TOPOLOGICAL ANTIFERROMAGNETIC MAGNONS
A Thesis Presented to the Department of Theoretical and Applied Physics,
African University of Science and Technology, Abuja
In partial ful�lment of the requirements for the award.
MASTER OF SCIENCE DEGREE
By
JAMIU ADAM AJIBOLA
Supervised by
PROF. AURELIEN MANCHON
African University of Science and Technology
www.aust.edu.ng
P.M.B 681, Garki, Abuja F.C.T
Nigeria.
July, 2019
TOPOLOGICAL ANTIFERROMAGNETIC MAGNONS
By
ADAM JAMIU AJIBOLA
A THESIS APPROVED BY THE DEPARTMENT OF THEORETICAL AND
APPLIED PHYSICS
RECOMMENDED:
..................................................
Supervisor: PROF. AURELIEN MANCHON
..................................................
Head: Department of Theoretical Physics
APPROVED:
............................................................
Chief Academic O�cer (Prof. C. E. Chidume)
..................................................
Date
ABSTRACT
We have studied a model for a non-collinear but coplanar antiferromagnetic spin tex-
ture on a two dimensional kagome lattice structure in the presence of Dzyaloshinskii-
Moriya Interaction (DMI). We observed some interesting topological properties in
our system i.e. the presence of non-trivial edge state in the wave function. This
non-trivial edge state, which mainly surfaced in the presence of the DMI, showed
robustness against the external magnetic �eld and thus can be further studied to
see how important transport properties can be computed.
i
ACKNOWLEDGMENTS
My acknowledgement and appreciation goes to my amiable supervisor Professor Au-
relien Manchon, the group leader of the Spintronics research group at KAUST, for
his guidance during this research work and the entire personnel in the group who at
some points supported me. I look forward to continuing my PhD research under your
mentorship. My Masters degree was fully sponsored by the Pan-African Materials
Institutes and the African Development Bank, I acknowledge this support. I also
appreciate the entire academic and non-academic personnel, students and colleagues
of the African University of Science and Technology and the Nelson Mandela Insti-
tution for the love and experience shared. To my head of department Prof. Kenfack
Anatole, my ever supporting faculty, Dr. AbdulHakeem Bello and my mentor, Dr.
Collins A. Akosa thank you for your support. All softwares and hardwares used for
this research were licensed to King Abdullah University of Science and Technology.
Finally, much appreciation to my beloved wife, Hassanat and precious daughter Fa-
timah, for their patience, perseverance and understanding, I love you so much. MAY
ALMIGHTY GOD REWARD YOU ALL ACCORDINGLY. Above all, I give glory
to Almighty Allah for His Grace, Favour and Mercy throughout the journey of my
life so far. ALHAMDULILLAH.
The journey indeed has just begun . . .
ii
DEDICATION
Glory be to the endower of knowledge, we have no knowledge except what He has
taught us; verily the All-knower, the All-wise [Quran 2:32]. This project is
dedicated to my parents Mr. and Mrs. Jamiu.
iii
CONTENTS
Abstract i
Acknowledgments ii
Dedication iii
Table of Contents iv
1 Introduction 1
1.1 THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . 4
1.1.1 Linear spin-wave theory . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Holstein - Primako� Transformation . . . . . . . . . . . . . . 5
1.1.3 Topological Insulators and the Edge States . . . . . . . . . . . 7
2 LITERATURE REVIEW 9
3 METHODOLOGY 12
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The Kagome Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2.1 Without Dzyaloshinskii Moriya Interaction . . . . . . . . . . . 14
3.2.2 With Dzyaloshinskii Moriya Interaction . . . . . . . . . . . . . 16
3.3 Construction of ribbon . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 RESULT AND DISCUSSION 22
iv
LIST OF FIGURES
1.1 A wave packet of magnon moving from hot to cold is de�ected by
DMI. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Condition for Anomalous Hall E�ect . . . . . . . . . . . . . . . . . . 3
1.3 Schematic diagram of edge states of a semi-in�nite plane of a 2D
system for (a) 2D ordinary insulator and (b) topological insulator [10]. 7
2.1 Collective spins excitation. . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 1200-coplanar antiferromagnets spin texture in the kagome lattice . . 13
3.2 Kagome lattice with lattice vectors a1 and a2. Dzaloshinskii-Mariya
vectors are aligned to the lattice plane and are denoted by red dots:
along -z (+z) for a clockwise (anticlockwise) chirality: C-B-A (A-B-
C)[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Kagome lattice showing the hopping direction (red and blue lines) . . 19
3.4 Edges of kagome lattice for two di�erent terminations. (a) smooth
termination (b) rough termination. . . . . . . . . . . . . . . . . . . . 21
4.1 Magnon bulk bands energy dispersion. (a) Without any component
of the DMI, J =0.5. (b) With the out-of-plane component of the DMI,
J = 0.5, D⊥ = 0.1 (c) With out-of-plane and in-plane components of
the DMI, J = 0.5, D⊥ = 0.1, D‖ = −0.25. (d) With h = -0.01. . . . . 23
4.2 Edge magnon in kagome lattice for N = 5 nanoribbon widths. The
parameters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = −0.01 . . . . . . . 25
4.3 Trivial edge magnon in kagome lattice for (a) J = 0.5, D⊥ = 0.0, D‖ =
0.0, h = −0.0(b)J = 0.5, D⊥ = 0.0, D‖ = 0.0, h = −0.01 . . . . . . . . 26
4.4 Edge magnon in kagome lattice for varied external �eld. The param-
eters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = -0.04 . . . . . . . . . . 26
v
CHAPTER 1
INTRODUCTION
The last two decades have witnessed tremendous work in the study of collective
excitations of electron's spins called spin waves[1],[2] . Spin waves are disturbance of
the magnetic ordering that travel through the magnetic material. In the quantum
picture, the excitations can be described by magnons. Magnons are chargeless low-
energy collective excitations of localized neighboring spins, with a �xed amount of
energy and play the vital role of an elemental magnetic carrier in numerous insulating
magnets. Due to their intrinsic bosonic nature, magnons can form a macroscopic
coherent state and propagate spin information over several millimeters,[3] much
farther than spin-polarized conduction electrons in metals. With these interesting
and fascinating properties, there are possibilities of utilizing magnons as a carrier to
propagate information in a unit of Bohr magneton in spin-based devices, these have
attracted considerable attention [2],[4],[5],[6] over the last decade.
Antiferromagnetic materials (AFMs) are materials in which electron spin asso-
ciated to individual atom at particular lattice point are ordered antiparallel relative
to each other such that their net magnetization is equal to zero. This phenomenon
occurs below a reference temperature, known as the Neel temperature (TN) , above
this temperature, the material loses its order and behaves just like a paramagnetic
material. Among the properties that make AFMs a promising material for new
functionalities include:- absence of stray �elds, robustness against magnetic �eld
perturbation, ultrafast dynamics [7]. Looking at these properties, in addition to the
fascinating properties of magnons mentioned above, it is only natural to be funda-
mentally interested in understanding the basis of magnon transport in insulating
antiferromagnets.
Unlike the charge-based electronics device, which dissipates heat, spin-based elec-
tronics produces very low dissipation by replacing charge currents with spin currents
(currents magnetic moments). However, in conducting materials, the spin current
is transported by charge carriers that experience dissipation and thereby contribute
to Joule heating. So, to achieve less dissipative materials, one considers magnetic
1
insulators, in which magnon is used to transport spin current. Among the promising
physical mechanisms adjunct to spin transport, spin Hall e�ect [8], which is the gen-
eration of a pure spin current transverse to an injected charge current, is particularly
interesting. This mechanism enables the generation of pure spin current which can
be used in spin-based devices. When spin current is carried by magnons, rather than
electrons, magnon Hall e�ect can also emerge and give new functionalities [2].
In 1879, Edwin Hall discovered the Hall E�ect, a phenomenon that occurs when
a current-carrying conductor is placed in a magnetic �eld in a direction perpen-
dicular to that of the �ow of current. Due the Lorentz force exerted by the mag-
netic �eld on the �owing electrons, a transverse electric �eld is created across the
conductor. The quantum-mechanical version of the Hall e�ect i.e., the quantum
Hall e�ect, is observed in a two-dimensional system subjected to low temperatures
and strong magnetic �elds, in which the Hall conductance undergoes quantum Hall
transitions to take on the quantized values. It was established that ferromagnetic
conductors which have nonzero intrinsic magnetization exhibit anomalous Hall e�ect
which is not a consequence of the Lorentz forces but of spin-orbit coupling (SOC)[2]
and therefore has nonzero transverse conductivity without external magnetic �eld.
Anomalous Hall E�ect (AHE) also exists in a "quantized version", called the Quan-
tum Anomalous Hall E�ect (QAHE), where charge edge currents are quantized [9].
In an insulating magnetic material with SOC present, [Fig. (1.2)], novel proper-
ties emerges which include topological states and unconventional magnetic ordering.
The presence of SOC in such a material presents itself as Dzyaloshinskii Moriya
Interaction to nontrivial topological magnetic states. Several interesting works have
proposed the use of collective magnetic excitations as a tool for understanding topo-
logical magnetic materials, [10], [2].
The Dzyaloshinskii Moriya Interaction (DMI) is an antisymmetric interaction
induced by SOC and it is responsible for the AHE observed in ferromagnets. About
six decades ago, the presence of DMI was demonstrated in non-centrosymmetric
systems [11],[12]. This antisymmetric interaction has great impact in the dynamics of
spin system, as it favours noncollinear magnetic con�guration and, as a consequence,
gives rise to topological spin texture. Futhermore, the presence of DMI brings about
non-trivial topologies of magnons which has promising applications.
In this research work, we considered an antiferromagnetic kagome lattice because
of its non-collinear magnetic structure which displays topological features. Topol-
ogy has been a concept used in mathematics for about three centuries . Topological
2
properties are the properties of a system that are preserved under continuous defor-
mations. These properties depend on the entire structure of the system rather than
on the local features. The non-trivial topology of a material means that at the edges
- where the continuity of the material is broken - the topological invariant is broken
too. In the case of topological insulators, it implies the presence of edge states that
can transport spin and/or charge currents. Now, the concept has been extended
to magnonic materials, where the non-trivial topology of the magnon states of the
volume implies the existence of magnonic edge states[13]. In the presence of lon-
gitudinal temperature gradient, magnon edge current produces transverse thermal
current. This e�ect was �rst observed experimentally by Onose et al in a pyrochlore
ferromagnet [2], [Fig.(1.1)] and latter in kagome ferromagnet [4], where the DMI
acts, as a vector potential.
Figure 1.1: A wave packet of magnon moving from hot to cold is de�ected by DMI.
[2]
In this thesis, we aim to investigate the role of the topology of the magnon
wave function in 1200 coplanar AFM spin texture (kagome lattice) in the presence
of Heisenberg interaction, DMI and exchange �eld. To track the presence of edge
states which will serve as a basis to compute the spin Seebeck e�ect, a crucial
quantity that is relevant to experiments. Also, this thesis will serve as a basis to
future investigations on the transport properties of the magnonic edge states.
Figure 1.2: Condition for Anomalous Hall E�ect
3
1.1 THEORETICAL BACKGROUND
1.1.1 Linear spin-wave theory
The linear spin-wave theory (LSWT) enables us to study the physical properties of
magnetic ordering at non-zero temperature. It was �rst introduced by F. Bloch and
Slater independently and later developed by Holstein - Primako� using the second
quantization bosonic operators [14]. Here, we introduce the linear spin-wave analysis
based on the work of S. Toth and B Lake [15] as well as Mook et al [16], given the
spin Hamiltonian in the form:
H =1
2
∑ij
STi JijSj (1.1)
The Heisenberg exchange Hamiltonian above describes a system of interacting spins
in an insulating magnet and can be used to study the behavior of magnons. As shown
in the equation above, the sum goes over all nearest neighbors' pairs i and j on the
lattice sites. With the spin operator Si and Sj at the lattice sites i and j respectively,
and the magnetic interaction or coupling term between both spins is de�ned by Ji,j.
This coupling constant is positive for the case of an antiferromagnetic ordering.
By treating these quantum mechanical spins as classical vectors, one obtains the
classical ground state, which can be minimized to obtain the ground state energy.
Once this is achieved, the number of spins in the magnetic unit cell and their local
directions are known. Now, we sum the Hamiltonian over the entire M sites of the
magnetic unit cell, referred to by the indices i, j, and a sum over the N unit cells of
the magnetic volume, indexed by l, n.
H =1
2
M∑i,j
N∑l,n
STl,iJijSn,j (1.2)
To investigate the magnon band structure, one needs to express each magnetic mo-
ment from its local frame of reference onto a global frame of reference. So, Using
the Cartesian coordinate system, each spin coordinate system is localized, thus we
rotate each of the spins on to a global coordinate system (x,y, z) by introducing a
rotation matrix. This will also make it easy to transform from the spin Hamiltonian
to the desired bosonic Hamiltonian. The associated rotation matrix is,
Rn = (xn, yn, zn) (1.3)
4
The associated transformation matrix for the n spin operator in the j unit cell reads:
Sn,j = RnSn (1.4)
Here, Sn,j is de�ned in the global cartesian system (x, y, z) while Snj is de�ned in
the local coordinate system (xj, yj, zj).
1.1.2 Holstein - Primako� Transformation
To study how magnons behave in our antiferromagnetic kagome system using the
LSWT, we �rst make the magnon operator appear explicitly in our Hamiltonian.
This is usually done using the Holstein-Primako� transformation [14], that is, the
mapping of the spin operators to the bosonic annihilation and creation operators,
Eq.(1.5). The spin operator S can be written explicitly in terms of annihilation and
creation operators as shown below:
Sn,j =
√
Sn2
(an,j + a†n,j)
−i√
Sn2
(an,j − a†n,j)(S − a†n,jan,j)
(1.5)
With this an,j(a†n,j) represents the bosonic annihilation (creation) operators and
obeys the commutation relation [ai, a†j] = δi,j. Also the operator n = a†n,jan,j counts
the number of bosons (magnons) placed at site j and it is often called the number
operator. The spin operators in the global reference frame can be expressed as
follow:
Sn,j =(√
Sn2
(u∗nan,j + una†n,j) + zn(S − a†n,jan,j)
)(1.6)
Where: un = xj + iyj
Next, we obtain the magnon Hamiltonian in momentum space by taking the
Fourier transform of the magnon operator from the position space to momentum
space using the Bloch state of a single magnon given by:
a(†)n,j =
1√M
∑i,j
e(−)i(rn+tj)a(†)n,k (1.7)
With rn as the position of the nth basis and tj the vector to the jth spin in the
basis. By substituting Eq.(1.7) into the Hamiltonian equation, Eq.(1.2), one gets
the bilinear Hamiltonian as:
H =1
2
∑k
Ψ†kHkΨk (1.8)
5
with Ψ†k as the row vector of the bosonic operators given by:
Ψ†k = (a(†)k,1, ..., a
(†)k,N , a−k,1, ..., a−k,N) (1.9)
the non-hermitian Hamiltonian Hk of the linear spin-wave matrix contains the fol-
lowing submatrices:
Hk =
(Ak − C Bk
B†k ATk − C
)(1.10)
elements of the sub-matrices of matrix read [16],
(A)l,n =
√SlSm2
uTnJlmk u∗n
(B)l,n =
√SlSm2
uTnJlmk un
(C)l,n = δl,n∑p
SpzTl J
ln0 zl
This Hamiltonian can be diagonalized by using paraunitary Bogoliubov transforma-
tion which gives the magnetic normal modes bnk with corresponding bands energies
εnk [17], where n here is the band index.
H =1
2
∑k
(b†k b−k
)Ek
(bk
b†−k
)=∑k
∑n
εnk(b†nkbnk + 1
2
)(1.11)
The basic idea of the paraunitary Bogoliubov transformation is to introduce a set of
operators such that the Hamiltonian has only terms proportional to
(bk
b†−k
). These
operators are de�ned below in terms of the coe�cient matrix Tk. With
b†k =(b†1,k..., b
†N,k
)(1.12)
(bk
b†−k
)= Tk
(ψk
ψ†−k
)(1.13)
Ek = T †kHkTk = daig (ε1,k, ...εN,k, ε1,−k, ...εN,−k) (1.14)
This transformation arises from the need to ensure the bosonic commutation relation
is satis�ed, i.e
T †kσ3Tk = σ3
6
Tkσ3T†k = σ3
where ,σ3= daig (IN×N ,−IN×N) showing that the Hamiltonian consists of two copies
of the same eigenvectors.
1.1.3 Topological Insulators and the Edge States
Topological insulators (TI) are nonmagnetic materials that have a bulk band gap like
an ordinary insulator but have protected conducting gapless states on their edges or
surfaces. These insulators are in the same category of topological phase as Quantum
Hall systems. In topological insulators, time-reversal symmetry is assumed, so it
does not require a magnetic �eld to produce edge state in a 2D material. Edge
states consist of pairs of states where up-spin and down-spin counter propagate due
to the combination of spin-orbit interactions and time-reversal symmetry [10], [13].
The spin-orbit coupling behaves like a spin-dependent magnetic �eld and gives rise
to spin-dependent quantum Hall e�ect.
Figure 1.3: Schematic diagram of edge states of a semi-in�nite plane of a 2D
system for (a) 2D ordinary insulator and (b) topological insulator [10].
Edge states arise at the interface between two insulators with di�erent topological
orders. These states reveal the topological phase transition between trivial and non-
trivial insulators. The trivial and non-trivial insulators themselves are characterized
by a Z2 index, v, (v = 0→ ordinary insulator , v = 1→ topological insulator ).
Now, because the edge states occur at the topological phase boundary, they are
determined by the topology of the two insulators on each side of the interface. As
such, they are robust against non-magnetic impurities, provided the topology of
the two insulators is una�ected by the impurities. This is because Z2 topological
number cannot change continuously when nonmagnetic impurities are added, the TI
remains invariant [10]. Edge states have been experimentally observed in materials
7
like CdTe/HgTe [18]. The topological properties of a system can be determined from
its Hamiltonian. A way to determine these properties of the system is by considering
the wave function. These properties are robust against perturbations as mentioned
above. This indicates that one could obtain currents without dissipation (like in
quantum Hall e�ect or in superconductivity).
8
CHAPTER 2
LITERATURE REVIEW
Antiferromagnetic spintronics is an aspect of solid state physics that deals with spin
transport in antiferromagnets. In the past few decades, this �eld of study has wit-
nessed tremendous development [7]. Each electron in a material possesses intrinsic
spin which behaves like a small magnet and has a magnetic moment. Antiferro-
magnets are materials which have their magnetic moments align anti-parallel to the
neighboring moments and this occurs below a certain temperature called the Neel
temperature (TN). Above this temperature, the material loses this alignment due
to thermal agitation. Despite having it net magnetization to be zero, it is di�cult
to harness for practical applications, antiferromagnets possess quite a number of
interesting properties, such as the absence of stray �elds, robust against magnetic
�elds, ultrafast dynamics; these properties and more make it a suitable candidate
for novel applications.
Generally, in a spin system, the total magnetic exchange interaction between
pair of spins have contributions from the symmetric and antisymmetric exchange
interactions. The symmetric exchange, also known as the Heisenberg exchange,
promotes collinear alignment between pairs of spins, while antisymmetric exchange
interactions favor an orthogonal orientation. This exchange is called Dzyaloshinskii-
Moriya interaction and arises in non-centrosymmetric environment in the presence of
spin-orbit coupling [19]. Both symmetric and antisymmetric exchange interactions
could be anisotropic.
In the classical picture of the ground state of a simple insulating ferromagnet
with all spins aligning parallel, a possible excitation occur when one spin is re-
versed. Also the spins can share the reversal as shown in the [Fig.(2.1)] below [20].
The elementary collective excitation of the magnetic system has a wave-like form
which is called magnon. Magnon is a quantum chargeless quasiparticle correspond-
ing to an elementary increase of energy and an elementary reduction of the total
magnetization. Just as phonon is a quantum elastic wave and photon a quantum of
electromagnetic wave, magnon is a quantum spin-wave.
9
Figure 2.1: Collective spins excitation.
There have been several experimental and theoretical reports on the AHE caused
by collective spin excitations (magnons), commonly called magnon Hall E�ect (MHE),
in both insulating ferromagnets and antiferromagnets. Experimental result for an
insulating ferromagnets with pyrochlore lattice structure showed transverse heat
current when a longitudinal temperature gradient was applied [2]. This transverse
current is associated to Hall E�ect [21]. Using both the semi-classical approach
and the linear response theory, Matsumoto and Murakami predicted a new form of
topological insulator, inspired from Onose's experiment [1]. They explained that the
MHE is due to non-compensated magnon motion along the boundary of the sample
(magnon edge current) and that this net current is a consequence of the topology of
the system, hence, the existence of topological magnon insulator [9]. The presence
of spin-orbit coupling which manifests as DMI is responsible for nontrivial topology
in kagome ferromagnets [21, 4] and honeycomb lattice [6] systems.
In a 2D like Dirac Hamiltonian, time reversal symmetry de�ned with sublattice
pseudo spins is preserved and in a magnon system, the Dirac points are robust
against magnon-magnon interaction [5] . The Dirac point is the crossing point of
the linear energy dispersion curves. Opening a gap at the Dirac points lead to
magnon edge states as the case of Haldane electronic system [22]. It was reported
that Mn3Sn a hexagonal antiferromagnet that exhibits non-collinear ordering at
the TN , shows large anomalous Hall e�ect [23] .i.e a phenomenon initially associated
with ferromagnets. This empirical evidence has since open a new page for the intense
research in the �eld of antiferromagnetic spintronics.
The AHE observed in ferromagnetic materials �nds its origin from the spin-orbit
10
coupling which favors certain scattering direction for up-spins and down-spins, each
moving on opposite edge. For a ferromagnet there is the uneven spin population at
these opposite edges, thereby setting up a transverse electric �eld in the absence of
external magnetic �eld.
11
CHAPTER 3
METHODOLOGY
3.1 Introduction
In this section, we describe how magnons propagate on 1200-coplanar antiferromag-
nets spin texture in kagome lattice in the presence of Heisenberg interaction, DMI
and external �eld. The quantum mechanical Heisenberg model Hamiltonian consid-
ered is
H = HH +HDM +Hext (3.1)
where,
HH = −∑i 6=j
J ijSi · Sj (3.2)
HDM = −∑i 6=j
Dij · Si × Sj (3.3)
Hext = −gµB∑i 6=j
H · Si (3.4)
With the Heisenberg interaction J ij = J ji being symmetric and the DMI Dij =
−Dji constrained by symmetry. In this system, we are considering both the in-plane
and out-of-plane components of the DMI given by: Dij = νij (D‖nxij, D‖n
zij, D⊥)
12
3.2 The Kagome Lattice
Figure 3.1: 1200-coplanar antiferromagnets spin texture in the kagome lattice
The kagome lattice has three atoms in its unit cell which are arranged at the corners
of an equilateral triangle with the side length equal half of the lattice constant (�gure
3.1). The lattice vectors in the Cartesian coordinate are
a1 = a2(x+
√3y)
a2 = a2(−x+
√3y)
13
Figure 3.2: Kagome lattice with lattice vectors a1 and a2. Dzaloshinskii-Mariya
vectors are aligned to the lattice plane and are denoted by red dots: along -z (+z)
for a clockwise (anticlockwise) chirality: C-B-A (A-B-C)[21]
Now we apply the linear spin-wave method discussed in chapter one to solve the
Hamiltonian of our system. In this system, each atom is surrounded by four nearest
neighbors. We started by performing the calculation for the Hamiltonian without
the DMI.
3.2.1 Without Dzyaloshinskii Moriya Interaction
Let H1, H2 and H3 be the Hamiltonian of each of the three atoms with their nearest
neighbors, and HH be the total Hamiltonian of the unit cell.
H1 =1
2J [SA · (SB1 + SB2 + SC1 + SC2)] (3.5)
H2 =1
2J [SB · (SA1 + SA2 + SC1 + SC2)] (3.6)
H3 =1
2J [SC · (SA1 + SA2 + SB1 + SB2)] (3.7)
HH = H1 +H2 +H3 (3.8)
We have to rotate the local coordinate system (xj, yj, zj) of each spin onto the
global coordinate system (x, y, z) to perform the Holstein-Primako�'s transforma-
tion. To do so we de�ned a local rotation matrix for each spin as: Rn = (xn, yn, zn).
14
In the x - z plane, we have:-
R =
−1
20 −
√3
2
0 1 0√
32
0 −12
(3.9)
Which gives the transformation as: Sn = RSn,j.In the global reference frame of the Cartesian coordinate, the Holstein Primako�'stransformation for each spin is:-
Sj,n =
√Sn2
(aj,n + a†j,n)
−i√Sn2
(aj,n − a†j,n)
(S − a†j,naj,n)
(3.10)
SA =
(√Sn
2(aj,A + a†j,A),−i
√Sn
2(aj,A − a†j,A), (S − a†j,Aaj,A)
)(3.11)
SB =
(−
1
2
√Sn
2(aC,A + a†B,j) +
√3
2(S − a†B,jaB,j),−i
√Sn
2(aB,j − a†B,j),−
√3
2
√Sn
2(aC,A + a†B,j)−
1
2(S − a†B,jaB,j
)(3.12)
SC =
(−
1
2
√Sn
2(aC,A + a†C,j) +
√3
2(S − a†C,jaC,j),−i
√Sn
2(aC,j − a†C,j),−
√3
2
√Sn
2(aC,A + a†C,j)−
1
2(S − a†C,jaC,j)
)(3.13)
Now inserting the Holstein Primako�'s transformation described in Eq.(3.11) �
(3.13) in the Hamiltonian and ignoring magnon-magnon interaction, i.e. only terms
in the quadratic form are considered, we have;
HH =3
2JZNS2 −
3JS
8
A,B,C∑α6=β
[a†α,ja†β,1 + a†α,ja
†β,2]−
3JS
8
A,B,C∑α6=β
[aα,jaβ,1 + aα,jaβ,2]
+JS
8
A,B,C∑α 6=β
[a†α,jaβ,1 + a†α,jaβ,2] +JS
8
A,B,C∑α6=β
[aα,ja†β,1 + aα,ja
†β,2] +
2JS
8
A,B,C∑α6=β
[a†α,jaα,j ]
Next, we perform the Fourier transformation of the operators to diagonalize the
Hamiltonian. We choose the vectors k to lie in the Brillouin zone associated with
each sub-lattice.
The Fourier transform is of the form:
aα,j =1√L
∑k
ak,αeik·(rα+rj)
15
a†α,j =1√L
∑k
a†k,αe−ik·(rα+rj) (3.14)
Where rj is the position of the jth basis and rα the vector of the spin in the α th
basis,
α, β = A,B,C, j = 1, 2...L
δαβ = rβ − rα (3.15)
We now evaluate each term present in the Hamiltonian, i.e.∑j
a†α,ja†β,j =
1
L
∑j
∑k
∑k′
a†k,αe−ik·(rα+rj)a†k′,βe
−ik′·(rβ+rj) (3.16)
=∑k
∑k′
δk′,ka†k′,βe
−i(k′rβ+krα) (3.17)
=∑k
a†k,αa†−k,βe
−ik·(rβ−rα) (3.18)
=∑k
a†k,αa†−k,βe
−ikδα,β (3.19)
By performing similar transformation to other components in the Hamiltonian,(shown in appendix). The Hamiltonian becomes:
HH = −3
2JZNS2 +
3JZS
4
A,B,C∑k,α6=β
[ak,αak,β + a†k,αa†k,β ]γk −
JZS
4
A,B,C∑k,α6=β
[a†k,αak,β + ak,αa†−k,β ]γk − 2JS
A,B,C∑k,α6=β
a†k,βak,α
(3.20)
With γk = 2z
∑δα,β
cos(k·δα,β) and the operators satisfying the commutation relation
[ak, a†k′ ] = δk′,k.
This Hamiltonian has the general form as shown in Eq.(3.20) detailed in appendix
B
HH = −3
2JZNS2 +
s
2
∑k
Ψ†HKHΨk (3.21)
Ψ†k = (a†A,k, a†B,k, a
†C,k, a
†A,−k, a
†B,−k, a
†C,−k) (3.22)
3.2.2 With Dzyaloshinskii Moriya Interaction
By adding the DMI terms derived in the appendix, we obtained the bilinear Hamil-
tonian in Eq.(3.27). Here we let H ′1, H′2 and H ′3 be the Hamiltonian of each of the
three atoms with their nearest neighbors, and HDM be the Hamiltonian contribution
from DMI.
H′1 =1
2[SA × (SB1 + SB2 + SC1 + SC2)] (3.23)
16
H′2 =1
2[SB × (SA1 + SA2 + SC1 + SC2)] (3.24)
H′3 =1
2[SC × (SA1 + SA2 + SB1 + SB2)] (3.25)
HDM = Dij · (H′1 +H′2 +H′3) (3.26)
H = H0 +S~2
2
∑k
Ψ†Hkψk (3.27)
We �nally obtain:
Hk =
(Ak − C BkB†k A†−k − C
)(3.28)
And the matrix representation has the form:
Ak =
0 (j − iD‖(cosθ +√
3sinθ))cAB (j + iD‖(cosθ −√
3sinθ))cAC
(j + iD‖(cosθ +√
3sinθ))cAB 0 (j + 2iD‖cosθ)cBC
(j − iD‖(cosθ −√
3sinθ))cAC (j − 2iD‖cosθ)cBC 0
(3.29)
A−k =
0 (j + iD‖(cosθ +√
3sinθ))cAB (j − iD‖(cosθ −√
3sinθ))cAC
(j − iD‖(cosθ +√
3sinθ))cAB 0 (j − 2iD‖cosθ)cBC
(j + iD‖(cosθ +√
3sinθ))cAC (j + 2iD‖cosθ)cBC 0
(3.30)
Bk =
0 (j3 − iD‖(3cosθ −√
3sinθ))cAB (j3 + iD‖(cosθ +√
3sinθ))cAC
(j3 − iD‖(cosθ −√
3sinθ))cAB 0 (j3 − 2√
3iD‖sinθ)cBC
(j3 + iD‖(cosθ +√
3sinθ))cAC (j3 − 2√
3iD‖sinθ)cBC 0
(3.31)
B†k =
0 (j3 + iD‖(3cosθ −√
3sinθ))cAB (j3 − iD‖(cosθ +√
3sinθ))cAC
(j3 + iD‖(cosθ −√
3sinθ))cAB 0 (j3 + 2√
3iD‖sinθ)cBC
(j3 − iD‖(cosθ +√
3sinθ))cAC (j3 + 2√
3iD‖sinθ)cBC 0
(3.32)
C =
j0 0 0
0 j0 0
0 0 j0
(3.33)
with
j0 = −4(√
3D⊥ + J)
j = (√
3D⊥ − J)
j3 = (√
3D⊥ − 3J) (3.34)
δAC = (1
2, 0)
17
δAB = (1
2,
√3
2)
δBC = (−1
2,−√
3
2) (3.35)
cαβ = cos(k·δαβ) (3.36)
To diagonalize this Hamiltonian, we introduce the paraunitary Bogoliubov trans-
formation as discussed in section (1.3) in chapter one.
Hk =1
2
∑k
(b†k b−k
)εk
(bk
b†−k
)=∑k
∑n
εnk(b†kb−k +
1
2) (3.37)
With b†k = (b†1,k, ..., b†N,k) (
bk
b†−k
)= T−1
k
(Ψk
Ψ†−k
)εk = T †kHkTk = diag(ε1,k, ..., εN,k, ε1,−k, ..., εn,−k)
This canonical transformation arises from the need to ensure the bosonic commu-
tation relation is satis�ed. This Hamiltonian diagonalization was done numerically
in the Bogoliubov transformation basis.
3.3 Construction of ribbon
In order to track the presence of edge states, we constructed a model for the nanorib-
bon from the bulk Hamiltonian. To build this model, we rewrote the cosine terms
in exponential form. The cosine comes from the assumption that the wave functions
are Bloch states. In a ribbon, the wave functions are Bloch states along the ribbon
direction, but are quantized along the transverse direction. Therefore, one manner
to represent this symmetry lowering is to write the cosine functions into exponen-
tial form, retaining only the exponential factors along the ribbon direction, while
discarding the exponents along the transverse direction. The ribbon was consid-
ered to be in�nite along the x-direction and �nite along the y-direction. The green
lines show the chain direction, the red and blue lines show the hopping along the
transverse direction [Fig.(3.3)]. We as well considered two di�erent types of edges
18
truncations [Fig.(3.4)] for several numbers of ribbons, N, ranging from N = 5, N =
30 to N = 80.
Figure 3.3: Kagome lattice showing the hopping direction (red and blue lines)
The expansion of the cosine functions of interests reads
cAB = cos
(−1
2kx +
√3
2ky
)=
1
2
[exp−
12ikxa expi
√3
2kya + exp−−
12ikxa exp−i
√3
2kya]
cBC = cos
(−1
2kx −
√3
2ky
)=
1
2
[exp−
12ikxa exp−i
√3
2kya + exp−−
12ikxa exp−−i
√3
2kya]
(3.38)
which allows us to write the bulk Hamiltonian in the form:
H = Hchain + τ expi√
32kxa +τ † exp−i
√3
2kxa (3.39)
In other words, the bulk Hamiltonian is separated into chains of along x, represented
byHchain, and hopping matrices along y, represented by τ . The expression of Hchain
and τ are given explicitly in the next
a. Ribbon Hamiltonian
19
HChain
=1 2
2J
0(J−iS
1)ex
p−
ikxa
42(J
+iS
2)cos(kxa
2)
0(J
3−iS
4)ex
p−
ikxa
42(J
3+iS
3)cos(kxa
2)
(J+iS
1)ex
pikxa
42J
0(J
+2D‖icosθ
)ex
p−
ikxa
4(J
3−iS
4)ex
pikxa
40
(J3−iD‖2√
3sinθ)ex
p−
ikxa
4
2(J−iS
2)cos(kxa
2)
(J−
2D‖icosθ
)ex
pikxa
42J
02(J
3+iS
3)cos(kxa
2)
(J3−iD‖2√
3sinθ)ex
pikxa
40
0(J
3+iS
4)ex
p−
ikxa
42(J
3−iS
3)cos(kxa
2)
2J
0(J
+iS
1)ex
p−
ikxa
42(J−iS
2)cos(kxa
2)
(J3
+iS
4)ex
pikxa
40
(J3
+iD‖2√
3sinθ)ex
p−
ikxa
4(J−iS
1)ex
pikxa
42J
0(J−
2D‖icosθ
)ex
p−
ikxa
4
2(J
3−iS
3)cos(kxa
2)
(J3
+iD‖2√
3sinθ)ex
pikxa
40
2(J
+iS
2)cos(kxa
2)
(J+
2D‖icosθ
)ex
pikxa
42J
0
(3.40)
b.Hoppingmatrix
τ=
1 2
0(J
+iS
1)ex
pikxa
40
0(J
3−iS
4)ex
pikxa
40
00
00
00
0(J−
2D‖icosθ
)ex
p−
ikxa
40
0(J
3+iD‖2√
3sinθ)ex
p−
ikxa
40
0(J
3+iS
4)ex
pikxa
40
0(J
+iS
1)ex
pikxa
40
00
00
00
0(J
3+iD‖2√
3sinθ)ex
p−
ikxa
40
0(J−
2D‖icosθ
)ex
p−
ikxa
40
(3.41)
20
Figure 3.4: Edges of kagome lattice for two di�erent terminations. (a) smooth
termination (b) rough termination.
c. The ribbon system
For the Hamiltonian of individual chains, Hchain, and the Hamiltonian of transverse
hopping, τ , we can reconstruct the Hamiltonian of the nanoribbon. For instance,
the Hamiltonian of a nanoribbon with N=5 chains, reads
H5 =
Hbdary τ 0 0 0
τ † Hchain τ 0 0
0 τ † Hchain τ 0
0 0 τ † Hchain τ
0 0 0 τ † Hbdary
(3.42)
21
CHAPTER 4
RESULT AND DISCUSSION
Let us �rst study the properties of the bulk antiferromagnet. We observed that when
the DMI components are added to the system, the band's degeneracy is lifted and
a bandgap is seen between the lowest band and the top bands. The band structure
showed particle-hole symmetry, there's one �at band and the dispersion is linear
close to kx = 0.5 Fig. (4.1a). When the out-of-plane DMI is turned on Fig. (4.1b),
the band degeneracy is lifted although no band gap due to touching points on the
bands. The in-plane DMI opens up gap at the touching points between the top bands
and the lowest band Fig. (4.1c). Although, the linear dispersion close to kx = 0.5
is maintained, it seems like a reminiscence of the linear dispersion obtained for a
collinear antiferromagnetic chain without anisotropy. But, the addition of external
�eld to the system, eliminates the Dirac cpoint at kx = 0.5. Thus, the coexistence
of DMI and magnetic gap Fig. (4.1d) suggests that topological edge states might
arise in this system. So, we now move on to the ribbon case in order to observe the
emergence of such states.
22
Figure 4.1: Magnon bulk bands energy dispersion. (a) Without any component of
the DMI, J =0.5. (b) With the out-of-plane component of the DMI, J = 0.5, D⊥ =
0.1 (c) With out-of-plane and in-plane components of the DMI, J = 0.5, D⊥ =
0.1, D‖ = −0.25. (d) With h = -0.01.
Let us now turn on the DMI, as depicted in Fig.4.2 for various nanoribbon widths.
Figures 4.2(a), (c) and (e) represent the one-dimensional band structure for non-zero
external �eld and out-of-plane DMI, while keep the in-plane DMI zero. For a narrow
ribbon [N=5, Fig. 4.2(a)], the band structure is gapped. This gap is systematically
quenched upon increasing the width as shown in Figs. 4.2(c) and 4.2(e). This band
structure is comparable to the one computed in Fig. (4.1c).
We now turn on the in-plane DMI. Again, a central gap is observed for the
23
narrow nanoribbon [N=5, Fig. 4.2(b)], which is progressively closed upon increasing
the ribbon width [Figs. 4.2(d) and 4.2(f)]. The remarkable feature is observed at
positive energy, around E = 2.5t. There, a bulk gap opens upon increasing the ribbon
width, as already observed in Fig. 4.1(d). However, while Fig. 4.1(d) exhibits a
bulk gap, Fig. 4.2(f) reveals the emergence of single gap state which corresponds to
the topological magnon edge state.
24
Figure 4.2: Edge magnon in kagome lattice for N = 5 nanoribbon widths. The
parameters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = −0.01
25
Figure 4.3: Trivial edge magnon in kagome lattice for (a) J = 0.5, D⊥ = 0.0, D‖ =
0.0, h = −0.0(b)J = 0.5, D⊥ = 0.0, D‖ = 0.0, h = −0.01
Figure 4.4: Edge magnon in kagome lattice for varied external �eld. The parame-
ters are J = 0.5, D⊥ = 0.1, D‖ = −0.25, h = -0.04
In conclusion, we have studied a model for a non-collinear and coplanar antiferro-
magnetic spin texture on a two dimensional kagome lattice. We observed topological
properties in our system i.e. the presence of non-trivial edge states in the energy
spectrum. This edge state showed robustness against the external magnetic �eld
[Fig.(4.4)]. For further study, we hope to use the linear response Kubo formula to
compute the spin Hall and spin thermal Hall conductivities, two important proper-
ties relevant to the experiment.
26
APPENDIX
Explicit expression of the magnon Hamiltonian given
in Eq. (3.20).
HH = H1 +H2 +H3
HH = −JS
8
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
(3aA1aB1 + 3aB1aaA1 + 3aA1aB2 + 3aB1aA2 + 3aA1aC1 + 3aC1aA1 + 3aA1aC2 + 3aC1aA2)
+(3a†A1a†B1 + 3a†B1a
†A1 + 3a†A1a
†B2 + 3a†B1a
†A2 + 3a†A1a
†C1 + 3a†C1a
†A1 + 3a†A1a
†C2 + 3a†C1a
†A2)
+(3aB1aC1 + 3aC1aB1 + 3aB1aC2 + 3aC1aB2 + 3a†B1a†C1 + 3a†C1a
†B1 + 3a†B1a
†C2 + 3a†C1a
†B2)
+(−a†A1aB1 − a†B1aA1 − a†A1aB2 − a†B1aA2 − a†A1aC1 − a†C1aA1 − a†A1aC2 − a†C1aA2)
+(−aA1a†B1 − aB1a
†A1 − aA1a
†B2 − aB1a
†A2 − aA1a
†C1 − aC1a
†A1 − aA1a
†C2 − aC1a
†A2)
+(a†B1aC1 − a†C1aB1 − a†B1aC2 − a†C1aB2 − aB1a†C1 − aC1a
†B1 − aB1a
†C2 − aC1a
†B2)
−2(−12S + 6aA1aA1 + 2aA2aA2 + 6aB1aB1 + 2aB2aB2 + 6aC1aC1 + 2aC2aC2)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣(4.1)
by taking 2cαβ = γk, and writing HH in matrix form gives:
HH = −3
2JZNS2 +
s
2
∑k
Ψ†HkHΨk (4.2)
HkH = J
−4 −cAB −cAC 0 3cAB 3cAC
−cAB −4 −cBC 3cAB 0 3cAC
−cAC −cBC −4 3cAB 3cBC 0
0 3cAB 3cAC −4 −cAB −cAC3cAB 0 3cAC −cAB −4 −cBC3cAB 3cBC 0 −cAC −cBC −4
(4.3)
The matrix is given as:
HkH =
(AkH − CH BkHB†kH A†−kH − CH
)(4.4)
AkH = J
0 −cAB −cAC−cAB 0 −cBC−cAC −cBC 0
(4.5)
BkH = J
0 3cAB 3cAC
3cAB 0 3cBC
3cAC 3cBC 0
(4.6)
CH = J
−4 0 0
0 −4 0
0 0 −4
(4.7)
27
Explicit expression of the magnon Hamiltonian given
in Eq. (3.27).
HDM = Dij · (H′1 +H′2 +H′3) (4.8)
Recall the expressions for SA, SB and SC from Eq. (3.11) - (3.13)
SA×SB =
∣∣∣∣∣∣∣∣x y z√
Sn2
(aA,j + a†A,j) −i√Sn2
(aA,j − a†A,j) (S − a†A,jaA,j)
− 12
√Sn2
(aB,j + a†B,j) +√
32
(S − a†B,jaB,j) −i√Sn2
(aB,j − a†B,j) −√
32
√Sn2
(aB,j + a†B,j)−12
(S − a†B,jaB,j)
∣∣∣∣∣∣∣∣(4.9)
SzASxB − S
xAS
zB =
√3
4S(a†A,ja
†B,j + aA,jaB,j + aA,ja
†B,j + a†A,jaB,j) +
√3
4S(2S − 2a†A,jaA,j − 2a†B,jaB,j)y (4.10)
SyASzB − S
zAS
yB = i
√3
4S(aA,jaB,j − a†A,ja
†B,j + aA,ja
†B,j − a
†A,jaB,j)x (4.11)
SxASyB − S
yAS
xB = −i
S
4(3aA,jaB,j − 3a†A,ja
†B,j − aA,ja
†B,j + a†A,jaB,j)z (4.12)
SA×SC =
∣∣∣∣∣∣∣∣x y z√
Sn2
(aA,j + a†A,j) −i√Sn2
(aA,j − a†A,j) (S − a†A,jaA,j)
− 12
√Sn2
(aC,j + a†C,j)−√
32
(S − a†C,jaC,j) −i√Sn2
(aC,j − a†C,j)√
32
√Sn2
(aC,j + a†C,j)−12
(S − a†C,jaC,j)
∣∣∣∣∣∣∣∣(4.13)
SzASxC − S
xAS
zC =
√3
4S(a†A,ja
†C,j + aA,jaC,j + aA,ja
†C,j + a†A,jaC,j) +
√3
4S(2S − 2a†A,jaA,j − 2a†C,jaC,j)y (4.14)
SyASzC − S
zAS
yC = i
√3
4S(aA,jaC,j − a†A,ja
†C,j + aA,ja
†C,j − a
†A,jaC,j)x (4.15)
SxASyC − S
yAS
xC = i
S
4(3aA,jaC,j − 3a†A,ja
†C,j − aA,ja
†C,j + a†A,jaC,j)z (4.16)
SA×SC =
∣∣∣∣∣∣∣∣x y z
− 12
√Sn2
(aB,j + a†B,j) +√
32
(S − a†B,jaB,j) −i√Sn2
(aB,j − a†B,j) −√
32
√Sn2
(aB,j + a†B,j)−12
(S − a†B,jaB,j)
− 12
√Sn2
(aC,j + a†C,j)−√
32
(S − a†C,jaC,j) −i√Sn2
(aC,j − a†C,j)√
32
√Sn2
(aC,j + a†C,j)−12
(S − a†C,jaC,j)
∣∣∣∣∣∣∣∣(4.17)
SzBSxC − S
xBS
zC =
√3
4S(a†B,ja
†C,j + aB,jaC,j + aB,ja
†C,j + a†B,jaC,j) +
√3
4S(S − a†B,jaB,j − a
†C,jaC,j)y (4.18)
SyBSzC − S
zBS
yC = i
√3
4S(2aB,jaC,j − 2a†B,ja
†C,j)x (4.19)
SxBSyC − S
yBS
xC = i
S
4(2aB,jaC,j − 2a†B,ja
†C,j)z (4.20)
The Fourier transform for each component is of the form:
SySz − SzSy = i
√3
2S
(ak,Aa−k,B − a†k,Aa
†−k,B + ak,Aa
†k,B − a
†k,Aak,B)cAB
+(ak,Aa−k,C − a†k,Aa†−k,C + ak,Aa
†k,C − a
†k,Aak,C)cAC
−(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC
x (4.21)
SxSy − SySx = iS
2
−(3ak,Aa−k,B − 3a†k,Aa
†−k,B − ak,Aa
†k,B + a†k,Aak,B)cAB
+(3ak,Aa−k,C − 3a†k,Aa†−k,C − ak,Aa
†k,C + a†k,Aak,C)cAC
+(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC
z (4.22)
SzSx − SxSz =
√3
2S
(ak,Aa−k,B + a†k,Aa
†−k,B + ak,Aa
†k,B + a†k,Aak,B)cAB
+(ak,Aa−k,C + a†k,Aa†−k,C + ak,Aa
†k,C + a†k,Aak,C)cAC
+(ak,Ba−k,C + a†k,Ba†−k,C + ak,Ba
†k,C + a†k,Bak,C)cBC
+(6S − 4a†k,Aak,A − 4a†k,Bak,B)cBC − 4a†k,Cak,C
y (4.23)
28
Explicit expression of the DMI given in Eq. (3.27).
Dij = vij(D‖nxij , D‖n
zij , D⊥)
(nxij , nzij) = (sinθx, cosθz) (4.24)
Where θ is the angle between the spin vectors at i and j, depending on the orien-tation of the bond i, j, the signs of the DMI components changes.
HDM = D‖nxij · i√
3
2S
(ak,Aa−k,B − a†k,Aa
†−k,B + ak,Aa
†k,B − a
†k,Aak,B)cAB
+(ak,Aa−k,C − a†k,Aa†−k,C + ak,Aa
†k,C − a
†k,Aak,C)cAC
−(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC
x
−D‖nzij · iS
2
−(3ak,Aa−k,B − 3a†k,Aa
†−k,B − ak,Aa
†k,B + a†k,Aak,B)cAB
+(3ak,Aa−k,C − 3a†k,Aa†−k,C − ak,Aa
†k,C + a†k,Aak,C)cAC
+(2ak,Ba−k,C − 2a†k,Ba†−k,C)cBC
z
+D⊥ ·√
3
2S
(ak,Aa−k,B + a†k,Aa
†−k,B + ak,Aa
†k,B + a†k,Aak,B)cAB
+(ak,Aa−k,C + a†k,Aa†−k,C + ak,Aa
†k,C + a†k,Aak,C)cAC
+(ak,Ba−k,C + a†k,Ba†−k,C + ak,Ba
†k,C + a†k,Bak,C)cBC
+(6S − 4a†k,Aak,A − 4a†k,Bak,B)cBC − 4a†k,Cak,C
y
AkDM =
0 (√
3D⊥ − iD‖(cosθ +√
3sinθ))cAB (√
3D⊥ + iD‖(cosθ −√
3sinθ))cAC
(√
3D⊥ + iD‖(cosθ +√
3sinθ))cAB 0 (√
3D⊥ + 2iD‖cosθ)cBC
(√
3D⊥ − iD‖(cosθ −√
3sinθ))cAC (√
3D⊥ − 2iD‖cosθ)cBC 0
(4.25)
A−kDM =
0 (√
3D⊥ + iD‖(cosθ +√
3sinθ))cAB (√
3D⊥ − iD‖(cosθ −√
3sinθ))cAC
(√
3D⊥ − iD‖(cosθ +√
3sinθ))cAB 0 (√
3D⊥ − 2iD‖cosθ)cBC
(√
3D⊥ + iD‖(cosθ +√
3sinθ))cAC (√
3D⊥ + 2iD‖cosθ)cBC 0
(4.26)
BkDM =
0 (√
3D⊥ − iD‖(3cosθ −√
3sinθ))cAB (√
3D⊥ + iD‖(cosθ +√
3sinθ))cAC
(√
3D⊥ − iD‖(cosθ −√
3sinθ))cAB 0 (√
3D⊥ − 2√
3iD‖sinθ)cBC
(√
3D⊥ + iD‖(cosθ +√
3sinθ))cAC (√
3D⊥ − 2√
3iD‖sinθ)cBC 0
(4.27)
B†kDM =
0 (√
3D⊥ + iD‖(3cosθ −√
3sinθ))cAB (√
3D⊥ − iD‖(cosθ +√
3sinθ))cAC
(√
3D⊥ + iD‖(cosθ −√
3sinθ))cAB 0 (√
3D⊥ + 2√
3iD‖sinθ)cBC
(√
3D⊥ − iD‖(cosθ +√
3sinθ))cAC (√
3D⊥ + 2√
3iD‖sinθ)cBC 0
(4.28)
CDM =
4√
3D⊥ 0 0
0 4√
3D⊥ 0
0 0 4√
3D⊥
(4.29)
HkDM =
(AkDM − CDM BkDMB†kDM A†−kDM − CDM
)(4.30)
which gives:-HDM =
S
2
∑k
Ψ†HkDMψk (4.31)
Adding Eq.(4.2) and Eq.(4.31) gives the bilinear bulk Hamiltonian in Eq.(3.27),
where H0 = 32JzNS2.
29
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