Hyperbolic Lattices in Circuit Quantum Electrodynamics
After close to two decades of research and development, superconducting circuits have emerged as a rich platform for both quantum computation and quantum simulation. Lattices of superconducting coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave photons, where weak interactions can be introduced either via non-linear resonator materials or strong interactions via qubit-resonator coupling. Here, we introduce a technique using networks of CPW resonators to create a new class of materials which constitute regular lattices in an effective hyperbolic space with constant negative curvature. We show numerical simulations of a class of hyperbolic 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-chip quantum simulation of materials science and interacting particles in curved space.
Euclidean space-time is the familiar geometry of non-relativistic physics. Its spatial dimensions are geometrically flat, having no inherent curvature or length scale. Within this paradigm, Newtonian gravity is completely described by a scalar potential. However, this description is insufficient to describe gravitational radiation or strong gravitational fields such as those found near stellar objects like black holes. These situations require a description in terms of general relativity in which gravity appears as variations in the metric of space-time. The resulting mass-dependent curvature modifies the properties of parallel lines and geodesics, and even though particles still move along the shortest paths between points, they no longer move along the intuitive straight lines of familiar Euclidean geometry. The equations of motion must now be formulated in terms of the metric tensor, which can no longer be written in a position-independent way. The behavior of classical astrophysical objects in this mass- and position-dependent metric has been studied and simulated extensively, but a consistent formulation of both gravitation and quantum mechanics remains elusive. Therefore, a table-top simulator which naturally incorporates curvature and non-classical degrees of freedom is of considerable interest.
Positive spatial curvature is naturally embeddable in the Euclidean laboratory frame by working with systems restricted to the surfaces of spheres. However, negative (hyperbolic) spatial curvature Cannon et al. (1997) cannot be isometrically embedded in Euclidean space, so it is much more difficult to attain. To date, experimental simulations of particles in negatively curved space have been restricted to hyperbolic metamaterials in which the dielectric constant is varied to reproduce the effects of a curved metric Leonhardt and Philbin (2006); Batz and Peschel (2008); Smolyaninov and Narimanov (2010); Genov et al. (2009); Chen et al. (2010); Bekenstein et al. (2017, 2015). However, these experiments are purely classical and only weakly interacting. Analogs of event horizons and Hawking radiation have been studied both experimentally and theoretically using acoustic waves Weinfurtner et al. (2011), ultrashort optical pulses Philbin et al. (2008), Bose-Einstein condensates (BECs) Steinhauer (2016); Carusotto et al. (2008), and exciton-polariton condensates Gerace and Carusotto (2012), but these techniques were not able to encode negative curvature, only event horizons.
Expanding upon previous work which realized Euclidean lattice models using the techniques of circuit quantum electrodynamics (cQED) and interconnected networks of superconducting microwave resonators Hoffman et al. (2011); Houck et al. (2012); Underwood et al. (2012); Schmidt and Koch (2013); Underwood et al. (2016); Fitzpatrick et al. (2017); Anderson et al. (2016), we present a novel scheme to generate photonic materials which constitute periodic lattices in the two-dimensional hyperbolic plane Cannon et al. (1997); Coxeter (1954). Classical photon-photon interactions can be added by incorporating non-linear materials into the resonators, and quantum mechanical interactions by introducing qubits Hoffman et al. (2011); Houck et al. (2012); Schmidt et al. (2010); Raftery et al. (2014); Fitzpatrick et al. (2017). The strongly non-classical properties of superconducting qubits and the large qubit-photon coupling rates available will allow cQED hyperbolic materials to access an entirely new regime of simulation of interacting quantum mechanics in curved space. Additionally, these systems realize much stronger curvatures than previously possible, with lattice spacings in excess of approximately times the curvature length.
Beyond their natural connection to general relativity, hyperbolic lattices also have significant applications in mathematics and computer science. Classification of these lattices and the study of their spectral properties relates to open problems in representation theory of non-commutative groups, graph theory, random walks, and automorphic forms Woess (1987); Aomoto and Kato (1988); Sunada (1990); Floyd and Plotnik (1987); Bartholdi and Ceccherini-Silberstein (2002); Strichartz (1989); McLaughlin (1986); Kuhn (1991); Agmon (1986). Computer scientists often study hyperbolic networks because they have several useful qualities for robust and efficient communication networks. For example, trees, which are naturally hyperbolic, are highly efficient at connecting a large number of nodes to a few central servers, and in fact the connectivity of the internet is a hyperbolic map Krioukov et al. (2010); Boguña et al. (2010). Additionally, there exist classes of hyperbolic lattices which, unlike Euclidean lattices, cannot be split in half by the removal of a small number of nodes Lipton and Tarjan (1979). They therefore arise frequently in the study of how to fortify a communication network against hostile tampering. This enhanced connectivity also leads to lower-overhead logical qubit encoding in surface codes Breuckmann and Terhal (2016); Breuckmann et al. (2017).
Here, we will concentrate on a set of examples which are hyperbolic generalizations of the kagome lattice, and present numerical studies of their non-interacting band structures, which display highly unusual features, in particular a spectrally isolated degenerate flat band. Finally, we present experimental measurements of a device which realizes a finite section of one of these lattices, a kagome lattice made with heptagons instead of hexagons.
The remainder of this paper is organized as follows. In Section II we describe cQED lattices, in particular those made from 2D coplanar waveguide (CPW) resonators, and review how they can be described by tight-binding Hubbard-like effective models where the energy of “atomic” sites is set by the resonance frequency of the microwave resonators, and the hopping rate is set by capacitive coupling between them. In Section III we show that networks of CPW resonators can be used to create non-flat lattices, in particular hyperbolic analogs of the kagome lattice. Numerical simulations and experimental measurements are presented in III.1 and III.2, respectively. Further details of the experimental device and numerical simulations are found in the Supplemental Materials.
Ii Circuit QED Lattices
The field of circuit quantum electrodynamics is a solid-state implementation of cavity quantum electrodynamics, in which artificial atoms, primarily superconducting qubits, are coupled to microwave resonators Blais et al. (2004); Koch et al. (2007); Reagor et al. (2016); Anderson et al. (2016); Paik et al. (2011). The strong coupling regime is readily achievable, and unlike atoms, qubits are lithographically defined, so strong qubit-resonator coupling can be maintained indefinitely. This, coupled with the ease of performing high-fidelity gate operations and relatively long coherence times, has enabled superconducting qubits to emerge as a promising candidate for universal digital quantum computation. However, a digital, gate-based, architecture is not the only useful one. Pattern recognition and machine learning routinely use architectures based on massively interconnected neural networks. Regular networks map naturally to lattice-based physics problem, and are therefore a convenient platform for quantum simulation of many-body physics Houck et al. (2012); Schmidt and Koch (2013).
Realizing this type of architecture by connecting a large number of resonators together via capacitive or inductive coupling results in artificial photonic materials. Such materials have been realized using both 2D CPW resonators Blais et al. (2004); Houck et al. (2012); Schmidt and Koch (2013); Underwood et al. (2016, 2016); Fitzpatrick et al. (2017) and 3D stub resonators Paik et al. (2011); Reagor et al. (2016); Anderson et al. (2016).
The lattices in this paper are made using CPW resonators, which can easily be etched from a single layer of superconducting film. They are a planar analog of a cylindrical coaxial cable, and consist of an electrically-isolated center pin surrounded by two ground planes on either side. Resonators are readily defined simply by removing a section of the center pin, and the external quality factor is set by the size of this capacitive gap. By microfabrication standards they are quite large, several mm in length, so they are typically fabricated with meanders to compactify them, as seen in Figure 1 (b)-(d).
Lattices, such as the one shown in Figure 1 (a), are formed by connecting many resonators together end to end Underwood et al. (2012, 2016); Houck et al. (2012); Schmidt and Koch (2013). The strength of the coupling is determined by the capacitance at the junction where the center pins converge. A close-up of a three-way junction is shown in Figure 1 (e). The hopping rate is naturally negative, and as a result, these systems exhibit band structures which are inverted from what is found in actual solid state systems, with rapidly oscillating Bloch waves at the low-energy end of the spectrum.
In the absence of interactions these lattices are well described by a tight-binding model, which can be written in the following form (with ):
where the first sum runs over all lattice sites and accounts for the on-site energy , and the second describes hopping between nearest neighbors () with a characteristic rate Houck et al. (2012); Schmidt and Koch (2013). The geometry of the lattice is encoded entirely in the structure of this last sum. Consider 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 to replacing this continuum model with a graph-based one such as that shown in Figure 2 (b). Each on-site wavefunction is replaced with a single complex variable that encodes its amplitude and phase. The graph has one edge for each allowed hopping transition, and they are weighted by the corresponding hopping rates. Once this assignment has been made, the tight-binding model is fully specified and, in fact, independent of how the graph is drawn. For example, the graph shown in Figure 2 (c) is identical to that in Figure 2 (b) despite the fact that it appears disordered.
Such a deformation is challenging in an ultracold-atom or trapped-ion quantum simulator because the effective is determined by the physical distance between atoms or ions and inherently changes when they are displaced. However, the situation is quite different for CPW lattices. In this case, an individual site is a naturally one dimensional object whose connections are determined by its end points. Like a coaxial cable, a CPW resonator can be bent, and as long as the length of the resonator is fixed, its properties in the tight-binding model are unchanged. Thus, despite the differences in their physical layout, the resonators in Figure 1 (b)-(d) are identical. Therefore, as long as the end capacitors are unchanged and the resonators can be deformed sufficiently to maintain constant total length, the tight-binding model is unchanged by moving the resonators or the locations of the couplers. It is this flexibility that makes it possible to construct curved-space lattices on flat, Euclidean substrates.
Iii Hyperbolic Lattices
Typically, crystallography deals with periodic lattices consisting of a unit cell, possibly with more than one site, and a tiling of that unit cell that fills all of space with no gaps and no overlaps Ashcroft and Mermin (1976). Geometry therefore strongly constrains the set of all possible unit cells. However, hyperbolic and spherical polygons are smaller and larger than their Euclidean counterparts, respectively (see Fig. 3), and the set of allowed lattices is different in curved spaces.
To see this more clearly, we will adopt a slightly non-standard approach to crystallography which generalizes to curved space more readily than the usual description in terms of unit cells and Bravais lattices. We will describe each lattice as a tiling of the plane with polygons such that each lattice site is at a vertex of the tiling, and two vertices are connected by an edge if they are nearest neighbors in the lattice. The tiling will consist of only a few distinct plaquettes, or tiles, and the geometry of the lattice can be classified in terms of the set of tiles and their tiling rule. For example, graphene will be described as a tiling of regular hexagons such that three of them touch at every vertex, as shown in Figure 3 (a).
In Euclidean space analogous tilings with three pentagons or three heptagons meeting at every vertex are forbidden. A sample vertex for each of these is shown in Figure 3 (d) and (g). Clearly the tiles do not fit. In the pentagon case there is a gap left between the tiles, and in the heptagon case the tiles are forced to overlap. However, both of these tilings can exist in curved space Coxeter (1954).
Unlike Euclidean space, spherical and hyperbolic space each have a natural length scale , set by , where is the Gaussian curvature Carroll (1997). The shape of polygons in curved space depends on their size relative to , and the larger they are, the more they deviate from their Euclidean counterparts Coxeter (1954); Cannon et al. (1997). Consequently, hyperbolic and spherical tilings can only exist for a specific size of tile, which cannot be scaled up or down without inducing gaps between the tiles or overlapping regions. Each tiling therefore has an intrinsic ratio between the curvature and the lattice spacing Coxeter (1954).
Space with positive (spherical) curvature is effectively smaller than flat space. For example the circumference of a disc of radius is . As a result, the gap in the tiling in Figure 3 (d) can be eliminated if the curvature is sufficiently strong. This is why the famous Buckminsterfullerene Kroto et al. (1985), or “soccer ball” tiling of regular pentagons and hexagons can be used to cover a sphere, but not the flat plane. When such a tiling is projected flat, edges have to be stretched in order to bridge the gaps, as shown in Figure 3 (e).
Hyperbolic space with negative curvature is the opposite. It is larger than flat space, and the circumference of a disc of radius is . As a result, the overlap in the tiles in Figure 3 (g) can be eliminated if the curvature is strong enough, and a heptagonal analog of graphene is possible. Projecting this lattice into the flat plane requires edges to be shortened to remove overlaps, as shown in Figure 3 (h).
In order to produce an effectively curved lattice in a flat quantum simulator, this stretching or compression of the tiling must be accomplished without changing the tight-binding model. Schematics of realizing this transformation using CPW resonators are shown in Figure 3 (f) and (i). In both cases the vertices of the curved tiling no longer appear in a highly regular pattern in the physical circuit, but the meanders in the resonators have been adjusted to maintain constant length despite the distortion, and the tight-binding model has been preserved.
The limitations of this technique are two-fold. First, the resonators must be maintained at constant length without overlapping on the substrate. Since the turns in a CPW cannot be made arbitrarily tight without destroying its waveguide properties, the achievable meander density cannot be made arbitrarily large. This imposes a limit on the maximum feasible distortion which typically rules out complete spherical tilings, such as Buckminsterfullerene, or radically curved hyperbolic tilings. However, finite sections of spherical tilings and of many hyperbolic lattices, such as the heptagon version of the kagome lattice, are readily achievable.
The second limit to the set of curved lattices achievable in circuit QED is the fact that the resonators are natively one-dimensional objects. Consequently, the most natural way to lay them down is to select a tiling and place one resonator on each edge, rather than on each vertex. The effective lattice which then appears in the tight-binding model is the medial lattice, or line graph, of the original layout, where particles live on the center of the edges and a hopping matrix element exists between effective sites if their edges share an end point. Achievable lattices are therefore limited to those that can be written as the medial of another. Figure 4 (a) and (b) show a Euclidean resonator lattice with an overlay of the layout lattice and effective lattice, respectively. The layout lattice is a hexagonal lattice, but the effective lattice is a kagome lattice.
The hyperbolic lattices studied in this paper are natural generalizations of the kagome lattice where the layout of the resonators is changed to a hyperbolic version of graphene, such as the heptagonal one shown in Figure 4 (c), resulting in an effective kagome-like lattice such as that shown in Figure 4 (d). We focus primarily on this heptagon version, which we will refer to as the heptagon-kagome lattice, because the numerical studies in the next section suggest that it belongs to a class of hyperbolic kagome-like lattices whose highly unusual band structure exhibits a spectrally isolated flat band. Additionally, the heptagon-graphene layout lattice has the weakest curvature of all possible tilings of the hyperbolic plane with a single tile, making it particularly easy to fabricate. However, even this weakest curvature is quite strong. The heptagon-graphene lattice has an inter-site spacing of , and the resulting heptagon-kagome effective lattice has Coxeter (1954). Conversely, if we take the inter-site spacing to be that of graphene, then they would have and , respectively. Plots of other hyperbolic graphene- and kagome-like lattices and their curvatures are shown in Supplemental Figure 9.
iii.1 Tight-Binding Simulations
In Euclidean crystallography, calculation of the tight-binding band structure of a lattice is straightforward once the unit cell, Bravais lattice, and hopping rates have been determined. Translation groups in Euclidean space are commutative, so representation theory guarantees that a Bloch-wave ansatz will yield eigenstates and eigenenergies as a function of momentum Ashcroft and Mermin (1976).
In hyperbolic space, however, the discrete translation groups are non-commutative. As a result, there is no natural analog of a Bravais lattice. In fact, simply writing down the locations of all the lattice sites and the directions of all the bonds is already a non-trivial mathematical problem. To date, no hyperbolic equivalent of Bloch theory exists, and there is no known general procedure for calculating band structures in either the nearly-free-electron, or tight-binding limits. Specialized methods are known for the cases of trees Chen et al. (1974) or Cayley graphs of the free products of cyclic groups McLaughlin (1986). However, these cannot be applied to lattices. The only universal method is numerical diagonalization of the hopping Hamiltonian. This is a brute force method which yields a list of eigenvectors and eigenvalues, but no classification of eigenstates 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 spectrum and the density of states (DOS).
We produce finite-size samples using a cylindrical construction in which we start with a single layout polygon and successively add shells of nearest-neighbor polygons. We compute the effective lattice of this truncated layout, neglecting all hopping matrix elements that would connect to resonators outside the simulation size, and obtain a hopping matrix in a localized, delta-function, basis. Numerical diagonalization of this matrix yields eigenenergies and eigenstates for each lattice type and simulation size. A sample energy spectrum for a three-shell piece of the Euclidean kagome lattice is shown in Figure 5 (c). It has significantly reduced information compared to the full band structure calculation plotted in Figure 5 (a) and (b), but the presence of the flat band, and accompanying delta-function spike in the DOS, at is clear. 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 flat band at and the remaining, presumably dispersive, bands filling the range from to . For the two tilings formed from odd-sided polygons a robust spectral gap is visible between the flat band and the remaining eigenstates. We have verified that this gap is independent of system size and decreases with the number of sides of the layout polygon. The other gaps visible in the spectrum appear to be finite-size artifacts that close with increasing system size. (See Supplemental Figure 7 for details.)
Flat bands, like those seen in Figure 5, are quite rare in band structures Bergman et al. (2008); Leykam et al. (2018). However, because the kinetic energy in these bands effectively vanishes, they are ripe for strongly correlated many-body physics and non-perturbative interactions, such as fractional Quantum Hall states arising from discrete Landau levels. However, a pure flat-band description is only valid if the interaction strength is smaller than the gap to the nearest dispersive band. This is very much the case with Landau levels where magnetic energies far outweigh tunneling energies and tight-binding band structure, but otherwise gapped flat bands are highly unusual. Among Euclidean lattices there are only a handful of known examples where real-space topology of the flat band allows a gap to exist Bergman et al. (2008); Leykam et al. (2018). However, there are entire classes of hyperbolic systems that display gapped flat bands, including the -gon-kagome lattices, and certain generalizations of trees.
In all cases, the flat band arises because of an infinite multiplicity of localized eigenstates. In the kagome lattice the smallest such state consists of a single occupied hexagonal loop with alternating sign on each site and vanishing occupation elsewhere. This state can easily be generalized to the octagon-kagome lattice, but not to the heptagon or nonagon versions where the odd number of sides introduces geometric frustration. In these cases, the flat band still consists of localized states in the form of an alternating closed loop, but it must now extend over two tiles, as seen in the inset of Figure 6 (c). Without a hyperbolic generalization of Bloch theory, the real-space topology argument which proves that the flat band of the kagome lattice is not gapped Bergman et al. (2008) cannot be readily extended to hyperbolic kagome lattices. As a result, while the flat-band eigenstates have been identified in all cases, the origin of the spectral gap isolating these states from the rest of the spectrum for -gons remains unknown.
We have constructed a device to realize a finite section of the heptagon-kagome lattice. It consists of one central heptagon and two shells of neighboring tiles. A schematic of the layout is shown in Figure 6 (a), where each resonator has been approximated by a single line, and the lengths have not been held fixed. The actual device is fabricated using photolithography to etch CPW resonators into a nm film of niobium on a 500 m sapphire substrate. A photograph of the device is shown in Figure 6 (b). The resonators are mm long with a fundamental resonance frequency of GHz and a second harmonic of GHz. The hopping rate is MHZ at the first harmonic and MHz at the second. The length of all resonators has been held fixed, and the varying meanders required to make up the excess are clearly visible. In addition to the lattice itself, the circuit also contains four measurement ports, visible in each corner, used to interrogate the lattice.
In order to simulate the transmission properties of the device, we use the tight-binding calculations described in III.1 with small additional levels of disorder in both the on-site energies and hopping rates. After numerical diagonalization, we obtain the frequency and wavefunction of each eigenmode of the lattice. Assuming a typical HWHM of , we generate a Lorentzian resonance profile for each mode, centered about its eigenenergy. We then compute the level of mode matching between the eigenmodes and the input and output ports using the numerical wavefunctions. To calculate the transmitted electric field versus probe frequency, we sum the transmission through all eigenmodes, weighted by their Lorentzian line shapes and spatial mode matching. In the absence of disorder, transmission through the flat band 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. The remainder of the spectrum gives rise to a dense region of sometimes overlapping resonances, as shown in Figure 6 (c).
An experimental plot of transmitted power near the second harmonic frequency of the device is shown in Figure 6 (d), along with plots of the wavefunctions of selected eigenstates. (The fundamental modes of the device obey a different tight-binding model due to the asymmetry of the mode function within each resonator Schmidt and Koch (2013). See the Supplementary Information for details.) The experimental data and simulation show good qualitative agreement, particularly at the bottom of the spectrum near the flat band and at high energy near the uniform state. This device demonstrates that hyperbolic lattices can be produced on chip using CPW resonators, and paves the way to the study of interactions in hyperbolic space, and to simulation of new models with non-constant curvature. In particular, the gapped flat band is an ideal host for non-perturbative interactions and strongly correlated many-body states Bergman et al. (2008).
In conclusion, we have shown that circuit QED lattices of two-dimensional CPW resonators can be used to produce artificial photonic materials which exist in an effective curved space. In particular, we conducted numerical tight-binding simulations of a class of hyperbolic analogs of the kagome lattice, and demonstrated that they display a flat band similar to that of their Euclidean counterpart. However, for the case of odd-sided polygons this band is isolated from the rest of the spectrum. We also constructed an experimental device which realizes a finite section of non-interacting heptagon-kagome lattice. Mathematical investigation into the origin of the gap is still ongoing.
While our present work is purely non-interacting, interactions can be included via incorporating kinetic inductance materials such as NbTiN into the resonators to obtain a classical non-linearity Annunziata et al. (2010); Vissers et al. (2015); Swenson et al. (2013); Rotzinger et al. (2014), or via the addition of qubits to each resonator. These methods may also be applied to computer science or mathematics problems such as the study of non-linear operators on trees or Cayley graphs of free products of cyclic groups, rather than the lattices discussed here. Alternatively, it is possible to appropriate the techniques of hyperbolic metamaterials in which the equations of motion are tailored to mimic the existence of a nontrivial metric by deliberate modulation of the dielectric constant. A discrete version of the same effect can be realized here by tailoring the hopping magnitude, providing a simple route to models of more moderate curvature and specific metrics such as the Schwartzchild solution. The promise of strong interactions in these lattices leads to an exciting frontier which may provide answers to questions at the interface of quantum mechanics, gravity, and condensed matter physics, as lattices with these properties cannot be fabricated from actual materials.
Acknowledgements.We thank Peter Sarnak, János Kollár, Rivka Bekenstein, Charles Fefferman, and Siddharth Parameswaran for helpful discussions. This work was supported by the NSF, the Princeton Center for Complex Materials DMR-1420521, and by the MURI W911NF-15-1-0397.
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Appendix A Supplementary Information
a.1 CPW Resonators and Circuit QED Lattices
The CPW microwave resonators used in this work are formed by electrically isolating a portion of the center pin. Networks are formed by butting the ends of the center 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 the description of the network, and will not be discussed here.
When the resonators are laid out in a periodic lattice the circuit naturally maps to a tight-binding model of a 2D crystalline solid. The type of lattice, square, kagome, etc., is determined by how the resonators are connected together, and the frequency of the resonator maps to the atomic binding energy of the conduction band. The capacitive coupling between resonators can be expressed as a nearest-neighbor hopping rate, , whose magnitude is set by the size and shape of the gap between the center pins. The sign of is determined by two factors: the energetics of coupling two cavities, and the sign of the on-site mode function.
The eigenmodes of the CPW resonators are standing waves with an antinode of the voltage at each end of the cavity. Quantum computing applications typically use the fundamental, half-wave, mode. However, when studying network or lattice physics, the inherent asymmetry of the half-wave mode complicates the sign of the hopping rates. In many cases this minus sign can be removed by a local gauge transformation which redefines positive and negative on alternating sites Schmidt and Koch (2013). However, for many of the hyperbolic lattices considered here, this gauge transformation is not possible, and it is impossible to consistently define all the hopping rates with the same sign. This complicates the mathematical description and, in some cases, induces additional geometric frustration. The second-harmonic, or full-wave, mode is symmetric and does not have this sign problem. In the body of this paper, we therefore restrict to full-wave modes, and all results, unless explicitly stated, will refer to this case.
wave Regardless of whether the on-site wavefunction is even or odd, it is energetically favorable for the voltages on either side of the coupling capacitors connecting the resonators to be of opposite sign. Circuit QED lattices therefore exhibit band structures which are inverted from what is found in actual solid state-systems, with rapidly oscillating Bloch waves at the low-energy end of the spectrum. For full-wave modes, this physics can be captured by a standard tight-binding model with negative . For half-wave modes, a global definition of the sign of the on-site wavefunction may no longer be possible, and care must be taken to define the sign of consistently Ashcroft and Mermin (1976); Schmidt and Koch (2013). We proceed by defining an orientation for each resonator and arbitrarily designate one end of each resonator “positive” and one end “negative”. For the tight-binding model to be physical the sign of must be consistent with the designated orientation and always favor opposite voltages on either side of the coupling capacitors. Therefore, when it describes hopping between two ends of the same sign and when describing hopping between two ends of different sign.
As hinted above, the choice of the orientation of each half-wave resonator is arbitrary and merely a bookkeeping convention. However, some choices are, of course, more convenient and illustrative than others. For some layout lattices, e.g. square or hexagonal, the orientation of each resonator can be chosen consistently such that all hopping matrix elements are negative, even for half-wave modes Schmidt and Koch (2013). In these cases, the asymmetry of the on-site wavefunction can be gauged away by this judicious choice of orientations, and the half-wave tight-binding model is isomorphic to the version with even on-site wavefunctions, in much the same way that the magnon spectrum of an unfrustrated antiferromagnet is identical to that of the corresponding ferromagnet. However, if the layout lattice contains plaquettes with an odd number of sides, then an orientation which fixes the sign of the hopping matrix elements cannot generally be found. Compared to their full-wave counterparts, these tight-binding models exhibit additional geometric frustration due to the phase windings forced by the half-wave mode function.
This effect can most easily be seen by considering the maximally excited state of a kagome-like layout lattice, which has eigenvalue . The defining features of this state are: uniform probability of occupation of all the sites, and equal voltages on either side of all coupling capacitors. For full-wave modes, both of these conditions are easily satisfied by uniform occupation of all cavities with a single phase. For half-wave modes, this is not the case. Consider a single layout plaquette of the lattice and define the orientations such that going from the positive to the negative end of a resonator corresponds to moving clockwise around the plaquette. In order maintain equal voltages on both sides of all coupling capacitors using half-wave modes, the sign of the on-site wavefunction must alternate on neighboring sites, which cannot be done consistently on a plaquette with an odd number of sides.
Hyperbolic kagome lattices made from even-sided polygons are free of this effect and the asymmetry of the half-wave mode function can always be gauged away. Kagome-like lattices made from odd-sided polygons, on the other hand, are frustrated. Remarkably, the flat-band is immune to these frustration effects because it is spanned by states which consist of loops with an even number of edges, such as the one shown in Figure 8 (g). The dispersive bands above , however, are affected, and the spectral gap which isolates the flat band in the full-wave case disappears from the half-wave model.
a.2 Tight-Binding Models
a.2.1 System Size Effects
Numerical diagonalization studies of finite-size tight-binding models of the heptagon-kagome lattice were conducted for a series of system sizes, starting with a smallest simulation consisting of a central layout heptagon and one shell of nearest-neighbor plaquettes. We compute the effective lattice of this finite layout, neglecting all links to sites outside the current system size. Larger systems are created iteratively by adding successive shells of nearest neighbor layout polygons to the edge of the smaller simulations and recomputing the effective lattice.
Numerical eigenenergy spectra for a series of system sizes are shown in Figure 7. The flat band is clearly visible even for the smallest simulation with only one shell of neighboring polygons. The spectral gap between the flat band the rest of spectrum is already present, but not readily distinguishable from the finite-size-induced gaps at higher energy. Within the size of simulation currently possible, the higher-lying gaps all close progressively with increasing system size, and it remains unclear whether any of them also remain open at infinite size. Since the spectral gap between the flat band and the rest of the energy spectrum is readily visible for systems with two or more layers of neighbor polygons, our physical device, shown in Figure 6, was constructed at this depth.
a.2.2 Numerical Eigenstates
Color plots of selected numerical eigenstates from a three-shell simulation of the heptagon-kagome lattice are shown in Figure 8. Due to the negative hopping amplitude, the uniform-phase configuration typically associated with the ground state of a lattice is, in fact, the highest excited state, shown in Figure 8 (a). Additionally, it exhibits an radial amplitude modulation due to confinement from the missing links beyond the edge of the sample, which constitute an effective hard-wall potential. The next eigenstates down, Figure 8 (b) and (c), bear a striking resemblance to the Laguerre-type 0,1 and 0,2 modes of a Euclidean harmonic oscillator or a particle in a cylindrical box. The qualitative similarity persists for many modes throughout the spectrum, such as the one shown in Figure 8 (d) which resembles a Laguerre 4,7 mode.
Conversely, the eigenstates also show several features which are strikingly different from the Euclidean case. Because of the hyperbolic nature of the lattice, a macroscopic fraction of the lattice sites are in the outermost ring, and the spectrum therefore contains a relatively large number of states like the one in Figure 8 (e) which reside primarily at the edge of the system. Another intriguing feature is the existence of states like the one in Figure 8 (f), which display azimuthal amplitude and phase modulation with different periods. The azimuthal-angle dependence of eigenstates of cylindrically symmetric, two-dimensional Euclidean problems can be written in either sine-cosine or complex exponential forms, but there is only one azimuthal quantum number, so the two sets of modes differ only by a rotation. The presence of two different azimuthal periods in Figure 8 (f) suggests that the hyperbolic problem requires a second azimuthal quantum number.
A flat-band eigenstate is shown in Figure 8 (g). The analogous state in the kagome lattice encircles a single hexagonal plaquette, but this type of behavior is not possible in a hyperbolic kagome-like lattice formed with odd-sided polygons because the sign flips cannot be consistently maintained. Therefore, the flat-band state in the heptagon kagome lattice (or any -gon kagome lattice) consists of a single loop across two plaquettes in which the phase flips by between every neighboring pair of sites. It is a localized eigenstate which is protected from hopping by destructive interference in the triangular plaquettes which border the loop. Since this state has compact support, translations of it are orthogonal and form a degenerate manifold of states whose multiplicity is proportional to the system size.
a.3 Lattice Curvatures
Compared to spherical or hyperbolic space, familiar Euclidean space is unique in that there is no intrinsic length scale. In particular, polygons have the same internal angles, regardless of size. Therefore, any Euclidean tiling can be scaled up or down in size and remain unchanged. This is not true in spherical or hyperbolic space where there is a natural length scale which is set by the Gaussian curvature Carroll (1997). In both cases, the shape of a polygon depends on its size relative to this length scale Coxeter (1954); Cannon et al. (1997). For example, consider a triangle on the surface of a sphere which has one vertex at the north pole and the other two on the equator separated by longitudinally. The total internal angle of this triangle is (each angle is ), but if it is reduced in size until it is much smaller than the radius of curvature of the sphere, it tends to a Euclidean triangle with a total internal angle of . Tiles in spherical space therefore get “fatter” as they increase in size and can cover gaps left between Euclidean tiles of the same shape. Hyperbolic polygons, however, have smaller internal angles at their vertices than their Euclidean counterparts. Therefore, hyperbolic tilings are precisely those for which the tiles would overlap if drawn according to Euclidean geometry. (See Figure 3.) As hyperbolic polygons become larger and larger, their internal angles decrease progressively, and they become “pointier”.
Because of this size-dependent geometry, whether or not a set of tiles can form a valid lattice in curved space depends on their size. The polygons must be precisely big enough so that the tiles fit exactly. Increasing the size of the polygons any further will cause gaps or overlaps to appear between the tiles, depending on the sign of the 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 an intrinsic curvature given by the required ratio of inter-site spacing and .
In order to determine these curvatures for the hyperbolic tilings considered in this paper we make use of the Poincaré disc model conformal mapping of the two-dimensional hyperbolic plane with curvature onto the Euclidean unit disc Cannon et al. (1997). For concreteness, we consider only hyperbolic tilings which are generalizations of graphene to polygons with a larger number of sides. These are precisely those that produce kagome-like effective models. The layout tiling will exist when a hyperoblic -gon can be produced which is bounded by geodesics and whose hyperbolic internal angle is , allowing copies of it to be tiled in a graphene-like way with three tiles meeting at every vertex. Since the Poincaré disc model is a conformal model which preserves angles, it suffices to determine the polygon size which results in an internal angle of when drawn in the unit disc. The geodesics of the Poincaré disc model are circles which intersect the unit disc at an angle of and the metric is known analytically, so the radius and edge length of the tile required to form the layout lattice can be readily determined. The corresponding inter-site spacing for the effective lattice is determined by computing the distance between the midpoints of the edges of the layout polygon.
For the heptagon-graphene layout lattice the inter-site spacing is determined to be Coxeter (1954), and the curvature of the resulting heptagon-kagome effective lattice is slightly weaker with an inter-site spacing of . Lattice spacing and Poincaré disc model plots of a series of graphene- and kagome-like lattices are shown in Figure 9. The ratio of the lattice spacing to increases with increasing number of sides of the layout polygon because the corresponding attempt at a Euclidean tiling has more and more overlap that needs to be removed by increasing the size of the tiles. This is clearly visible in the plots as the polygon at the origin occupies a larger and larger fraction of the unit disc. In fact, the heptagon-graphene tiling has smallest ratio of lattice spacing to curvature of any tiling of the hyperbolic plane by a single polygon, regardless of vertex coordination number. Correspondingly, the heptagon-kagome has the weakest curvature of all the resulting effective lattices, making this pair natural entry points to the study of hyperbolic lattices.