Q13|ψ2〉 (〈ψ3|Q2|z〉〈W |Q1 − 〈ψ1|Q2|z〉〈x|Q1)

〈ψ1|Q2|U〉〈ψ3|Q2|z〉 − 〈ψ1|Q2|z〉〈x|Q1|ψ2〉.

Thus the solution for H1 is

H1 =Q2|z〉〈x|Q1

〈ψ3|Q2|z〉+

(〈ψ3|Q2|z〉(EFB −H0)|ψ1〉 − 〈ψ3|Q2|z〉Q3|u〉 − 〈x|Q1|ψ2〉Q2|z〉)〈ψ3|Q2|z〉 ((〈ψ1|(EFB −H0)|ψ1〉 − 〈ψ1|Q3|u〉) 〈ψ3|Q2|z〉 − 〈ψ1|Q2|z〉〈x|Q1|ψ2〉)

× (〈ψ3|Q2|z〉〈ψ2|(EFB −H0)− 〈ψ3|Q2|z〉〈w|Q3 − 〈ψ1|Q2|z〉〈x|Q1) .(C.70)

Using the same procedure, we get the solution for H2, H3 as

H2 = −Q3|y〉〈x|Q1

〈ψ2|Q3|y〉+

(〈ψ2|Q3|y〉Q3|u〉+ 〈x|Q1|ψ3〉Q3|y〉)〈ψ2|Q3|y〉 (〈ψ2|Q3|y〉〈ψ1|Q3|u〉+ 〈x|Q1|ψ3〉〈ψ1|Q3|y〉)

× (〈ψ2|Q3|y〉〈ψ3|(EFB −H0)− 〈ψ2|Q3|y〉〈v|Q2 + 〈ψ1|Q3|y〉〈x|Q1) ,

H3 = −Q2|z〉〈y|Q3

〈ψ1|Q2|z〉+

(〈ψ1|Q2|z〉Q2|v〉+ 〈y|Q3|ψ2〉Q2|z〉) (〈ψ1|Q2|z〉〈w|Q3 + 〈ψ3|Q2|z〉〈y|Q3)

〈ψ1|Q2|z〉 (〈ψ1|Q2|z〉〈ψ3|Q2|v〉+ 〈y|Q3|ψ2〉〈ψ3|Q2|z〉).

(C.71)

C.3.2 U=(2,1,0) case

This case is shown in Fig. 5.4 (d). Putting the values U2 = 1, s = 0 to Eq. (5.3),

we get the eigenvalue problem and destructive interference conditions in this case as

followsH1|ψ2〉 = (EFB −H0)|ψ1〉,

〈ψ1|H1 = 〈ψ2|(EFB −H0),

H1|ψ1〉 = 0,

〈ψ2|H1 = 0,

H2|ψ1〉 = 0,

〈ψ2|H2 = 0,

H3|ψ1〉 = 0,

〈ψ2|H3 = 0,

H2|ψ2〉+H†3|ψ1〉 = 0,

〈ψ1|H2 + 〈ψ2|H†3 = 0.

(C.72)


Using destructive interference conditions, we eliminateH1, H2, H3 to get the following

CLS constraints

〈ψ2|(EFB −H0)|ψ1〉 = 0,

〈ψ1|(EFB −H0)|ψ1〉 = 〈ψ2|(EFB −H0)|ψ2〉 .(C.73)

The destructive interference conditions (third to eighth equations in (C.72)) suggest

thatH1 = Q2|a〉〈b|Q1,

H2 = Q2|c〉〈d|Q1,

H3 = Q2|e〉〈f |Q1,

(C.74)

where |a〉, |b〉, |c〉, |d〉, |e〉, |f〉 are arbitrary vectors. Then, the last two equations in

(C.72) giveQ2|c〉〈d|Q1|ψ2〉 = −Q1|f〉〈e|Q2|ψ1〉,

〈ψ1|Q2|c〉〈d|Q1 = −〈ψ2|Q1|f〉〈e|Q2 .(C.75)

This indicates thatQ2|a〉 ∝ Q1|b〉, ∀a, b,

Q1|c〉 ∝ Q2|d〉, ∀c, d.(C.76)

The above condition implies that |a〉, |b〉, |c〉, |d〉 are perpendicular to |ψ1〉, |ψ2〉 at the

same time. Therefore, we can write H2, H3 as

H2 = Q12|c〉〈d|Q12,

H3 = Q12|e〉〈f |Q12 .(C.77)

Then the last destructive interference conditions become

H2|ψ2〉 = 0,

〈ψ1|H2 = 0,

H3|ψ2〉 = 0,

〈ψ1|H3 = 0.

(C.78)

For given |ψ1〉, |ψ2〉, H0, EFB satisfying constraints (C.73), the eigenvalue prob-

lem (C.72) then becomes an inverse eigenvalue problem of H1 as

H1|ψ2〉 = (EFB −H0)|ψ1〉,

〈ψ1|H1 = 〈ψ2|(EFB −H0) ,(C.79)


which gives the following solution

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉. (C.80)

The final solution is

H1 =(EFB −H0)|ψ1〉〈ψ2|(EFB −H0)

〈ψ1|(EFB −H0)|ψ1〉,

H2 = Q12|c〉〈d|Q12,

H3 = Q12|e〉〈f |Q12.

(C.81)


Appendix D

Supplementary materials for thenon-Hermitian flatband generator

D.1 CLS-based generator

This method is based on compact localized states as a direct extension of the method

introduced in Ref. [97]. Unlike the Hermitian case, the non-Hermitian FB does not

necessarily host CLSs [97, 98]. In the case that the non-Hermitian FB does host a CLS

of class U [97, 98], then the band is fully flat, i.e. both real and imaginary parts are k

independent. More precisely, the CLS ΨCLS = (~ψ1, ~ψ2, . . . , ~ψU) is the eigenvector of

the U × U tri-diagonal block matrix

HU =

H0 Hl 0 0 . . . 0

Hr H0 Hl 0 . . . 0

0. . . . . . . . . . . . ...

......

0 . . . 0 Hr H0 Hl

0 . . . 0 0 Hr H0

, (D.1)

with eigenenergy EFB = E1 + iE2, E1, E2 ∈ R. The following destructive interfer-

ence condition must be satisfied

Hl~ψ1 = Hr

~ψU = 0. (D.2)

Therefore, a necessary condition for the existence of a non-Hermitian CLS reads

detHl = detHr = 0. (D.3)

181

182 Supplementary materials for the non-Hermitian flatband generator

We rewrite the CLS problem (D.1–D.2) as

Hl~ψ2 = (EFB −H0)~ψ1, (D.4)

Hr~ψj−1 +Hl

~ψj+1 = (EFB −H0)~ψj, 2 ≤ j ≤ U − 1, (D.5)

Hr~ψU−1 = (EFB −H0)~ψU , (D.6)

Hl~ψ1 = Hr

~ψU = 0, (D.7)

~ψj = 0 , j < 0, j > U. (D.8)

The non-Hermitian FB generator is the set of all possible matrices H0, Hr, Hl that

satisfy Eqs. (D.4–D.8).

We consider a two-band problem, i.e. ν = 2 sites per unit cell. In this case, using

the same argument as the previous section,H0 can take the form as given in (6.4). With

the proper choice of CLS ΨCLS and FB energy EFB, solving Eqs. (D.4–D.8) becomes

an inverse eigenvalue of finding Hr, Hl.

D.1.1 U = 1 case

In this case, the eigenvalue problem (D.4–D.8) becomes

H0~ψ = λ~ψ,

Hr~ψ = 0,

Hl~ψ = 0.

(D.9)

Suppose ~ψ = (x, y) 6= 0, then (D.9) becomes

νy = λx,

µy = λy,

ax+ by = λx,

cx+ dy = λy,

fx+ gy = λx,

hx+ ly = λy.

(D.10)

We solve (D.9) for different forms of H0.

D.1 CLS-based generator 183

Degenerate H0 In this case µ = ν = 0 and (D.10) yields the following solution:

λ = 0,

a = −yxb,

c = −yxd,

f = −yxg,

h = −yxl.

Thus the hopping matrices, CLS, and FB energy are

H0 =

(0 0

0 0

),

Hl =

(− yxg g

− yxl l

),

Hr =

(− yxb b

− yxd d

),

~ψ = (x, y),

λ = 0.

Abnormal H0 In this case µ = 0, ν = 1 and (D.10) gives

y = 0,

a = 0,

c = 0,

f = 0,

h = 0,


which gives the following hopping matrices, CLS, and FB energy:

H0 =

(0 1

0 0

),

Hl =

(0 g

0 l

),

Hr =

(0 b

0 d

),

~ψ = x, 0,

λ = 0.

Non-degenerate H0 In this case µ = 1, ν = 0 and (D.9) has the following solution:

H0 =

(0 0

0 1

),

Hl =

(f 0

g 0

),

Hr =

(a 0

c 0

),

~ψ = (0, 1),

λ = 1 .

Here, y is normalized to be 1.

D.1.2 U=2 case

In this case the eigenvalue problem (D.4–D.8) becomes

H0ψ1 +Hlψ2 = λψ1,

H0ψ2 +Hrψ1 = λψ2,

Hlψ1 = 0, Hrψ2 = 0.

(D.11)

We can parameterize Hr, Hl in the following way

Hr =

(a b

c bca

), Hl =

(f g

h ghf

), (D.12)

D.1 CLS-based generator 185

which satisfies the destructive interference conditions by definition. Then we can

choose ψ1, ψ2 to be zero eigenvectors of Hl, Hr, respectively, as

ψ1 =

(− gf

1

), ψ2 =

(− ba

1

). (D.13)

Then (D.11) becomes (− bf

a+ g + ν

− bha

+ ghf

+ µ

)=

(−gλ

f

λ

),(

−agf

+ b+ νbca− cg

f+ µ

)=

(− bλ

a

λ

).

(D.14)

Solving the above we get

c =a2µ− a2λ

aν + bλ,

f = −a− λ,

g = −(a+ λ)(ab+ aν + bλ)

a2,

h =a2(λ− µ)

aν + bλ.

(D.15)

Putting the corresponding values of µ, ν in (D.15) we get solutions for degenerate,

non-degenerate, and abnormal cases. The band structure for solution (D.15) is

EFB = λ,

Ek =aeik(aν + bµ)

aν + bλ+e−ik(−bµ(a+ λ)− aν(a+ µ))

aν + bλ− λ+ µ.

D.1.3 U=3 case

In this case we haveH0ψ1 +Hlψ2 = λψ1,

H0ψ2 +Hrψ1 +Hlψ3 = λψ2,

H0ψ3 +Hrψ2 = λψ3,

Hlψ1 = 0, Hrψ3 = 0.

(D.16)

We use the same parameterization as in (D.12) and choose the CLS as

ψ1 =

(− gf

1

), ψ2 =

(α

β

), ψ3 =

(− ba

1

), (D.17)


which satisfies the destructive interference conditions by definition. Then (D.16) be-

comes (αf + βg + νβghf

+ αh+ µ

)=

(−gλ

f

λ

),(

− bfa− ag

f+ βν + b+ g

a(−cg+βfµ+gh)+bf(c−h)af

)=

(λ

βλ

),(

aα + bβ + νbβca

+ αc+ µ

)=

(− bλ

a

λ

).

(D.18)

Solving the above equation we have

b =

(−λ±

√λ2 − 4af

)(fν + gλ) + 2afg

2f 2,

c =(λ− µ)

(λ(λ±

√λ2 − 4af

)− 2af

)2(fν + gλ)

,

h =f 2(µ− λ)

fν + gλ,

α =gλ(f − a)−

(2afν ± g(a− f)

√λ2 − 4af

)2af 2

,

β = −λ(a+ f)± (a− f)√λ2 − 4af

2af.

(D.19)

This solution (D.19) is the same as the solution (D.39) with the band calculation

method. By inserting corresponding values of µ, ν, we can get solutions for differ-

ent cases.

When a = −f − λ, the solution (D.19) for the U = 3 case reduces to solution

(D.15) for the U = 2 case.

D.1.4 Inverse eigenvalue method for CLS approach

We can write the inverse eigenvalue problem for the U = 2 case as

Hr|ψ2〉 = (λ−H0|ψ1〉,

Hl|ψ1〉 = (λ−H0) |ψ2〉,

Hr|ψ1〉 = 0, Hl|ψ2〉 = 0,

(D.20)

D.2 Solving completely flat bands 187

with solution

Hr =(λ−H0) |ψ1〉〈ψ2|〈ψ2|Q1|ψ2〉

Q1,

Hl =(λ−H0) |ψ2〉〈ψ1|〈ψ1|Q2|ψ1〉

Q2.

(D.21)

If we use the following ansatz

ψ1 = |θ〉 =

(cos θ

sin θ

), ψ2 = α|ϕ〉 = α

(cosϕ

sinϕ

), (D.22)

the solution is given for parameters θ, ϕ, α, λ, and the FB energy is EFB = λ.

D.2 Solving completely flat bands

A completely flat band has k-independent real and imaginary parts. Our starting point

solving this case is Eq. (6.7), which is

xk + yk = µ+ eik(a+ d) + e−ik(f + l),

xkyk = e2ik detHr + e−2ik detHl

+ (νf − µh)eik + (νa− µc)e−ik

+ df − cg − bh+ al.

(D.23)

We assume one of xk, yk or both are k independent and solve (D.23) to find the FB

Hamiltonian.

D.2.1 Both bands are completely flat

In this case, both xk, yk in (D.23) are k independent. We assume xk = x and yk = y,

then (D.23) becomes

x+ y = µ+ eik(a+ d) + e−ik(f + l),

xy = e2ik detHr + e−2ik detHl + (νf − µh)eik

+ (νa− µc)e−ik + df − cg − bh+ al.

(D.24)


Requiring the polynomial of eik to vanish gives the following equations:

a+ d = 0,

f + l = 0,

det Hr = ad− bc = 0,

det Hl = fl − hg = 0,

νf − µh = 0,

νa− µc = 0,

xy = df − cg − bh+ al,

x+ y = µ.

(D.25)

From these it follows d = −a, l = −f , and either f = a = 0 or h = c = 0, or none.

Solving (D.25) for degenerate, non-degenerate, and abnormal cases separately, we

can get Hl, Hr that gives both bands as completely flat.

Degenerate case: Here we have µ = ν = 0, and bc + a2 = 0, hg + f 2 = 0, and

y = −x. Therefore, c = −a2/b, h = −f 2/g, and

Hr =

(a b

−a2

b−a

), (D.26)

Hl =

(f g

−f2

g−f

), (D.27)

x2 = 2af − a2g

b− bf 2

g. (D.28)

Non-degenerate H0 In this case, µ = 1, ν = 0 and h = c = 0, ad = fl = 0, and

y = 1 − x. Therefore, a = d = f = l = 0 and either b or c are zero and either h or g

are zero, giving

Hr =

(0 b

0 0

) (0 0

c 0

), (D.29)

Hl =

(0 g

0 0

) (0 0

h 0

), (D.30)

x(1− x) = −cg − bh. (D.31)

There are four possible solutions of bc = 0, hg = 0. Interestingly, two out of the four

imply x = 0, y = 1 or x = 1, y = 0, that are very likely U = 1 class.

D.2 Solving completely flat bands 189

Abnormal H0 Here, µ = 0, ν = 1, f = a = 0, and y = −x. Thus d = l = 0 and

bc = hg = 0, and

Hr =

(0 b

0 0

) (0 0

c 0

), (D.32)

Hl =

(0 g

0 0

) (0 0

h 0

), (D.33)

x2 = cg + bh. (D.34)

D.2.2 One band is completely flat

Requiring either x or y in (D.23) will yield detHr = 0, detHl = 0, and therefore we

can parameterize Hr, Hl as

Hr =

(a b

c bca

), Hr =

(f g

h ghf

), (D.35)

which makes Hr, Hl to be singular by definition. We assume that only xk = x is flat,

so then (D.23) becomes

x+ yk =bceik

a+ aeik + e−ik

(gh

f+ f

)+ µ),

xyk =(ag − bf)(ah− cf)

af+ eik(aµ− cν)

+ e−ik(fµ− hν).

(D.36)

This results in the following equations:

yk =bceik

a+ aeik + e−ik

(gh

f+ f

)+ µ− x,

yk =eik(aµ− cν) + e−ik(fµ− hν)

x,

+(ag − bf)(ah− cf)

afx.

(D.37)


Consequently, equating powers of eik, we find

bc

a+ a =

aµ− cνx

,

gh

f+ f =

fµ− hνx

µ− x =(ag − bf)(ah− cf)

x(af)

(D.38)

Solving (D.38) gives

b =

(−x±

√x2 − 4af

)(fν + gx) + 2afg

2f 2,

c =(x− µ)

((x2 ± x

√x2 − 4af

)− 2af

)2(fν + gx)

,

h =f 2(µ− x)

fν + gx.

(D.39)

Then the band structure is

EFB = x,

Ek =2(fν + gµ)

(−(x− µ) +

(aeik + fe−ik

))2(fν + gx)

∓eikν(x− µ)

(√x2 − 4af − x

)2(fν + gx)

.

(D.40)

Putting corresponding values of µ, ν into (D.39) and (D.40), we can get solutions

for degenerate, non-degenerate, and abnormal cases with corresponding band struc-

tures.

D.3 Solving partially flat bands 191

D.3 Solving partially flat bands

We assume that xk = x1 + ix2, y = y1 + iy2, and x1, x2, y1, y2 ∈ R, so then expanding

(D.23) yieldsxk + yk = x1 + y1 + i (x2 + y2) = µ

+ cos(k)(a+ d+ f + l)

− i sin(k)(a+ d− f − l),

xkyk = x1y1 − x2y2 + i (x2y1 + x1y2)

= al − bh− cg + df

+ (aµ− cν + fµ− hν) cos(k)

+ (detHl + detHr) cos(2k)

+ i((−aµ+ cν + fµ− hν) sin(k)

+ (detHl − detHr) sin(2k)).

(D.41)

Equating real and imaginary parts of (D.41) gives

x1 + y1 = µ+ (a+ d+ f + l) cos(k), (D.42)

x2 + y2 = −(a+ d− f − l) sin(k), (D.43)

x1y1 − x2y2 = al − bh− cg + df

+(aµ− cν + fµ− hν) cos(k) (D.44)

+(detHl + detHr) cos(2k),

x2y1 + x1y2 = (−aµ+ cν + fµ− hν) sin(k) (D.45)

+(detHl − detHr) sin(2k).

Solving Eqs. (D.42–D.45) under the condition that some of x1, x2, y1, y2 are k

independent, we get the solution for partially flat bands.

D.3.1 Real parts of both bands are flat

In this case, x1, y1 are k independent, so Eqs. (D.42–D.43) give

y1 = µ− x1 + (a+ d+ f + l) cos(k),

y2 = −x2 − (a+ d− f − l) sin(k).(D.46)

We put this equation into (D.44–D.45) and solve for x2. Then, requiring x1 to be

k independent by setting the coefficients of k-independent terms to zero, we get the


following equations:

a+ d+ f + l = 0,

a− c+ f − h = 0,

ad+ fl − bc− gh =(X + Y ) (X + Z)

2(µ− 2x1)2,

−ad+ bc+ fl − gh = 0,

(D.47)

where X = x1(−a− d+ f + l), Y = aµ− cν − fµ+ hν, Z = cν + dµ− hν − lµ.

Solving (D.47) for degenerate, non-degenerate, and abnormal cases separately, we

get the Hr, Hl that gives the real parts of both bands as flat.

Degenerate case In this case µ = ν = 0 and (D.47) becomes

a+ d+ f + l = 0,

1

8(a+ d− f − l)2 + bc+ gh = ad+ fl,

−ad+ bc+ fl − gh = 0,

1

8(a+ d− f − l)2 + al + df + x2

1 = bh+ cg.

(D.48)

If we consider x1 as a parameter, then the solution for (D.48) is

H0 =

(0 0

0 0

),

Hl =

(−f ±2

√A+B

(a−d)2

−±2√A+B

4b2−a− d− f

),

Hr =

(a b

− (a−d)2

4bd

),

where A = b2x21 ((d− a)(a+ d+ 2f) + x2

1) , B = b(a−d)(a+d+2f)−2bx21. Then

the band structure is

xk = −x1 + i(a+ d) sin(k), (D.49)

yk = x1 + i(a+ d) sin(k). (D.50)

On the other hand, if we consider x1 as a function of Hr, Hl, then the solution for


(D.48) is

H0 =

(0 0

0 0

), (D.51)

Hl =

(f − (a+d+2f)2

4h

h −a− d− f

), (D.52)

Hr =

(a b

− (a−d)2

4bd

), (D.53)

x1 = ±(a− d)(a+ d+ 2f) + 4bh

4√b√h

, (D.54)

and the band structure is

xk = −√bh((a− d)(a+ d+ 2f) + 4bh)2

4bh

+4ibh(a+ d) sin(k)

4bh, (D.55)

yk =

√bh((a− d)(a+ d+ 2f) + 4bh)2

4bh

+4ibh(a+ d) sin(k)

4bh. (D.56)

Non-degenerate case In this case µ = 1, ν = 0 and (D.47) becomes

a+ d+ f + l = 0, (D.57)

a+ f = 0, (D.58)

(X + a− f) (X + d− l)2 (1− 2x1) 2

+ bc+ gh = ad+ fl (D.59)

− ad+ bc+ fl − gh = 0, (D.60)

x1 (1− x1) (a− d− f + l)2

2 (1− 2x1)2 = −(bh+ cg + x1) (D.61)

+ al + df + x21 ,


where X = x1(−a− d+ f + l). Then the solution is

H0 =

(0 0

0 1

),

Hr =

(−f − (x1−1)x1(f−l)2

c(1−2x1)2

c −l

),

Hl =

(f (x1−1)x1(C+D)

2c(1−2x1)2

c(C−D)2(f−l)2 l

),

(D.62)

where C =√

4(f − l)2 + (1− 2x1)2 (1− 2x1) , D = 2(f − l)2 + (1− 2x1)2.


xk = x1 −2i sin(k) (x1(f + l)− f)

2x1 − 1,

yk = −2i sin(k) (x1(f + l)− l)2x1 − 1

− x1 + 1 .

(D.63)

Obviously, the real parts x1, 1− x1 are k independent, i.e flat.

Abnormal case In this case µ = 0, ν = 1 and (D.47) becomes

a+ d+ f + l = 0, (D.64)

− c− h = 0, (D.65)

(X + c− h) (X − c+ h)

8x21

= ad+ fl − bc− gh (D.66)

− ad+ bc+ fl − gh = 0, (D.67)

bh+ cg +(c− h)2

8x21

=1

8(a+ d− f − l)2 (D.68)

+ al + df + x21 ,

where X = x1(−a− d+ f + l). Then the solution is

H0 =

(0 1

0 0

),

Hl =

(f

h2−x21(d+f−x1)2

4hx21

h x1 − d

),

Hr =

(−f − x1

x21(d+f+x1)2−h24hx21

−h d

).

(D.69)


The band structure is

xk = −x1 + i sin(k)

(d− f +

h

x1

− x1

),

yk = x1 − i sin(k)(x1 (−d+ f + x1) + h)

x1

.

(D.70)

D.3.2 Real part of one band is flat

In this case, either x1 or y1 is k independent in Eqs. (D.42)–(D.45). If we assume x1 is

k independent and solve equations (D.42)–(D.45) for x1, then according to (D.42) we

havex1 = µ,

y1 = (a+ d+ f + l)Cos(k).(D.71)

(In a similar way we can assume y1 is k independent, and then y1 = µ, x1 = (a+ d+

f + l)Cos(k).) Equations (D.42–D.45) become

x2 + y2 = sin(k)(a+ d− f − l),

x2y2 = −al + bh+ cos(k)(ν(c+ h) + dµ+ lµ)

+ cg − df − (detHl + detHr) cos(2k),

µy2 = sin(k)(aµ− cν − fµ+ hν)

− x2 cos(k)(a+ d+ f + l),

+ (detHr − detHl) sin(2k).

(D.72)

For convenience, we make the following replacement of variables:

detHr = ad− bc,

detHl = fl − gh,

V = a+ d+ f + l,

X = a+ d− f − l,

Y = aµ− cν + fµ− hν,

Z = aµ− cν − fµ+ hν,

W = al − bh− cg + df.

(D.73)


Equation (D.72) then becomes

x1 + y1 = V cos(k) + µ, (D.74)

x2 + y2 = X sin(k), (D.75)

x1y1 − x2y2 = (detHl + detHr) cos(2k) (D.76)

+ Y cos(k) +W,

x2y1 + x1y2 = −(detHl − detHr) sin(2k). (D.77)

+ Z sin(k)

We can solve (D.77) for three variables: x2, y2, and a third variable that is one of

X, Y, Z, U, V,W . We require the third variable to be k independent by zeroing the

coefficients of all k-dependent terms. This gives a set of equations, and solving them

gives the solution for this case (the real part of one band is flat). The following are our

results.

Degenerate case: In this case µ = 0, ν = 0 and the solution is

Hr =

(a b

(a+f)(b(d+f)+(d−a)g)(b+g)2

d

),

Hl =

(f g

(a+f)(g(a+l)+b(l−f))(b+g)2

l

).

(D.78)


E1 = −2i sin(k)(bf − ag)

b+ g,

E2 =

(eik(b(a+ d+ f) + dg)

)b+ g

,

+

(e−ik(ag + bl + fg + gl)

)b+ g

.

(D.79)

Abnormal case: In this case µ = 0, ν = 1 and the solution is

Hr =

(a b

0 d

), Hl =

(−a g

0 l

). (D.80)

The band structure isE1 = 2ia sin(k),

E2 = deik + e−ikl + 1.(D.81)


Note that when a = −f , the solution (D.78) for the degenerate case reduces to the

abnormal case (D.80).

Non-degenerate case: In this case µ = 1, ν = 0 and the solution is

Hr =

(a 0

c d

), Hl =

(f 0

h −d

). (D.82)

The band structure isE1 = 1 + 2id sin(k),

E2 = e−ik(f + ae2ik

).

(D.83)

D.3.3 Imaginary parts of both bands are flat

In this case, x2, y2 are k independent in Eqs. (D.42)–(D.45). Using the same procedure

as in Section D.3.1, solving (D.42)–(D.45) for y2 and requiring y1 to be k independent,

we obtaina+ d = f + l,

bc+ fl = ad+ gh,

16x22(bc− ad) = −(aµ+ ν(h− c)− fµ)2

− x22(a+ d+ f + l)2,

µx2(a+ d+ f + l) = 2x2(aµ− ν(c+ h) + fµ),

8x42 + 2µ2x2

2 = 8x22(al − bh− cg + df)

− x22(a+ d+ f + l)2,

+ (aµ+ ν(h− c)− fµ)2.

(D.84)

Degenerate case In this case µ = ν = 0 and (D.84) becomes

a+ d = f + l,

bc+ fl = ad+ gh,

16x22(bc− ad) = −x2

2(a+ d+ f + l)2,

x22(a+ d+ f + l)2 = 8x2

2(al − bh− cg + df)

− 8x42.

(D.85)


The solution is

H0 =

(0 0

0 0

),

Hl =

fb(F+2x22)

(a−d)2

F−2x224b

a+ d− f

,

Hr =

(a b

− (a−d)2

4bd

),

(D.86)

where F = −2√x2

2(d− a)(a+ d− 2f) + x42 + (d− a)(a + d− 2f). Then the band

structure is

xk =2b(a− d)2(a+ d) cos k

2b(a− d)2

− 2√−b2x2

2(a− d)4

2b(a− d)2,

yk =2b(a+ d)(a− d)2 cos k

2b(a− d)2

+2√−b2x2

2(a− d)4

2b(a− d)2.

(D.87)

Non-degenerate case: In this case µ = 1, ν = 0 and (D.84) becomes

a+ d = f + l,

bc+ fl = ad+ gh,

x22

(16(bc− ad) + (a+ d+ f + l)2

)+ (a− f)2 = 0,

x2(a− d+ f − l) = 0,

x22

(8(al − bh− cg + df)− (a+ d+ f + l)2 − 2

)+ (a− f)2 = 8x4

2.

(D.88)

The solution is

H0 =

(0 0

0 1

),

Hl =

f −G(a−f)2

(4x22+1)(a−f)4

16x22Ga

,

Hr =

a b

−(4x22+1)(a−f)2

16bx22f

,

(D.89)


where G = 2√b2x2

2 (x22 − (a− f)2) + b(a− f)2 − 2bx2

2. Then the band structure is

xk = − K

4bGx22(a− f)2

+ (a+ f) cos(k) +1

2,

yk =K

4bGx22(a− f)2

+ (a+ f) cos(k) +1

2,

(D.90)

where

K =

[bGx2

2(a− f)4

(4x2

2

(b2(a− f)4 − 4ibG(a− f) sin(k)

+ bG(1− 2(a− f)2

)+G2

)+ b2(a− f)4

− 2bG(a− f)2 cos(2k) +G2

)]1/2

.

Abnormal case In this case µ = 0, ν = 1 and (D.84) becomes

a+ d = f + l,

bc+ fl = ad+ gh,

(c− h)2 = −16x22(bc− ad)

− x22(a+ d+ f + l)2,

x2(c+ h) = 0,

(c− h)2 − 8x42 = x2

2(a+ d+ f + l)2

− 8x22(al − bh− cg + df).

(D.91)

The solution is

H0 =

(0 1

0 0

),

Hl =

(d− ix2

c4x22− x22

c

−c d+ ix2

),

Hr =

(d − c

4x22

c d

).

(D.92)


xk = 2d cos k −√c2x2

2 (c sin(k) + ix22) 2

cx22

,

yk = 2d cos k +

√c2x2

2 (c sin(k) + ix22) 2

cx22

.

(D.93)


D.3.4 Imaginary part of one band is flat

Using the same method as the case in which the real part of one band is flat, we can

also solve the case in which the imaginary part of one band is flat. To do so, we

require x2 (y2) to be k independent in Eqs. (D.42)–(D.45). The only possibility is

x2 = 0, y2 = (a+ d− f − l), and then following the same steps as the real part of one

band case we get the following results.

Degenerate case:

Hr =

(a b

(a−f)(g(a−d)+b(d−f))(b−g)2 d

),

Hl =

(f g

(a−f)(g(a−l)+b(l−f))(b−g)2 l

).

(D.94)


E1 =2 cos(k)(bf − ag)

b− g,

E2 =eik(b(a+ d− f)− dg)

b− g,

+e−ik(ag + bl − fg − gl)

b− g.

(D.95)

Abnormal case:

Hr =

(a b

0 d

), Hl =

(a g

0 l

). (D.96)

The band structure isE1 = 2a cos(k),

E2 = e−ik(l + de2ik

).

(D.97)

Non-degenerate case In this case, the solution is the same as the abnormal case, and

the band structure isE1 = 2a cos(k),

E2 = deik + e−ikl + 1.(D.98)


D.3.5 Modulus of a band is flat

Suppose x = reiθk in Eq. (D.24), then

y = eik(a+ d) + e−ik(f + l) + µ− reiθk ,

y =1

re−iθk

(eik(aµ− cν) + al − bh− cg + df

+ detHle−2ik + detHre

2ik + e−ik(fµ− hν)).

(D.99)

If we assume θk = mk, m 6= 0, then

ei(m−1)k(f + l) + ei(m+1)k(a+ d) + µeimk − re2imk

=1

r

(e−ik(fµ− hν)

+ al − bh− cg + df + eik(aµ− cν)

+ detHle−2ik + detHre

2ik).

(D.100)

By equating the same powers of eik in (D.100), we can show that, when m > 1 or

m < −1, the only possible solution for (D.100) is r = 0. Therefore, only when

m = ±1 do we have a non-trivial solution.

m=1 case

In this case, equating the coefficients of the same powers of eik on the two sides of

(D.100) we get

f + l =1

r(al − bh− cg + df) ,

µ =1

r(aµ− cν) ,

a+ d− r =1

rdetHr,

detHl = 0,

fµ− hν = 0.

(D.101)

Degenerate case: In this case µ = 0, ν = 0 and (D.101) gives the following solu-

tion:

Hr =

(a b

− (a−r)(r−d)b

d

),

Hl =

(f bf

a−rl(a−r)b

l

).

(D.102)



E1 = eikr,

E2 = e−ik(ae2ik + de2ik + f − e2ikr + l

).

(D.103)

Non-degenerate case: In this case ν = 0, µ = 1 and (D.101) gives

Hr =

(r 0

c d

),

Hl =

(0 0

h l

),

(D.104)

with band structureE1 = eikr,

E2 = deik + e−ikl + 1.(D.105)

Abnormal case: In this case µ = 0, ν = 1 and (D.101) gives

Hr =

(r b

0 d

),

Hl =

(0 g

0 l

),

(D.106)

with band structureE1 = eikr,

E2 = deik + e−ikl + 1.(D.107)

m = −1 case

Similar to the m = 1 case, by equating the same powers of eik in (D.100), we have

f + l − r =1

rdetHl,

1

r(fµ− hν) = µ,

a+ d =1

r(al − bh− cg + df) ,

detHr = 0,

aµ− cν = 0.

(D.108)


Degenerate case: In this case µ = 0, ν = 0 and (D.100) gives

Hr =

(a badb

d

),

Hl =

(f b(f−r)

aa(l−r)b

l

).

(D.109)


E1 = e−ikr,

E2 = e−ik(ae2ik + de2ik + f + l − r

).

(D.110)

Non-degenerate case: In this case ν = 0, µ = 1 and (D.100) gives the following

solution:

Hr =

(0 0

c d

),

Hl =

(r 0

h l

).

(D.111)

Then the band structure isE1 = e−ikr,

E2 = deik + e−ikl + 1.(D.112)

Abnormal case: In this case ν = 1, µ = 0 and (D.100) gives

Hr =

(0 b

0 d

),

Hl =

(r g

0 l

).

(D.113)

Then the band structure isE1 = e−ikr,

E2 = e−ik(l + de2ik

).

(D.114)

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