Hyperbolic Lattices in Circuit Quantum Electrodynamics

Alicia J. Kollar,1, 2 Mattias Fitzpatrick,1 and Andrew A. Houck1

1Department of Electrical Engineering, Princeton University, Princeton, NJ 08540, USA2Princeton Center for Complex Materials, Princeton University, Princeton, NJ 08540, USA

(Dated: February 28, 2018)

After close to two decades of research and development, superconducting circuits have emerged asa rich platform for both quantum computation and quantum simulation. Lattices of superconductingcoplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwavephotons, where weak interactions can be introduced either via non-linear resonator materials orstrong interactions via qubit-resonator coupling. Here, we introduce a technique using networks ofCPW resonators to create a new class of materials which constitute regular lattices in an effectivehyperbolic space with constant negative curvature. We show numerical simulations of a class ofhyperbolic analogs of the kagome lattice which show unusual densities of states with a spectrally-isolated degenerate flat band. We also present an experimental realization of one of these lattices,exhibiting the aforementioned band structure. This paper represents the first step towards on-chipquantum simulation of materials science and interacting particles in curved space.

I. INTRODUCTION

Euclidean space-time is the familiar geometry of non-relativistic physics. Its spatial dimensions are geometri-cally flat, having no inherent curvature or length scale.Within this paradigm, Newtonian gravity is completelydescribed by a scalar potential. However, this descrip-tion is insufficient to describe gravitational radiation orstrong gravitational fields such as those found near stel-lar objects like black holes. These situations require adescription in terms of general relativity in which grav-ity appears as variations in the metric of space-time. Theresulting mass-dependent curvature modifies the proper-ties of parallel lines and geodesics, and even though par-ticles still move along the shortest paths between points,they no longer move along the intuitive straight lines offamiliar Euclidean geometry. The equations of motionmust now be formulated in terms of the metric tensor,which can no longer be written in a position-independentway. The behavior of classical astrophysical objects inthis mass- and position-dependent metric has been stud-ied and simulated extensively, but a consistent formula-tion of both gravitation and quantum mechanics remainselusive. Therefore, a table-top simulator which naturallyincorporates curvature and non-classical degrees of free-dom is of considerable interest.

Positive spatial curvature is naturally embeddable inthe Euclidean laboratory frame by working with systemsrestricted to the surfaces of spheres. However, negative(hyperbolic) spatial curvature [1] cannot be isometricallyembedded in Euclidean space, so it is much more difficultto attain. To date, experimental simulations of particlesin negatively curved space have been restricted to hyper-bolic metamaterials in which the dielectric constant isvaried to reproduce the effects of a curved metric [2–8].However, these experiments are purely classical and onlyweakly interacting. Analogs of event horizons and Hawk-ing radiation have been studied both experimentally andtheoretically using acoustic waves [9], ultrashort opticalpulses [10], Bose-Einstein condensates (BECs) [11, 12],

and exciton-polariton condensates [13], but these tech-niques were not able to encode negative curvature, onlyevent horizons.

Expanding upon previous work which realized Eu-clidean lattice models using the techniques of circuitquantum electrodynamics (cQED) and interconnectednetworks of superconducting microwave resonators [14–20], we present a novel scheme to generate photonicmaterials which constitute periodic lattices in the two-dimensional hyperbolic plane [1, 21]. Classical photon-photon interactions can be added by incorporating non-linear materials into the resonators, and quantum me-chanical interactions by introducing qubits [14, 15, 19,22, 23]. The strongly non-classical properties of super-conducting qubits and the large qubit-photon couplingrates available will allow cQED hyperbolic materials toaccess an entirely new regime of simulation of interactingquantum mechanics in curved space. Additionally, thesesystems realize much stronger curvatures than previouslypossible, with lattice spacings in excess of approximately0.5 times the curvature length.

Beyond their natural connection to general relativ-ity, hyperbolic lattices also have significant applicationsin mathematics and computer science. Classification ofthese lattices and the study of their spectral propertiesrelates to open problems in representation theory of non-commutative groups, graph theory, random walks, andautomorphic forms [24–32]. Computer scientists oftenstudy hyperbolic networks because they have several use-ful qualities for robust and efficient communication net-works. For example, trees, which are naturally hyper-bolic, are highly efficient at connecting a large numberof nodes to a few central servers, and in fact the connec-tivity of the internet is a hyperbolic map [33, 34]. Addi-tionally, there exist classes of hyperbolic lattices which,unlike Euclidean lattices, cannot be split in half by theremoval of a small number of nodes [35]. They thereforearise frequently in the study of how to fortify a communi-cation network against hostile tampering. This enhancedconnectivity also leads to lower-overhead logical qubit en-

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coding in surface codes [36, 37].Here, we will concentrate on a set of examples which

are hyperbolic generalizations of the kagome lattice, andpresent numerical studies of their non-interacting bandstructures, which display highly unusual features, in par-ticular a spectrally isolated degenerate flat band. Finally,we present experimental measurements of a device whichrealizes a finite section of one of these lattices, a kagomelattice made with heptagons instead of hexagons.

The remainder of this paper is organized as follows.In Section II we describe cQED lattices, in particularthose made from 2D coplanar waveguide (CPW) res-onators, and review how they can be described by tight-binding Hubbard-like effective models where the energyof “atomic” sites is set by the resonance frequency ofthe microwave resonators, and the hopping rate is setby capacitive coupling between them. In Section III weshow that networks of CPW resonators can be used tocreate non-flat lattices, in particular hyperbolic analogsof the kagome lattice. Numerical simulations and exper-imental measurements are presented in III A and III B,respectively. Further details of the experimental deviceand numerical simulations are found in the SupplementalMaterials.

II. CIRCUIT QED LATTICES

The field of circuit quantum electrodynamics is a solid-state implementation of cavity quantum electrodynam-ics, in which artificial atoms, primarily superconductingqubits, are coupled to microwave resonators [20, 38–41].The strong coupling regime is readily achievable, and un-like atoms, qubits are lithographically defined, so strongqubit-resonator coupling can be maintained indefinitely.This, coupled with the ease of performing high-fidelitygate operations and relatively long coherence times, hasenabled superconducting qubits to emerge as a promis-ing candidate for universal digital quantum computation.However, a digital, gate-based, architecture is not theonly useful one. Pattern recognition and machine learn-ing routinely use architectures based on massively inter-connected neural networks. Regular networks map natu-rally to lattice-based physics problem, and are thereforea convenient platform for quantum simulation of many-body physics [15, 17].

Realizing this type of architecture by connecting alarge number of resonators together via capacitive orinductive coupling results in artificial photonic materi-als. Such materials have been realized using both 2DCPW resonators [15, 17–19, 38] and 3D stub resonators[20, 40, 41].

The lattices in this paper are made using CPW res-onators, which can easily be etched from a single layerof superconducting film. They are a planar analog of acylindrical coaxial cable, and consist of an electrically-isolated center pin surrounded by two ground planes oneither side. Resonators are readily defined simply by re-

moving a section of the center pin, and the external qual-ity factor is set by the size of this capacitive gap. By mi-crofabrication standards they are quite large, several mmin length, so they are typically fabricated with meandersto compactify them, as seen in Figure 1 (b)-(d).

Lattices, such as the one shown in Figure 1 (a), areformed by connecting many resonators together end toend [15–18]. The strength of the coupling is determinedby the capacitance at the junction where the center pinsconverge. A close-up of a three-way junction is shownin Figure 1 (e). The hopping rate is naturally negative,and as a result, these systems exhibit band structureswhich are inverted from what is found in actual solidstate systems, with rapidly oscillating Bloch waves atthe low-energy end of the spectrum.

In the absence of interactions these lattices are welldescribed by a tight-binding model, which can be writtenin the following form (with h = 1):

HTB = ω0

∑i

a†iai − t∑<i,j>

(a†iaj + a†jai), (1)

where the first sum runs over all lattice sites and accountsfor the on-site energy ω0, and the second describes hop-ping between nearest neighbors (〈i, j〉) with a character-istic rate t < 0 [15, 17]. The geometry of the lattice isencoded entirely in the structure of this last sum. Con-sider the example of a two-dimensional square lattice.A color map of the potential is shown in Figure 2 (a).Taking the tight-binding approximation is equivalent toreplacing this continuum model with a graph-based onesuch as that shown in Figure 2 (b). Each on-site wave-function is replaced with a single complex variable thatencodes its amplitude and phase. The graph has oneedge for each allowed hopping transition, and they areweighted by the corresponding hopping rates. Once thisassignment has been made, the tight-binding model isfully specified and, in fact, independent of how the graphis drawn. For example, the graph shown in Figure 2 (c)is identical to that in Figure 2 (b) despite the fact thatit appears disordered.

Such a deformation is challenging in an ultracold-atomor trapped-ion quantum simulator because the effectivet is determined by the physical distance between atomsor ions and inherently changes when they are displaced.However, the situation is quite different for CPW lat-tices. In this case, an individual site is a naturally onedimensional object whose connections are determined byits end points. Like a coaxial cable, a CPW resonatorcan be bent, and as long as the length of the resonatoris fixed, its properties in the tight-binding model are un-changed. Thus, despite the differences in their physicallayout, the resonators in Figure 1 (b)-(d) are identical.Therefore, as long as the end capacitors are unchangedand the resonators can be deformed sufficiently to main-tain constant total length, the tight-binding model is un-changed by moving the resonators or the locations of the

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FIG. 1. (a) Picture of a Euclidean lattice of CPW microwave resonators, modified from [15]. (b)-(d) Three mathematicallyidentical resonators with different shapes but the same resonance frequencies and hopping rates. (e) Close-up of a capacitivecoupler like the ones used in (a)-(d) to connect three resonators together. The effective hopping rate at this junction is set bythe capacitance between the three arrow-shaped center pins.

FIG. 2. (a) Color plot of the lattice potential of a regular 2D square lattice. (b) The corresponding regular tight-bindinggraph of the potential in (a). (c) Alternate drawing of the tight-binding graph in (a). Despite visible displacement of the nodes,the hopping Hamiltonian is unmodified because the hopping rates, indicated by the color of the graph edges, are identical to(b). (d) New tight-binding graph with the same nodes as (c) but with hopping rates dependent on the distance between thenodes. (e) Highly disordered lattice potential which gives rise to the tight-binding graph in (d). In systems where the effectivehopping rate is determined by the distance between sites, any displacement of the lattice sites results in a disordered model withmodified properties. In CPW lattices, however, the hopping rates are determined by the geometry of the coupling capacitors ateach end of the resonators. In some cases, such as the curved-space lattices shown in Figure 3 (d)-(i), the regular tight-bindinggraph is impossible to produce in 2D flat space, whereas a mathematically identical but distorted-looking graph like (c) canbe fabricated using CPW resonators. This would not be possible in systems where hopping rates are solely determined by thedistance between sites.

couplers. It is this flexibility that makes it possible to con-struct curved-space lattices on flat, Euclidean substrates.

III. HYPERBOLIC LATTICES

Typically, crystallography deals with periodic latticesconsisting of a unit cell, possibly with more than one site,and a tiling of that unit cell that fills all of space with no

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FIG. 3. Schematic diagram of Euclidean and non-Euclidean lattices in circuit QED. (a) One vertex of a successful attemptto tile the Euclidean plane with regular hexagons. (b) Resulting hexagonal lattice. (c) Euclidean circuit QED lattice with theresonators laid out in regular hexagons. Because this is a valid Euclidean tiling the resonator network is highly regular, andall resonators look the same. (Photograph modified from [15]) (d) One vertex of a failed attempt to tile the Euclidean planewith regular pentagons. A gap is left between the tiles, so this tiling is only valid in spherical space (positive curvature). (e)Projection of a spherical soccer-ball lattice into the Euclidean plane. Some tiles must be stretched to cover the missing space.(f) Schematic of a circuit QED lattice which realizes the soccer ball tiling. Resonator shapes are modified at different points inthe lattice to bridge the stretched distances while preserving the hopping rates and on-site energies. (g) One vertex of a failedattempt to tile the Euclidean plane with regular heptagons. The tiles overlap, so this tiling is only valid in hyperbolic space(negative curvature). (h) Conformal projection of a hyperbolic heptagon lattice into the Euclidean plane. (i) Schematic of acircuit QED lattice which realizes a section of the hyperbolic lattice in (h). Resonator shapes are modified at different pointsin the lattice to permit tighter packing while preserving the hopping rates and on-site energies.

gaps and no overlaps [42]. Geometry therefore stronglyconstrains the set of all possible unit cells. However,hyperbolic and spherical polygons are smaller and largerthan their Euclidean counterparts, respectively (see Fig.3), and the set of allowed lattices is different in curvedspaces.

To see this more clearly, we will adopt a slightly non-standard approach to crystallography which generalizesto curved space more readily than the usual description interms of unit cells and Bravais lattices. We will describeeach lattice as a tiling of the plane with polygons suchthat each lattice site is at a vertex of the tiling, and

two vertices are connected by an edge if they are nearestneighbors in the lattice. The tiling will consist of only afew distinct plaquettes, or tiles, and the geometry of thelattice can be classified in terms of the set of tiles andtheir tiling rule. For example, graphene will be describedas a tiling of regular hexagons such that three of themtouch at every vertex, as shown in Figure 3 (a).

In Euclidean space analogous tilings with three pen-tagons or three heptagons meeting at every vertex areforbidden. A sample vertex for each of these is shownin Figure 3 (d) and (g). Clearly the tiles do not fit. Inthe pentagon case there is a gap left between the tiles,

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and in the heptagon case the tiles are forced to overlap.However, both of these tilings can exist in curved space[21].

Unlike Euclidean space, spherical and hyperbolic spaceeach have a natural length scale R, set by 1/

√|K|, where

K is the Gaussian curvature [43]. The shape of polygonsin curved space depends on their size relative to R, andthe larger they are, the more they deviate from theirEuclidean counterparts [1, 21]. Consequently, hyperbolicand spherical tilings can only exist for a specific size oftile, which cannot be scaled up or down without inducinggaps between the tiles or overlapping regions. Each tilingtherefore has an intrinsic ratio between the curvature andthe lattice spacing [21].

Space with positive (spherical) curvature is effectivelysmaller than flat space. For example the circumferenceof a disc of radius r is 2πR sin(r/R) < 2πr. As a result,the gap in the tiling in Figure 3 (d) can be eliminatedif the curvature is sufficiently strong. This is why thefamous Buckminsterfullerene [44], or “soccer ball” tilingof regular pentagons and hexagons can be used to covera sphere, but not the flat plane. When such a tilingis projected flat, edges have to be stretched in order tobridge the gaps, as shown in Figure 3 (e).

Hyperbolic space with negative curvature is the oppo-site. It is larger than flat space, and the circumference ofa disc of radius r is 2πR sinh(r/R) > 2πr. As a result,the overlap in the tiles in Figure 3 (g) can be eliminatedif the curvature is strong enough, and a heptagonal ana-log of graphene is possible. Projecting this lattice intothe flat plane requires edges to be shortened to removeoverlaps, as shown in Figure 3 (h).

In order to produce an effectively curved lattice in aflat quantum simulator, this stretching or compressionof the tiling must be accomplished without changing thetight-binding model. Schematics of realizing this trans-formation using CPW resonators are shown in Figure 3(f) and (i). In both cases the vertices of the curved tilingno longer appear in a highly regular pattern in the physi-cal circuit, but the meanders in the resonators have beenadjusted to maintain constant length despite the distor-tion, and the tight-binding model has been preserved.

The limitations of this technique are two-fold. First,the resonators must be maintained at constant lengthwithout overlapping on the substrate. Since the turnsin a CPW cannot be made arbitrarily tight without de-stroying its waveguide properties, the achievable meanderdensity cannot be made arbitrarily large. This imposesa limit on the maximum feasible distortion which typi-cally rules out complete spherical tilings, such as Buck-minsterfullerene, or radically curved hyperbolic tilings.However, finite sections of spherical tilings and of manyhyperbolic lattices, such as the heptagon version of thekagome lattice, are readily achievable.

The second limit to the set of curved lattices achievablein circuit QED is the fact that the resonators are nativelyone-dimensional objects. Consequently, the most naturalway to lay them down is to select a tiling and place one

FIG. 4. (a) Microscope image of a Euclidean lattice devicewith the corresponding hexagonal layout graph overlaid inblue. Resonators correspond to edges of the graph and three-way capacitors to vertices. (b) The same device with theeffective, kagome, tight-binding graph overlaid in red. Res-onators now correspond to vertices of the graph and non-zerohopping matrix elements to the edges. A hyperbolic heptago-nal layout graph and the resulting effective kagome-like graphare shown in (c) and (d) respectively. Plots in (c) and (d) arevisualized in the Poincare disc model [1, 45, 46].

resonator on each edge, rather than on each vertex. Theeffective lattice which then appears in the tight-bindingmodel is the medial lattice, or line graph, of the origi-nal layout, where particles live on the center of the edgesand a hopping matrix element exists between effectivesites if their edges share an end point. Achievable lat-tices are therefore limited to those that can be writtenas the medial of another. Figure 4 (a) and (b) show aEuclidean resonator lattice with an overlay of the layoutlattice and effective lattice, respectively. The layout lat-tice is a hexagonal lattice, but the effective lattice is akagome lattice.

The hyperbolic lattices studied in this paper are nat-ural generalizations of the kagome lattice where the lay-out of the resonators is changed to a hyperbolic versionof graphene, such as the heptagonal one shown in Fig-ure 4 (c), resulting in an effective kagome-like latticesuch as that shown in Figure 4 (d). We focus primar-ily on this heptagon version, which we will refer to as theheptagon-kagome lattice, because the numerical studiesin the next section suggest that it belongs to a class of hy-perbolic kagome-like lattices whose highly unusual bandstructure exhibits a spectrally isolated flat band. Ad-ditionally, the heptagon-graphene layout lattice has theweakest curvature of all possible tilings of the hyperbolicplane with a single tile, making it particularly easy to

6

fabricate. However, even this weakest curvature is quitestrong. The heptagon-graphene lattice has an inter-sitespacing of 0.566R, and the resulting heptagon-kagome ef-fective lattice has 0.492R [21]. Conversely, if we take theinter-site spacing to be that of graphene, then they wouldhave R = 2.51 A and R = 2.88 A, respectively. Plots ofother hyperbolic graphene- and kagome-like lattices andtheir curvatures are shown in Supplemental Figure 9.

A. Tight-Binding Simulations

In Euclidean crystallography, calculation of the tight-binding band structure of a lattice is straightforward oncethe unit cell, Bravais lattice, and hopping rates have beendetermined. Translation groups in Euclidean space arecommutative, so representation theory guarantees that aBloch-wave ansatz will yield eigenstates and eigenener-gies as a function of momentum [42].

In hyperbolic space, however, the discrete translationgroups are non-commutative. As a result, there is no nat-ural analog of a Bravais lattice. In fact, simply writingdown the locations of all the lattice sites and the direc-tions of all the bonds is already a non-trivial mathemat-ical problem. To date, no hyperbolic equivalent of Blochtheory exists, and there is no known general procedurefor calculating band structures in either the nearly-free-electron, or tight-binding limits. Specialized methods areknown for the cases of trees [47] or Cayley graphs of thefree products of cyclic groups [30]. However, these can-not be applied to lattices. The only universal methodis numerical diagonalization of the hopping Hamiltonian.This is a brute force method which yields a list of eigen-vectors and eigenvalues, but no classification of eigen-states by a momentum quantum number. It, therefore,cannot be used to directly obtain the dispersion relations.Instead, it is more useful as a measure of the energy spec-trum and the density of states (DOS).

We produce finite-size samples using a cylindrical con-struction in which we start with a single layout polygonand successively add shells of nearest-neighbor polygons.We compute the effective lattice of this truncated lay-out, neglecting all hopping matrix elements that wouldconnect to resonators outside the simulation size, and ob-tain a hopping matrix in a localized, delta-function, basis.Numerical diagonalization of this matrix yields eigenen-ergies and eigenstates for each lattice type and simula-tion size. A sample energy spectrum for a three-shellpiece of the Euclidean kagome lattice is shown in Figure5 (c). It has significantly reduced information comparedto the full band structure calculation plotted in Figure5 (a) and (b), but the presence of the flat band, and ac-companying delta-function spike in the DOS, at −2|t| isclear. Spectra for the hyperbolic heptagon-, octagon-,and nonagon-kagome lattices are shown in Figure 5 (d)-(f). All four spectra share several similar features: a flatband at −2|t| and the remaining, presumably dispersive,bands filling the range from −2|t| to 4|t|. For the two

tilings formed from odd-sided polygons a robust spectralgap is visible between the flat band and the remainingeigenstates. We have verified that this gap is indepen-dent of system size and decreases with the number ofsides of the layout polygon. The other gaps visible in thespectrum appear to be finite-size artifacts that close withincreasing system size. (See Supplemental Figure 7 fordetails.)

Flat bands, like those seen in Figure 5, are quite rarein band structures [48, 49]. However, because the ki-netic energy in these bands effectively vanishes, they areripe for strongly correlated many-body physics and non-perturbative interactions, such as fractional QuantumHall states arising from discrete Landau levels. How-ever, a pure flat-band description is only valid if the in-teraction strength is smaller than the gap to the nearestdispersive band. This is very much the case with Landaulevels where magnetic energies far outweigh tunneling en-ergies and tight-binding band structure, but otherwisegapped flat bands are highly unusual. Among Euclideanlattices there are only a handful of known examples wherereal-space topology of the flat band allows a gap to exist[48, 49]. However, there are entire classes of hyperbolicsystems that display gapped flat bands, including the(2n+ 1)-gon-kagome lattices, and certain generalizationsof trees.

In all cases, the flat band arises because of an infinitemultiplicity of localized eigenstates. In the kagome lat-tice the smallest such state consists of a single occupiedhexagonal loop with alternating sign on each site andvanishing occupation elsewhere. This state can easily begeneralized to the octagon-kagome lattice, but not to theheptagon or nonagon versions where the odd number ofsides introduces geometric frustration. In these cases, theflat band still consists of localized states in the form ofan alternating closed loop, but it must now extend overtwo tiles, as seen in the inset of Figure 6 (c). Without ahyperbolic generalization of Bloch theory, the real-spacetopology argument which proves that the flat band ofthe kagome lattice is not gapped [48] cannot be read-ily extended to hyperbolic kagome lattices. As a result,while the flat-band eigenstates have been identified in allcases, the origin of the spectral gap isolating these statesfrom the rest of the spectrum for (2n + 1)-gons remainsunknown.

B. Experiment

We have constructed a device to realize a finite sec-tion of the heptagon-kagome lattice. It consists of onecentral heptagon and two shells of neighboring tiles. Aschematic of the layout is shown in Figure 6 (a), whereeach resonator has been approximated by a single line,and the lengths have not been held fixed. The actualdevice is fabricated using photolithography to etch CPWresonators into a 200 nm film of niobium on a 500 µmsapphire substrate. A photograph of the device is shown

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FIG. 5. (a) Band structure of the kagome lattice for t < 0 with periodic boundary conditions. The flat band is the lowest energyband at E = −2|t|. Two dispersive bands appear in the range [−2|t|, 4|t|] and touch the flat band at the Γ point at |k| = 0.(b) Slice through the kagome band structure showing the band touch. These Bloch-theory calculations cannot be generalizedto hyperbolic lattices, where the only available technique is numerical diagonalization on finite-sized lattices with hard-wallboundary conditions. (c) Finite-size numerical eigenenergy spectrum for the kagome lattice. The flat band and the band touchwith the dispersive bands are clearly visible, but the structure of the dispersive bands is lost. (d)-(f) Analogous numericaleigenenergy spectra for hyperbolic versions of the kagome lattice using heptagons, octagons, and nonagons, respectively. Theflat band is gapped for the case of odd-sided layout polygons, the heptagons and nonagons in (d) and (f).

in Figure 6 (b). The resonators are 7.5 mm long with afundamental resonance frequency of 8 GHz and a secondharmonic of 16 GHz. The hopping rate is −73.7 MHZ atthe first harmonic and −147.4 MHz at the second. Thelength of all resonators has been held fixed, and the vary-ing meanders required to make up the excess are clearly

visible. In addition to the lattice itself, the circuit alsocontains four measurement ports, visible in each corner,used to interrogate the lattice.

In order to simulate the transmission properties of thedevice, we use the tight-binding calculations describedin III A with small additional levels of disorder in both

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FIG. 6. (a) Resonator layout (dark blue) and effective lattice (light blue) for a circuit that realizes two shells of the heptagon-kagome lattice. Orange circles indicate three-way capacitive couplers. (b) Photograph of a physical device which realizesthe layout and effective graphs in (a). The device was fabricated in a 200 nm niobium film on a sapphire substrate andconsists of 140 CPW resonators with fundamental resonance frequencies of 8 GHz, second harmonic frequencies of 16 GHz,and hopping rates of −147.4 MHz at the second harmonic. Four additional CPW lines have been included at each corner ofthe device to couple microwaves into and out of the device for transmission measurements. Short stubs protruding inward fromthe outermost three-way couplers in the device are high frequency λ/4 resonators which maintain a consistent loading of thesites in the outer ring– ensuring uniform on-site energies. (c) Simulated and actual transmission (S21) for the device in (b)demonstrating reasonable agreement between theory and experiment. Eigenstates at various frequencies are included as insets.The diameter of the circles indicates the magnitude of the wavefunction and the color its sign. The flat-band state (far left) isincluded for reference although it does not show up prominently in transmission.

the on-site energies and hopping rates. After numericaldiagonalization, we obtain the frequency and wavefunc-tion of each eigenmode of the lattice. Assuming a typicalHWHM of 1.4 MHz, we generate a Lorentzian resonanceprofile for each mode, centered about its eigenenergy. Wethen compute the level of mode matching between theeigenmodes and the input and output ports using thenumerical wavefunctions. To calculate the transmittedelectric field versus probe frequency, we sum the trans-mission through all 140 eigenmodes, weighted by theirLorentzian line shapes and spatial mode matching. Inthe absence of disorder, transmission through the flatband states experiences perfect destructive interference,

so despite its prominent position in the density of states,the flat band is all but absent from the transmission. Theremainder of the spectrum gives rise to a dense region ofsometimes overlapping resonances, as shown in Figure 6(c).

An experimental plot of transmitted power near thesecond harmonic frequency of the device is shown in Fig-ure 6 (d), along with plots of the wavefunctions of se-lected eigenstates. (The fundamental modes of the deviceobey a different tight-binding model due to the asymme-try of the mode function within each resonator [17]. Seethe Supplementary Information for details.) The experi-mental data and simulation show good qualitative agree-

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ment, particularly at the bottom of the spectrum nearthe flat band and at high energy near the uniform state.This device demonstrates that hyperbolic lattices can beproduced on chip using CPW resonators, and paves theway to the study of interactions in hyperbolic space, andto simulation of new models with non-constant curva-ture. In particular, the gapped flat band is an ideal hostfor non-perturbative interactions and strongly correlatedmany-body states [48].

IV. CONCLUSION

In conclusion, we have shown that circuit QED latticesof two-dimensional CPW resonators can be used to pro-duce artificial photonic materials which exist in an effec-tive curved space. In particular, we conducted numericaltight-binding simulations of a class of hyperbolic analogsof the kagome lattice, and demonstrated that they dis-play a flat band similar to that of their Euclidean coun-terpart. However, for the case of odd-sided polygons thisband is isolated from the rest of the spectrum. We alsoconstructed an experimental device which realizes a fi-nite section of non-interacting heptagon-kagome lattice.Mathematical investigation into the origin of the gap isstill ongoing.

While our present work is purely non-interacting, in-teractions can be included via incorporating kinetic in-ductance materials such as NbTiN into the resonators toobtain a classical χ3 non-linearity [50–53], or via the ad-

dition of qubits to each resonator. These methods mayalso be applied to computer science or mathematics prob-lems such as the study of non-linear operators on treesor Cayley graphs of free products of cyclic groups, ratherthan the lattices discussed here. Alternatively, it is pos-sible to appropriate the techniques of hyperbolic meta-materials in which the equations of motion are tailoredto mimic the existence of a nontrivial metric by delib-erate modulation of the dielectric constant. A discreteversion of the same effect can be realized here by tai-loring the hopping magnitude, providing a simple routeto models of more moderate curvature and specific met-rics such as the Schwartzchild solution. The promise ofstrong interactions in these lattices leads to an excitingfrontier which may provide answers to questions at theinterface of quantum mechanics, gravity, and condensedmatter physics, as lattices with these properties cannotbe fabricated from actual materials.

ACKNOWLEDGMENTS

We thank Peter Sarnak, Janos Kollar, Rivka Beken-stein, Charles Fefferman, and Siddharth Parameswaranfor helpful discussions. This work was supported by theNSF, the Princeton Center for Complex Materials DMR-1420521, and by the MURI W911NF-15-1-0397.

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V. SUPPLEMENTARY INFORMATION

A. CPW Resonators and Circuit QED Lattices

The CPW microwave resonators used in this work areformed by electrically isolating a portion of the centerpin. Networks are formed by butting the ends of thecenter pins up against each other, as shown in Figure 1(e). Larger coordination numbers than three are possible,but the circuit layout tends to produce significant next-nearest-neighbor hopping which greatly complicates thedescription of the network, and will not be discussed here.

When the resonators are laid out in a periodic latticethe circuit naturally maps to a tight-binding model of a2D crystalline solid. The type of lattice, square, kagome,etc., is determined by how the resonators are connectedtogether, and the frequency of the resonator maps to theatomic binding energy of the conduction band. The ca-pacitive coupling between resonators can be expressed asa nearest-neighbor hopping rate, t, whose magnitude isset by the size and shape of the gap between the centerpins. The sign of t is determined by two factors: theenergetics of coupling two cavities, and the sign of theon-site mode function.

The eigenmodes of the CPW resonators are standingwaves with an antinode of the voltage at each end ofthe cavity. Quantum computing applications typicallyuse the fundamental, half-wave, mode. However, whenstudying network or lattice physics, the inherent asym-metry of the half-wave mode complicates the sign of thehopping rates. In many cases this minus sign can be re-moved by a local gauge transformation which redefinespositive and negative on alternating sites [17]. However,for many of the hyperbolic lattices considered here, thisgauge transformation is not possible, and it is impossibleto consistently define all the hopping rates with the samesign. This complicates the mathematical description and,in some cases, induces additional geometric frustration.The second-harmonic, or full-wave, mode is symmetricand does not have this sign problem. In the body of thispaper, we therefore restrict to full-wave modes, and allresults, unless explicitly stated, will refer to this case.

wave Regardless of whether the on-site wavefunctionis even or odd, it is energetically favorable for the volt-ages on either side of the coupling capacitors connect-ing the resonators to be of opposite sign. Circuit QEDlattices therefore exhibit band structures which are in-verted from what is found in actual solid state-systems,with rapidly oscillating Bloch waves at the low-energyend of the spectrum. For full-wave modes, this physicscan be captured by a standard tight-binding model withnegative t. For half-wave modes, a global definition ofthe sign of the on-site wavefunction may no longer bepossible, and care must be taken to define the sign of tconsistently [17, 42]. We proceed by defining an orienta-tion for each resonator and arbitrarily designate one endof each resonator “positive” and one end “negative”. Forthe tight-binding model to be physical the sign of t must

be consistent with the designated orientation and alwaysfavor opposite voltages on either side of the coupling ca-pacitors. Therefore, t < 0 when it describes hoppingbetween two ends of the same sign and t > 0 when de-scribing hopping between two ends of different sign.

As hinted above, the choice of the orientation of eachhalf-wave resonator is arbitrary and merely a bookkeep-ing convention. However, some choices are, of course,more convenient and illustrative than others. For somelayout lattices, e.g. square or hexagonal, the orientationof each resonator can be chosen consistently such that allhopping matrix elements are negative, even for half-wavemodes [17]. In these cases, the asymmetry of the on-sitewavefunction can be gauged away by this judicious choiceof orientations, and the half-wave tight-binding model isisomorphic to the version with even on-site wavefunc-tions, in much the same way that the magnon spectrumof an unfrustrated antiferromagnet is identical to that ofthe corresponding ferromagnet. However, if the layoutlattice contains plaquettes with an odd number of sides,then an orientation which fixes the sign of the hoppingmatrix elements cannot generally be found. Compared totheir full-wave counterparts, these tight-binding modelsexhibit additional geometric frustration due to the phasewindings forced by the half-wave mode function.

This effect can most easily be seen by considering themaximally excited state of a kagome-like layout lattice,which has eigenvalue 4|t|. The defining features of thisstate are: uniform probability of occupation of all thesites, and equal voltages on either side of all coupling ca-pacitors. For full-wave modes, both of these conditionsare easily satisfied by uniform occupation of all cavitieswith a single phase. For half-wave modes, this is not thecase. Consider a single layout plaquette of the lattice anddefine the orientations such that going from the positiveto the negative end of a resonator corresponds to mov-ing clockwise around the plaquette. In order maintainequal voltages on both sides of all coupling capacitorsusing half-wave modes, the sign of the on-site wavefunc-tion must alternate on neighboring sites, which cannotbe done consistently on a plaquette with an odd numberof sides.

Hyperbolic kagome lattices made from even-sided poly-gons are free of this effect and the asymmetry of thehalf-wave mode function can always be gauged away.Kagome-like lattices made from odd-sided polygons, onthe other hand, are frustrated. Remarkably, the flat-band is immune to these frustration effects because it isspanned by states which consist of loops with an evennumber of edges, such as the one shown in Figure 8(g). The dispersive bands above E = −2|t|, however,are affected, and the spectral gap which isolates the flatband in the full-wave case disappears from the half-wavemodel.

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FIG. 7. Numerical eigenenergy spectra for the heptagon-kagome lattice versus system size. The smallest layout tiling consistsof a central heptagon and a shell of seven immediate neighbors. Each successive tiling includes another shell of immediateneighbors of the previous one. The spectra are plotted for one (red), two (blue), three (cyan), and four (gold) shells ofneighbors. The density of states converges rapidly with number of shells despite relatively small system sizes and effectivehard-wall confinement. The theoretical plots in this paper are from three-shell simulations. The experimental device consistsof two shells.

B. Tight-Binding Models

1. System Size Effects

Numerical diagonalization studies of finite-size tight-binding models of the heptagon-kagome lattice were con-ducted for a series of system sizes, starting with a small-est simulation consisting of a central layout heptagon andone shell of nearest-neighbor plaquettes. We compute theeffective lattice of this finite layout, neglecting all linksto sites outside the current system size. Larger systemsare created iteratively by adding successive shells of near-est neighbor layout polygons to the edge of the smallersimulations and recomputing the effective lattice.

Numerical eigenenergy spectra for a series of systemsizes are shown in Figure 7. The flat band is clearlyvisible even for the smallest simulation with only one shellof neighboring polygons. The spectral gap between theflat band the rest of spectrum is already present, but notreadily distinguishable from the finite-size-induced gapsat higher energy. Within the size of simulation currentlypossible, the higher-lying gaps all close progressively withincreasing system size, and it remains unclear whetherany of them also remain open at infinite size. Since thespectral gap between the flat band and the rest of the

energy spectrum is readily visible for systems with twoor more layers of neighbor polygons, our physical device,shown in Figure 6, was constructed at this depth.

2. Numerical Eigenstates

Color plots of selected numerical eigenstates from athree-shell simulation of the heptagon-kagome lattice areshown in Figure 8. Due to the negative hopping am-plitude, the uniform-phase configuration typically asso-ciated with the ground state of a lattice is, in fact, thehighest excited state, shown in Figure 8 (a). Addition-ally, it exhibits an radial amplitude modulation due toconfinement from the missing links beyond the edge ofthe sample, which constitute an effective hard-wall po-tential. The next eigenstates down, Figure 8 (b) and (c),bear a striking resemblance to the Laguerre-type 0,1 and0,2 modes of a Euclidean harmonic oscillator or a particlein a cylindrical box. The qualitative similarity persistsfor many modes throughout the spectrum, such as theone shown in Figure 8 (d) which resembles a Laguerre4,7 mode.

Conversely, the eigenstates also show several featureswhich are strikingly different from the Euclidean case.

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FIG. 8. Color plots of selected numerical eigenstates of three shells of the heptagon-kagome lattice. States are ordered fromhighest to lowest energy, and are plotted by placing a circle on each lattice site of the effective lattice. The size of the circleindicates the amplitude of the state on that site, and the color its phase. (a) The maximally excited state. This state isuniform in phase, but its amplitude varies radially due to the effective confinement from the missing links at the boundary ofthe simulation. (b), (c) Examples of the two next-highest states. They bear a striking resemblance to Laguerre-Gaussian orparticle-in-a-cylindrical box modes found in flat Euclidean space. (d), (e), (f) Selected intermediate excited states. Notice thatthe state in (f) shows both amplitude and phase modulation in the azimuthal direction, with independent periods. This is veryunusual compared to the solutions to similar Euclidean problems. (g) The localized eigenstate of compact support that formsthe flat band.

Because of the hyperbolic nature of the lattice, a macro-scopic fraction of the lattice sites are in the outermostring, and the spectrum therefore contains a relativelylarge number of states like the one in Figure 8 (e) whichreside primarily at the edge of the system. Another in-triguing feature is the existence of states like the onein Figure 8 (f), which display azimuthal amplitude andphase modulation with different periods. The azimuthal-angle dependence of eigenstates of cylindrically symmet-ric, two-dimensional Euclidean problems can be writtenin either sine-cosine or complex exponential forms, butthere is only one azimuthal quantum number, so the twosets of modes differ only by a π/4 rotation. The presenceof two different azimuthal periods in Figure 8 (f) suggeststhat the hyperbolic problem requires a second azimuthalquantum number.

A flat-band eigenstate is shown in Figure 8 (g). Theanalogous state in the kagome lattice encircles a singlehexagonal plaquette, but this type of behavior is notpossible in a hyperbolic kagome-like lattice formed withodd-sided polygons because the sign flips cannot be con-sistently maintained. Therefore, the flat-band state inthe heptagon kagome lattice (or any 2n+ 1-gon kagomelattice) consists of a single loop across two plaquettesin which the phase flips by π between every neighboringpair of sites. It is a localized eigenstate which is protectedfrom hopping by destructive interference in the triangu-lar plaquettes which border the loop. Since this state hascompact support, translations of it are orthogonal and

form a degenerate manifold of states whose multiplicityis proportional to the system size.

C. Lattice Curvatures

Compared to spherical or hyperbolic space, familiarEuclidean space is unique in that there is no intrinsiclength scale. In particular, polygons have the same in-ternal angles, regardless of size. Therefore, any Euclideantiling can be scaled up or down in size and remain un-changed. This is not true in spherical or hyperbolic spacewhere there is a natural length scale R which is set bythe Gaussian curvature K = ±1/R2 [43]. In both cases,the shape of a polygon depends on its size relative to thislength scale [1, 21]. For example, consider a triangle onthe surface of a sphere which has one vertex at the northpole and the other two on the equator separated by π/2longitudinally. The total internal angle of this triangleis 3π/2 (each angle is π/2), but if it is reduced in sizeuntil it is much smaller than the radius of curvature ofthe sphere, it tends to a Euclidean triangle with a totalinternal angle of π. Tiles in spherical space therefore get“fatter” as they increase in size and can cover gaps leftbetween Euclidean tiles of the same shape. Hyperbolicpolygons, however, have smaller internal angles at theirvertices than their Euclidean counterparts. Therefore,hyperbolic tilings are precisely those for which the tileswould overlap if drawn according to Euclidean geometry.

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FIG. 9. Poincare disc model conformal projections of hyperbolic lattices. (a)-(d) Graphene-like lattices formed from heptagons,octagons, nonagons, and dodecagons, respectively. (e)-(h) The corresponding kagome-like effective lattices which arise when(a)-(d) are used as the layout lattice. (i) A table of inter-site spacings for graphene-like hyperbolic layout lattices and their

medial lattices, the kagome-like effective lattices. All distances are given in terms of the curvature length R = 1/√|K|. As

the number of sides of the layout polygon increases, the intrinsic curvature of the tiling also grows, and the polygons becomemarkedly larger.

(See Figure 3.) As hyperbolic polygons become largerand larger, their internal angles decrease progressively,and they become “pointier”.

Because of this size-dependent geometry, whether ornot a set of tiles can form a valid lattice in curved spacedepends on their size. The polygons must be precisely bigenough so that the tiles fit exactly. Increasing the sizeof the polygons any further will cause gaps or overlapsto appear between the tiles, depending on the sign ofthe curvature. Therefore, for fixed curvature, each non-Euclidean lattice can only exist for a specific tile size.Conversely each lattice can be though of as having anintrinsic curvature given by the required ratio of inter-site spacing and R.

In order to determine these curvatures for the hyper-bolic tilings considered in this paper we make use ofthe Poincare disc model conformal mapping of the two-dimensional hyperbolic plane with curvature −1 ontothe Euclidean unit disc [1]. For concreteness, we con-sider only hyperbolic tilings which are generalizations

of graphene to polygons with a larger number of sides.These are precisely those that produce kagome-like effec-tive models. The layout tiling will exist when a hyperob-lic n-gon can be produced which is bounded by geodesicsand whose hyperbolic internal angle is 2π/3, allowingcopies of it to be tiled in a graphene-like way with threetiles meeting at every vertex. Since the Poincare discmodel is a conformal model which preserves angles, itsuffices to determine the polygon size which results in aninternal angle of 2π/3 when drawn in the unit disc. Thegeodesics of the Poincare disc model are circles which in-tersect the unit disc at an angle of π/2 and the metricis known analytically, so the radius and edge length ofthe tile required to form the layout lattice can be readilydetermined. The corresponding inter-site spacing for theeffective lattice is determined by computing the distancebetween the midpoints of the edges of the layout polygon.

For the heptagon-graphene layout lattice the inter-sitespacing is determined to be 0.566R [21], and the curva-ture of the resulting heptagon-kagome effective lattice is

15

slightly weaker with an inter-site spacing of 0.492R. Lat-tice spacing and Poincare disc model plots of a series ofgraphene- and kagome-like lattices are shown in Figure9. The ratio of the lattice spacing to R increases withincreasing number of sides of the layout polygon becausethe corresponding attempt at a Euclidean tiling has moreand more overlap that needs to be removed by increasingthe size of the tiles. This is clearly visible in the plots

as the polygon at the origin occupies a larger and largerfraction of the unit disc. In fact, the heptagon-graphenetiling has smallest ratio of lattice spacing to curvatureof any tiling of the hyperbolic plane by a single polygon,regardless of vertex coordination number. Correspond-ingly, the heptagon-kagome has the weakest curvature ofall the resulting effective lattices, making this pair natu-ral entry points to the study of hyperbolic lattices.