Hyperbolic matter in electrical circuits with tunable complex phases

Curved spaces play a fundamental role in many areas of modern physics, from cosmological length scales to subatomic structures related to quantum information and quantum gravity. In tabletop experiments, negatively curved spaces can be simulated with hyperbolic lattices. Here we introduce and experimentally realize hyperbolic matter as a paradigm for topological states through topolectrical circuit networks relying on a complex-phase circuit element. The experiment is based on hyperbolic band theory that we confirm here in an unprecedented numerical survey of finite hyperbolic lattices. We implement hyperbolic graphene as an example of topologically nontrivial hyperbolic matter. Our work sets the stage to realize more complex forms of hyperbolic matter to challenge our established theories of physics in curved space, while the tunable complex-phase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.

Experimental Hamiltonian engineering and quantum simulation have become essential pillars of physics research, realizing artificial worlds in the laboratory with full control over tunable parameters and far-reaching applications from quantum many-body systems to highenergy physics and cosmology.Fundamental insights into the interplay of matter and curvature, for instance close to black hole event horizons or due to interparticle interactions [1][2][3], have been gained from the creation of synthetic curved spaces using photonic metamaterials [4,5].The recent ground-breaking experimental implementation of hyperbolic lattices [4,7,8] in circuit quantum electrodynamics [9][10][11] and topolectrical circuits [12][13][14][15] constitutes another milestone in emulating curved space, separating the spatial manifold underlying the Hamiltonian entirely from its matter content to engineer broad classes of uncharted systems [6,7,16,18].Conceptually, recent mathematical insights into hyperbolic lattices from algebraic geometry promise to inspire a fresh quantitative perspective onto curved space physics in general [2,3,22].
Hyperbolic lattices emulate particle dynamics that are equivalent to those in negatively curved space.They are two-dimensional lattices made from regular p-gons such that q lines meet at each vertex, denoted {p, q} for short, with (p − 2)(q − 2) > 4 [4].Such tessellations can only exist in the hyperbolic plane.In contrast, the Euclidean square and honeycomb lattices, {4, 4} and {6, 3}, are characterized by (p − 2)(q − 2) = 4. Particle propagation on any of these lattices is described by the tightbinding Hamiltonian H = −J i,j (c † i c j + c † j c i ), with c † i the creation operator of particles at site i, J the hopping amplitude, and the sum extending over all nearest neighbors.In all previous experiments [4,7,8], hyperbolic lattices have been realized as finite planar graphs, or flakes, consisting of bulk sites with coordination number q surrounded by boundary sites with coordination number < q.The ratio of bulk over boundary sites, as a fundamental property of hyperbolic space, is of order unity no matter how large the graph.Thus a large bulk system with negligible boundary, in contrast to the Euclidean case, can never be realized in a flake geometry.Instead, bulk observables on flakes always receive substantial contributions from excitations localized on the boundary.The isolation of bulk physics is thus crucial for understanding the unique properties of hyperbolic lattices.
In this work, we overcome the obstacle of the boundary and create a tabletop experiment that emulates genuine hyperbolic matter, which we define as particles propagating on an imagined infinite hyperbolic lattice, using topolectrical circuits with tunable complex-phase elements.This original method creates an effectively infinite hyperbolic space without the typical extensive holographic boundary-our system consists of pure bulk matter instead.The setup builds on hyperbolic band theory, which implies that momentum space of two-dimensional hyperbolic matter is four-, six-or higher-dimensional, as we confirm here numerically for finite hyperbolic lattices with both open and periodic boundary conditions.We introduce and implement hyperbolic graphene and discuss its topological properties and Floquet physics.Our  2), and the energy bands.Momentum k = (k1, k2) is an external parameter.d In topolectrical circuits, a complex-phase element imprints tunable Bloch phases along edges connecting neighboring sites.The circuit element is directed, with e iφ imprinted in one direction, and e −iφ in the other.This leads to Hermitian matrices H(k).e, f Unit-cell circuits for the {8, 4} (e) and {8, 3} (f ) hyperbolic lattices.The Bravais lattice is the {8, 8} lattice in either case, with 4 and 16 sites in the unit cell, respectively.In these lattices, Bloch waves carry a four-dimensional momentum k = (k1, k2, k3, k4).
work paves the way for theoretical studies of more complex hyperbolic matter systems and their experimental realization.

Infinite hyperbolic lattices as unit-cell circuits
The key to simulating infinite lattices is to focus on the wave functions of particles on the lattice.In Euclidean space, Bloch's theorem states that under the action of the two translations generating the Bravais lattice, denoted T 1 and T 2 , a wave function ψ k (z i ) transforms as Here z i is any site on the lattice, k = (k 1 , k 2 ) is the crystal momentum with µ = 1, 2, and e ikµ is the complex Bloch phase factor.In crystallography, we split the lattice into its Bravais lattice and a reference unit cell of N sites with coordinates z n , n ∈ {1, . . ., N }.The full wave function is obtained from the values in the unit cell by successive application of Eq. (1).Furthermore, the energy bands on the lattice in the tight-binding limit, ε n (k), are the eigenvalues of the N ×N Bloch-wave Hamiltonian matrix H(k).In the latter, the matrix entry at position (n, n ) is the sum of all Bloch phases for hopping between neighboring sites z n and z n after endowing the unit cell with periodic boundaries.(See Methods for an explicit construction algorithm of H(k).)The approach is visualized in Figs.1a and 1b for the {6, 3} honeycomb lattice with N = 2 unit cell sites.The associated 2 × 2 Bloch-wave Hamiltonian is with eigenvalues ε ± (k) = ±J|1+e ik1 +e ik2 |.This models the band structure of graphene in the non-interacting limit [9,23].
Recent theoretical insights into hyperbolic band theory (HBT) and non-Euclidean crystallography revealed that this construction also applies to hyperbolic lattices, as many of them split into Bravais lattices and unit cells [1,2].There are two crucial differences between two-dimensional Euclidean and hyperbolic lattices.First, the number of hyperbolic translation generators is larger than two, denoted T 1 , . . ., T 2g , with integer g > 1.Second, hyperbolic translations do not commute, T µ T µ = T µ T µ .Nonetheless, Bloch waves transforming as in Eq. ( 1) can be eigenfunctions of the Hamiltonian H on the infinite lattice.These solutions are labelled by 2g momentum components k = (k 1 , . . ., k 2g ) from a higher-dimensional momentum space.The dimension of momentum space is defined as the number of generators of the Bravais lattice.The associated energy bands ε n (k) are computed from the Bloch-wave Hamiltonian H(k) in the same manner as described above.
We are lead to the important conclusion that Blochwave Hamiltonians H(k) of both Euclidean and hyperbolic {p, q} lattices are equivalent to unit-cell circuits with N vertices of coordination number q. Bloch phases e iφ(k) are imprinted along certain edges in one direction and e −iφ(k) in the opposite direction, see Fig. 1d.Examples are visualized in Figs.1c, e, f.The infinite extent of space is implemented through distinct momenta k.Due to the non-commutative nature of hyperbolic translations, other eigenfunctions of H in higher-dimensional representations exist besides Bloch waves.They are labelled by an abstract k, where ψ k in Eq. (1) has d > 1 components and Bloch phases e iφ(k) are d×d unitary matrices.Presently very little is known about these states [3,22], but we demonstrate in this work that ordinary Bloch waves capture large parts of the spectrum on hyperbolic lattices.

Tunable complex phases in electrical networks
Topolectrical circuit networks are an auspicious experimental platform for implementing unit-cell circuits.In topolectrics, tight-binding Hamiltonians defined on finite lattices are realized by the graph Laplacian of electrical networks [12][13][14].Wave functions and their corresponding energies can be measured efficiently at every lattice site.While the real-valued edges in unit-cell circuits can be implemented using existing technology [14], we had to develop a tunable complex-phase element to imprint the non-vanishing Bloch phases e iφ(k) .Importantly, while circuit elements existed before that realize a fixed complex phase e iφ along an edge [8,26], changing the value of e iφ required to dismantle the circuit and modify the element.In contrast, the phase e iφ of the element constructed here can be tuned by varying external voltages applied to the circuit.In the future, this highly versatile circuit element can be applied in multifold physical settings beyond realizing hyperbolic matter, including synthetic dimensions and synthetic magnetic flux threading.
The schematic structure of the circuit element is shown in Fig. 2. It contains four analog multipliers, the impedance of which is chosen to be either resistive (for the bottom two multipliers) or inductive (for the top two multipliers).As detailed in Methods, their outputs are connected in such a way that the circuit Laplacian of the element reads where I 1 and I 2 are the currents flowing into the circuit from the points at potentials V 1 and V 2 , respectively.The diagonal entries merely result in a constant shift of the admittance spectrum.The off-diagonal entries are controlled by external voltages V a and V b according to V b /V a =tan φ, so φ is tunable, with resolution limited only by the resolution of the sources that provide those voltages.Equation (3) therefore realizes a Bloch-wave term with φ = φ(k).

Validity of Bloch-wave assumption
Unit-cell circuits of hyperbolic lattices only capture the Bloch-wave eigenstates of the hyperbolic translation group.To test how well this approximates the full energy spectrum on infinite lattices resulting from both Bloch waves and higher-dimensional representations, we compare the predictions of HBT for the density of states (DOS) to results obtained from exact diagonalization on finite {p, q} lattices with up to several thousand vertices and either open boundary conditions (flakes) or periodic boundary conditions (regular maps).In the case of flake geometries [4,18], the boundary effect on the DOS can be partly eliminated by considering the bulk-DOS [5,6,28], defined as the sum of local DOS over all bulk sites (see Methods).To implement periodic boundary conditions, we utilize finite graphs known as regu- e iφ 1 2 Tunable complex-phase element.a Hermitian hopping term a ± i b which is to be implemented between two nodes 1 and 2 in an electric circuit.b Symbol for the circuit element corresponding to the hopping term with e iφ ∝ a + ib.The impedance representation is given by Eq. ( 3) with a = cos(φ)/(iωL) and b = sin(φ)/(iωL).c Implementation of the circuit element using four analog multipliers (represented by the circles with a cross symbol).We choose R = ωL.The voltages Va and V b tune the phase φ = arctan V b /Va.This circuit implements the complex coupling from node 1 to 2 with phase e iφ as well as the back-direction from 2 to 1 with phase e −iφ .
lar maps [8,29,31,32], which are {p, q} tessellations of closed hyperbolic surfaces with constant coordination number q that preserve all local point-group symmetries of the lattice.
For the comparison, we consider lattices of type {7, 3}, {8, 3}, {8, 4}, {10, 3}, and {10, 5}.This selection is motivated by the possibility to split these lattices into unit cells and Bravais lattices, and hence to construct the Bloch-wave Hamiltonian H {p,q} (k) [1].Our extensive numerical analysis, presented in Suppl.Info.Secs.I-III, shows that both bulk-DOS on large flakes and DOS on large regular maps converge to universal functions determined by p and q.We find that HBT yields accurate predictions of the DOS for lattices {7, 3}, {8, 3}, and {10, 3}, see Fig. 3. Generally, the agreement between HBT and regular maps is better than for flake geometries, likely since no subtraction of boundary states is needed.For some regular maps, called Abelian clusters [3], HBT is exact and all single-particle energies on the graph read ε n (k i ) with certain quantized momenta k i .We explore their connection to higher-dimensional Euclidean lattices in Suppl.Info.Sec. S III.
For the {8, 4} and {10, 5} lattices, we find that the bulk-DOS on hyperbolic flakes deviates more significantly from the predictions of HBT.This may originate from (i) the omission of higher-dimensional representations or (ii) enhanced residual boundary contributions to the approximate bulk-DOS.The latter is due to the larger boundary ratio for {8, 4} and {10, 5} lattices (see Suppl.Info.Table S2).Despite the deviation, studying Bloch waves on these lattices, and their contribution to band structure or response functions, is an integral part of understanding transport in these hyperbolic lattices.Investigating the extent to which higher-dimensional representations mix with Bloch waves (selection rules) will shed light on their role in many-body or interacting hyperbolic matter in the future.
Note that the unit-cell circuits can be adapted to simulate non-Abelian Bloch states.One such option is to use a specific irreducible representation as an ansatz for constructing the corresponding non-Abelian eigenstates [22,33].If the representation is d-dimensional, then the non-Abelian Bloch Hamiltonian can be emulated by building a circuit with d degrees of freedom on each node, giving a total of N d nodes in the unit cell circuit.

Hyperbolic graphene
We define hyperbolic graphene as the collection of Bloch waves on the {10, 5} lattice, realized by its unit-cell circuit depicted in Fig. 4a.The {10, 5} lattice has two sites in its unit cell and four independent translation generators, resulting in the Bloch-wave Hamiltonian Info.Sec. S I for explicit construction).The two energy bands read ε ± (k) = ±J|h(k)|.Hyperbolic graphene mirrors many of the enticing properties of graphene on the {6, 3} lattice (henceforth assumed non-interacting with only nearest-neighbor hopping).Both systems belong to a larger family of {2(2g + 1), 2g + 1} Bravais lattices with two-site unit cells and 2g translation generators [1].
Restricting the sum in Eq. ( 5) to two complex phases, we obtain Eq. ( 2).In fact, hyperbolic graphene contains infinitely many copies of graphene through setting The most striking resemblance between hyperbolic graphene and its Euclidean counterpart is the emergence of Dirac particles at the band crossing points.These form a nodal surface S in momentum space, determined by the condition h(k) = 0.This is a complex equation and thus results in a manifold of real co-dimension two.Whereas this implies isolated Dirac points in graphene, the nodal surface of Dirac excitations in hyperbolic graphene is two-dimensional because momentum space is four-dimensional, see Fig. 4b.The associated Dirac Hamiltonian is derived in Suppl.Info.Sec. S IV.At each Dirac point k 0 ∈ S, momentum space splits into a tangential and normal plane.Within the latter, a π Berry phase can be computed along a loop enclosing the Dirac point, protected by the product of time-reversal and inversion symmetries [34,35].Therefore, hyperbolic graphene is a synthetic topological semimetal and a platform to study topological states of matter.Its momentum-space topology is the natural four-dimensional analogue of two-dimensional graphene and three-dimensional nodal-line semimetals [36].
We experimentally realized the unit-cell circuit for hyperbolic graphene in topolectrics with four tunable complex-phase elements.The circuit represents the Hamiltonian H {10,5} (k) at any desired point in the fourdimensional Brillouin zone.We measured the band structure in the two-dimensional plane defined by k = (k 1 , k 2 , 2π/3, 0) for varying k 1 , k 2 , which contains exactly two Dirac points, see Fig. 4c.We also obtained the accompanying eigenstates.In extended band-touching region of the model in momentum space, in contrast to the isolated nodal points in Euclidean graphene.
To visualize the nontrivial topology of hyperbolic graphene, we write the eigenstates as |ψ ± k = (1, ±e iα k ).The phase α k changes by 2π upon encircling a Dirac node in the normal plane, creating a momentum-space vortex, and |ψ ± k picks up a Berry phase of π (see Methods).We numerically compute the lower-energy eigenstates |ψ − k in the two-dimensional plane defined by k = (k 1 , k 2 , 0, π) and observe a vortex-antivortex pair, see Fig. 4e.While the nontrivial Berry phase in graphene implies zeroenergy boundary modes, the bulk-boundary correspondence in hyperbolic graphene is complicated by the mismatch of position-and momentum-space dimensions, see Suppl.Info.Sec. S V.
By periodic tuning of the complex-phase elements, it is also possible to imitate the effect of irradiation of charged carriers in hyperbolic lattices.In this context, recall that graphene irradiated by circularly polarized light, modelled by electric field E(t) = ∂a(t)/∂t and vector potential a(t) = a 0 (sin(ωt), cos(ωt)), where ω is the frequency of light, realizes a Floquet system with topologically nontrivial band gaps [11,12].In the unit-cell picture, this can be simulated by a fast periodic driving of the external momentum on time scales much shorter than the measurement time, parametrized as k µ (t) = k µ − A sin(ωt + ϕ µ ), with driving amplitude A = ea 0 and phase shift ϕ µ .We theoretically demonstrate that hyperbolic graphene with such k µ (t) exhibits characteristic gap opening in the Floquet regime, though the gap size varies over the nodal region in contrast to graphene (see Fig. 4f).Notably, part of the nodal region remains approximately gapless within the energy resolution of the experiment (see Methods), bearing potential to study exotic transport phenomena far from equilibrium.

Discussion
This work paves the way for several highly exciting future research directions in both experimental and theoretical condensed matter physics.Experimentally, the tunable complex-phase element developed here can be utilized in topolectrical networks to simulate Hamiltonians with topological ground states, such as the recently discovered hyperbolic topological band insulators [28,39] or hyperbolic Hofstadter butterfly models [6,32].In particular, local probes in electric circuits provide access to the complete characterization of the Bloch eigenstates, giving the necessary input to compute any topological invariant.We have shown how synthetic extra dimensions can be emulated efficiently through tunable complex phase elements, which may be used in conjunction with ordinary one-or two-dimensional lattices to create effectively higher-dimensional Euclidean or hyperbolic models.Electric circuits also admit measurements of the time-resolved evolution of states, thus giving access to various non-equilibrium phenomena beyond the Floquet experiment discussed in the text.Additionally, together with nonlinear, non-Hermitian or active circuit elements [40][41][42], interaction effects beyond the singleparticle picture can be captured in these models, allowing for experimental engineering of a wide range of Hamiltonians.
Theoretically, hyperbolic matter constitutes a paradigm for topological states of matter with many surprising and unique physical features, which are hinted at by the original energetic and topological properties of hyperbolic graphene with Dirac particles in four-dimensional momentum space.By joining multiple unit-cell circuits, multi-layer settings can be emulated: for instance, using two real-valued connections to join the same sublattice sites of hyperbolic graphene realizes AA-stacked bilayer hyperbolic graphene.Such studies will shed more light on the subtle interplay between lattice structure and energy bands, a topic that recently came into the focus of many researchers with the fabrication of moiré materials [43].The mismatch of position-and momentum-space dimensions requires to re-evaluate many properties of Dirac particles in the context of hyperbolic graphene such as the bulk-boundary correspondence discussed earlier, or Klein tunneling and Zitterbewegung, which have been observed in one-dimensional Euclidean condensed matter systems [44][45][46] and discussed for graphene [23,47].

Bloch-wave Hamiltonian matrix
One can construct the Bloch-wave Hamiltonian matrix H(k) of a {p, q} hyperbolic lattice if it can be decomposed into a {p B , q B } Bravais lattice with a unit cell of N sites, denoted {z n } n=1,...,N .The matrix is constructed as follows.(i) Initially set all entries of the matrix H(k) to zero.(ii) For each unit cell site z n , determine the q neighboring sites z i .(iii) For each neighbor z i , determine the translation T (i) such that z i = T (i) z m for some z m in the unit cell.(iv) If T (i) = 1, add 1 to H nm (k), otherwise add the Bloch phase e iφ(k) that is picked up when going from z n to z i .(v) Multiply the matrix by −J.The detailed procedure for the lattices considered in this work is documented in Suppl.Info Sec. S I A list of known hyperbolic lattices with their corresponding Bravais lattices and unit cells is given in Ref. [1].

Hamiltonian of real-space hyperbolic lattices
The Hamiltonians of hyperbolic lattices with open boundary conditions (flakes) were generated by the shellconstruction method used in Refs.[4] and [1].One obtains the Poincaré coordinates of the lattice sites and the adjacency matrix A, where A ij is 1 if sites i and j are nearest neighbours and 0 otherwise.The tight-binding Hamiltonian in first-quantized form is then H = −JA, where J is the hopping amplitude.The adjacency matrices of hyperbolic lattices with periodic boundary condition (regular maps) were identified from mathematical literature [8] and are listed in Suppl.Info Table S3.A larger set of hyperbolic regular maps has been identified in Ref. [32].

Bulk-DOS of hyperbolic flakes
To effectively remove the boundary contribution to the total DOS of a hyperbolic flake, we define the bulk-DOS as the sum of the local DOS over all bulk sites through Here, Λ bulk is the set of lattice sites with coordination number equal to q and N is the set of eigenstates with energies between and +δ .In the DOS comparison, we use the normalized integrated DOS (or spectral staircase function) This quantity is approximately independent of system size (number of shells), see Suppl.Info.Fig S2 .Note that the energy spectrum of a {p, q} lattice is in the range [−q, q].

Dirac nodal region of hyperbolic graphene
The Bloch-wave Hamiltonian of hyperbolic graphene can be written as where where Here e µ is the unit vector in the direction of k µ .For the detailed derivation, see Suppl.Info.Sec. S IV.

Berry phase in hyperbolic graphene
We write Eq. ( 8) as with The eigenstates are |ψ ± k = (1, ±e iα k ).The relative phase α k undergoes a 2π rotation around any given node Q ∈ S, implying a π Berry phase.One can verify this numerically by taking a chain of momenta {k 1 , k 2 , . . ., 2π], and then using the lower-energy state to compute the Berry phase, given by γ = Im ln( in the discrete formulation [48].

Floquet band gaps in hyperbolic graphene
With tunable complex-phase elements, it is possible to drive individual momentum components of hyperbolic graphene periodically, realizing the time-dependent Hamiltonian where A is the driving amplitude, ω is the frequency, and ϕ µ are offsets in the periodic drive.Applying Floquet theory [10] and degenerate perturbation theory [13] near a Dirac node k ∈ S, we determine the effective Hamiltonian in the limit A 1 and ω J, to order O(A 4 ), to be (12) Here J 0 (A) is the zeroth Bessel function of the first kind and The factor of J 0 (A) in the first term of Eq. ( 12) slightly shifts the location of the node while the second term opens up a k-dependent gap ∆(k).Clearly, if the phases ϕ µ are identical, ∆(k) is trivial.For a generic set of phases ϕ µ , however, there exists a one-dimensional subspace of S where ∆(k) = 0, implying that the nodes remain gapless up to O(A 4 ).See Suppl.Info.Sec.VI for a more detailed derivation and discussion of the Floquet equations relevant for this work.

Tunable complex-phase element
In the following we specify the components used in the circuit shown in Fig. 2 and derive Eq. ( 3).More technical details together with more detailed illustrations are given in Suppl.Info.Secs.VII and VIII.
The complex-phase element as shown in Fig. 2 features four AD633 analog multipliers by Analog Devices Inc.The transfer function of these multipliers is given by W = (X1−X2)•(Y1−Y2) 10 V + Z, where W is the output, X 1 , X 2 , Y 1 , Y 2 are the inputs (with X 2 and Y 2 inverted), and Z is an additional input.Note that 10 V is the reference voltage for the analog multipliers.The other components include the SRR7045-471M inductors, with a nominal inductance of 470 µH at 1kHz, which were selected to minimize variance in the inductance.To achieve tunability of the resistance value, the resistors connected to the bottom multipliers are the 50 Ω PTF6550R000BYBF resistor and the 50 Ω Bourns 3296W500 potentiometer in series.
To derive the circuit Laplacian of the complex-phase element as defined in Eq. ( 3), we consider the voltage drops over individual inductors and resistors in Fig. 2. First let us consider the pair on the left.The voltage drops are determined by the output voltages of the left multipliers and therefore equal to Va V1 10 V − V 2 and for the inductor and resistor respectively.The current I 2 is then the negated sum of these voltage drops, each multiplied by the respective admittance: The relationship between the current I 1 and the applied voltages can be derived in the same fashion, yielding One then obtains Eq. ( 3) by further choosing R = ωL and applying voltage signals of 10 V sin(φ) and 10 V cos(φ) to V a and V b respectively.
where n(z i ) is the set of q neighbours of site i.Without loss of generality, let us focus on the equation for an arbitrary site i.Some of its neighbours belong to its unit cell, denoted U i , and others belong to nearby unit cells.For clarity of notation, we re-label the coordinates of the sites in U i as u 1 ≡ z i .Due to the translation symmetry of the Bravais lattice, one can use U i as a reference unit cell and express any neighbouring site m ∈ n(u αm for some index α m ∈ {1, 2, ..., N } and some translation operator γ m ∈ Γ.The choice of (γ m , α m ) is unique.If m happens to be in U i , then γ m is the identity element, but in general it is a product of generators.Thus Eq. (S3) for site i can be written as Since H is invariant under Γ, all of its eigenstates belong to the irreducible representations of Γ.Here we focus on the U (1) representations.In other words, our eigenstates satisfy the Bloch condition [2] where k µ is the generalized crystal momentum corresponding to generator T µ .Based on how |ψ transforms under the generators, it is labelled by k = (k 1 , k 2 , ..., k 2g ), noting that only 2g generators are independent.Applying Eq. (S5) recurrently gives the Bloch condition for a generic translation operator γ = T µ1 T µ2 ...T µ , ψ k (γz) = e ikµ 1 e ikµ 2 ...e ikµ ψ k (z) = e i(kµ 1 +kµ 2 +...+kµ ) ψ k (z).(S6) Equation (S6) allows us to write Eq. (S4) in terms of the wavefunction coefficients within the reference unit cell U i : e iφ k (γm) ψ(u (i) αm ) ≡ −J m∈n(u with complex phase factors dependent on γ m as prescribed by Eq. (S6).We repeat the same procedure for the other sites u N in U i until we obtain a total of N equations, giving rise to a N × N adjacency matrix A(k) describing the complex-valued edges connecting sites within U i .Due to the translation symmetry of the Bravais lattice, we would obtain an identical A(k) (up to a change of basis) had we chosen a difference site i to begin with.Therefore the spectral problem of the tight-binding model on the infinite {p, q} hyperbolic lattice is now reduced to diagonalizing the k-dependent Hamiltonian H(k) = −JA(k).Note that by replacing the Fuchsian group with Z × Z, the above derivation applies to two-dimensional Euclidean lattices and reproduces the conventional band theory obtained via Fourier transformations.
While Eq. (S7) may seem complicated, in practice it is straightforward to construct A(k) for a given hyperbolic lattice if its Bravais lattice and unit cells are known.One starts by writing down the Poincaré-disk coordinates of all the sites in the central unit cell and constructing the PSU(1, 1) matrix representation of the Fuchsian group generators T µ (see Ref. [1] for detailed discussion on the geometry of hyperbolic lattices and the Fuchsian group generators).Then for each unit-cell site u n , one performs a numerical ground search for the specific product of generators that, when applied to some u m in the unit cell, yields a site that has the right hyperbolic distance from u n to be a nearest neighbor.(For each site u n , q neighbors exist.)These products of generators then give rise to the complex phases according to Eq. (S6).
As an example, let us consider the {10, 5} hyperbolic lattice, which can be decomposed into a {p B , q B } = {10, 5} Bravais lattice with 2 sites in each unit cell, labelled u 1 and u 2 as shown in Fig. S1.Their coordinates in the Poincaré disk are u 1 = r 0 e iπ/10 and u 2 = r 0 e i3π/10 with r 0 = cos(3π/10)/ cos(π/10).The Fuchsian group generators are and Here R(θ) = e iθ/2 0 0 e −iθ/2 is the rotation matrix.Their action on the complex coordinate z is defined as {7, 3} lattice -The Bravais lattice is {14, 7} with 56 sites in each unit cell (see Fig. S1).There are six independent Fuchsian-group generators, resulting in a six-dimensional hyperbolic Brillouin zone.The nonzero entries of the 56×56 Bloch-wave Hamiltonian are listed in Table S1.

SUPPLEMENTARY SECTION S II EXTENSIVE BOUNDARY OF HYPERBOLIC FLAKES
The boundary of a hyperbolic lattice with open boundary condition is not negligible even in the limit of large system size (unlike Euclidean lattices).This can be understood in the continuum limit.A hyperbolic circle of radius r on the Poincaré disk has circumference C H = 2πR sinh(r/R) and area A H = 4πR 2 sinh 2 (r/2R), where R = 1/ √ −K and K is the Gaussian curvature.The ratio C H /A H = coth(r/2R)/R approaches 1/R as r → ∞.Table S2 lists the number of total/boundary sites for the lattices we considered.The boundary sites consist of those with coordination number less than q.The boundary-to-total ratio indeed approaches a constant as system size increases.Furthermore, this ratio is higher for lattices with higher curvature per plaquette, which we derive below.
Each p-sided polygon in a {p, q} lattice can be divided into 2p right triangles of the same size by lines passing through the center of the polygon.The angles in each right triangle are π/2, π/p, and π/q.The area of a hyperbolic triangle is given by where θ is the total internal angle and R is the curvature radius.Here θ = π/2 + π/p + π/q.The total area of the polygon is then where we have used R = 1/ √ −K.Rearranging the equation gives the curvature per plaquette/polygon:

SUPPLEMENTARY SECTION S III DOS COMPARISONS BETWEEN HBT AND FINITE LATTICES
The Bloch-wave Hamiltonians are constructed under the assumption that the energy eigenstates of hyperbolic lattices behave like Bloch waves, such that they acquire a U(1) phase factor from one unit cell to the other.Due to the non-Abelian nature of the Fuchsian translation group Γ, eigenstates which transform as higher-dimensional representations of Γ can also be present.Exactly how much of the full energy spectrum is captured by the Bloch eigenstates is an ongoing research problem [3].One obvious approach to test the validity of the Bloch-wave assumption is to compare the energy spectra obtained by exact diagonalization of real-space, finite-sized hyperbolic lattices with those obtained from the Bloch-wave Hamiltonians.The main challenge is to eliminate the significant boundary effect.For a finite two-dimensional Euclidean lattice, the boundary becomes negligible at large system size.On the other hand, the boundary ratio of a hyperbolic lattice remains significant regardless of the system size (see Supplementary Sec. S II and Table S2).
In this supplementary section, we report two methods for isolating the bulk physics of finite hyperbolic lattices.We show that the resulting bulk density-of-states (bulk-DOS) is generally in good agreement with the DOS computed from the Bloch-wave Hamiltonians.We also discuss possible causes for the discrepancies.Hamiltonians and finite lattices in flake geometry.In the bottom panels, the boundary effect of the latter is effectively removed by using the bulk-DOS defined in Eq. (S21).For all lattices in consideration, both total DOS and bulk-DOS are independent of the system size, indicated by the number of shells used in the lattice construction.(Note that the 2-shell {10, 5} lattice has no bulk sites, so its bulk-DOS is undefined.)The efficacy of the boundary removal is most apparent in the low-energy region, where the linear growth in the integrated DOS is restored as dictated by Weyl's law [7].For lattices {7, 3}, {8, 3}, and {10, 3}, the bulk-DOS of finite lattices agrees very well with the band-theoretical prediction.On the other hand, lattices {8, 4} and {10, 5} demonstrate a stronger discrepancy.FIG.S3.Alternative definition of bulk.Keeping a smaller bulk region by defining the outermost two shells as the boundary improves the DOS comparison for lattices {7, 3} and {8, 3} significantly.However for lattices with higher curvature, corresponding to rapid inflation of sites in successive shells, this definition results in nearly featureless bulk-DOS data.
For each lattice in consideration, the Bloch-wave Hamiltonian is constructed following the procedure detailed in Supplementary Section "Hyperbolic Bloch-Wave Hamiltonians", where we also explicitly define all the Hamiltonians used in this work.The energy spectrum is a compilation of the eigenvalues of the Bloch-wave Hamiltonian on a fine grid of k-points in the Brillouin zone.It is then used to compute the normalized DOS for comparison with finite lattices.
As shown in Supplementary Fig. S2, the agreement between ρ bulk ( ) and DOS obtained from the Bloch-wave Hamiltonian is excellent for lattices {7, 3}, {8, 3}, and {10, 3}.The agreement is not as good for {8, 4} and {10, 5}, but is nevertheless a significant improvement over the comparison without boundary effect removed.The differences in the comparison are generally caused by a combination of (i) omission of eigenstates in higher-dimensional representations of Γ and (ii) contribution to ρ bulk ( ) by edge states penetrating deep into the bulk.The larger discrepancy for {8, 4} and {10, 5} lattices may be due to the high boundary ratios of their flakes (see Table S2), rendering them unsuitable for comparison with a purely bulk theory.We remark that one can opt for an alternative definition of the bulk region to obtain different bulk-DOS results.For example, defining the boundary region as the outermost two shells gives a smaller bulk region.As shown in Supplementary Fig. S3, this definition improves the agreement in lattices {7, 3} and {8, 3} but generates nearly featureless bulk-DOS for lattices {10, 3}, {8, 4}, and {10, 5}.For the {7,3} lattice, the FIG.S4.HBT vs. Hyperbolic Regular Maps.We identified several regular maps for lattices {7, 3}, {8, 3}, and {10, 3} from the online database of Conder [8].These regular maps are finite and boundary-less hyperbolic lattices embedded into high-genus surfaces.We compare the DOS obtained from their eigenvalues to band-theoretical predictions computed from Bloch-wave Hamiltonians.The comparison shows close agreement with the exception of additional finite-size-induced gaps in the DOS of regular maps.We average over several regular maps as they tend to have (likely accidental) degeneracies and finite-sized gaps in the energy spectra that we do not expect to represent the behavior of the infinite lattice.However, the agreement between HBT and each individual regular map is of comparable quality to the data shown here.FIG.S5.HBT vs. Higher-dimensional Euclidean lattices.The eigenvalues on graphs obtained from higher-dimensional Euclidean lattices (blue) agree exactly with the predictions from HBT (red).We plot the eigenvalues εi (in units of J) vs. i = 1, . . ., N , where N is the number of vertices on the graph.Each plot is labelled by the {p, q} lattice that is approximated through its unit cell in a Euclidean lattice of dimension 2g, the number of lattice points in each Euclidean direction, L, and the number of sites given by N = N L 2g , where N is the number of sites in the unit cell.
To construct the higher-dimensional Euclidean lattices, we start from a {p, q} lattice with unit dell D = {z 1 , . . ., z N } with n sites and a 2g-dimensional Bravais lattice.We specify the adjacency matrix (A ij ) through determining all nearest-neighbor bonds (i, j) with A ij = 1.Note that the higher-dimensional Euclidean lattice is an undirected graph, so no complex phase factors appear in the Hamiltonian, only 1s and 0s.Any lattice site v i is uniquely determined by the site in the unit cell z n and the Bravais lattice vectors σ = (σ 1 , . . ., σ 2g ) ∈ N 2g .We write in the following.We write i ↔ j if sites v i and v j are nearest neighbors and A ij = 1.We define 1 = (1, 0, 0, . . ., 0) and similarly μ for µ = 1, . . ., 2g.We use L µ sites in each direction µ of the 2g-dimensional lattice and impose periodic boundary conditions so that σ µ + L µ = σ µ for each µ, hence σ µ ∈ {1, . . ., L}.The quantization of each momentum component appearing in the eigenvalues ε(k i ) is then given by with k i = (k 1 , . . ., k 2g ).For all of the following lattices we confirm that the eigenvalues ε i = ε(k i ) of the N × N adjacency matrix on the higher-dimensional Euclidean lattice agrees exactly with the prediction from H {p,q} (k) under the quantization condition from Eq. (S24), see Fig. S5.{8, 3}-lattice.The unit cell has 16 sites {z 1 , . . ., z 16 } and momentum space is four-dimensional.The Euclidean with hopping amplitude J set to 1.The energy bands are The nodal (or band-touching) region thus satisfies We eliminate k 4 using cos 2 k 4 + sin 2 k 4 = 1 and obtain the following equation: Having three variables, this equation defines a two-dimensional surface, which is the nodal region S as projected onto the three-dimensional hyperplane (k 1 , k 2 , k 3 ) (see Fig. 4b of the main text).We now show that H {10,5} (k) is approximated by a Dirac Hamiltonian at every node Q ∈ S. Expanding d x (k) and d x (k) at k = Q + q for small q gives In the basis of vectors where e µ are the standard Cartesian unit vectors, the Hamiltonian near Q, describes relativistic Dirac particles with anisotropic velocities given by |u(Q)| and |v(Q)|.We confirmed that u(Q) and v(Q) are nonzero and linearly independent for Q ∈ S.

SUPPLEMENTARY SECTION S V BULK-BOUNDARY CORRESPONDENCE
In its idealized version of fermions hopping on a honeycomb lattice, semi-metallic graphene is a topological semimetal with zero-energy boundary states [9].The reason for this is that for any one-dimensional cut through the twodimensional Brillouin zone (avoiding a Dirac point), the Bloch wave Hamiltonian realizes a one-dimensional topological insulator in class AIII with protected boundary states in position space.We confirm this behavior in a numerical diagonalization of a {6, 3} flake: while the bulk DOS is small near zero energy, the edge DOS (defined as the difference between total BOS and bulk DOS) shows a pronounced peak at zero energy, see Fig. S6 Note that for this argument to work, the equality of dimension of position and momentum space are crucial.
The bulk topology of hyperbolic graphene is the four-dimensional analogue of graphene, as demonstrated by the π Berry phase around each Dirac node in the band-touching manifold.However, a similar theoretical construction of cuts in momentum space remains inconclusive, since position and momentum space have different dimensions.While first studies on the topological properties of hyperbolic lattices have appeared recently, the interplay between position and momentum space invariants remains an open problem.
We address the presence of boundary states in hyperbolic graphene with an unbiased numerical analysis.By using a finite-sized {10, 5} flake with 7040 sites, we compare bulk DOS and edge DOS, see Fig. S6.We observe that there is no pronounced peak of edge states at zero energy.While some energy ranges are strongly populated with edge states, these regimes do not coincide with regions of small bulk DOS so that their topological interpretation is questionable.On the other hand, we cannot fully exclude the possibility that topological boundary modes are present, as they might be obscured by the inherent inaccuracy in the separation of total DOS into bulk and edge contributions.This is particularly true for {10, 5}-flakes, which have an enormous fraction of boundary sites.Left.We compute the bulk-DOS and edge-DOS on a finite {6, 3}-flake with 2400 sites.While there is a reduced bulk-DOS at zero energy E = 0, due to the system being semi-metallic, a sharp peak of edge-DOS is visible at E = 0, corresponding to a topological boundary mode.Right.A similar analysis of a {10, 5}-flake, i.e. hyperbolic graphene, with 7040 sites does not yield the same pattern.While pronounced peaks of the edge-DOS are visible, they do not appear in energy regions of reduced bulk-DOS, and so their topological nature cannot be inferred from this analysis.
Because of the periodicity of H(t) and u(t), we can Fourier transform this equation to frequency space by Fourier decomposition where ω = 2π/T .Plugging this into Eq.(S41) yields There are infinitely many equations, but in general a truncated set of Fourier harmonics is sufficient to approximate the Floquet states and their quasi-energies to arbitrary accuracy.
Supplementary Discussion: Time-Periodic Hyperbolic Graphene Given the Bloch-wave Hamiltonian of hyperbolic graphene in Eq. ( 4), we add time-periodic terms to the momentum components where J is the nearest neighbour hopping amplitude, A is the driving amplitude, ω is the frequency, and ϕ 1 , ..., ϕ 4 are phase shifts in the sinusoidal terms.This model is inspired by previous studies on irradiated graphene [11,12], where the vector potential of a circularly polarized light, a(t) = a 0 (sin(ωt), cos(ωt)), modifies the momentum as k x → k x − ea 0 sin(ωt) and k y → k y − ea 0 cos(ωt).
To solve for the Floquet states, we compute the Fourier components of H {10,5} (k, t) as where we used with J m the Bessel function of the first kind.Note that in the limit A 1, the integrals are proportional to A |m| , so H (m) ∼ O(A |m| ).We work in the limit of small driving amplitude A and keep terms up to O(A 2 ).Rewriting Eq. (S44) in matrix form gives where we have truncated the matrix to only contain Fourier harmonics −2 ≤ n ≤ 2. This truncation does not affect the subsequent calculations using degenerate perturbation theory.The unperturbed Hamiltonian is where and the perturbation is where The unperturbed quasi-energy spectrum consists of many identical copies of the H {10,5} energy spectrum, ±|ε(k)|, shifted by nω.For k near the nodal region S, each pair of levels are nearly degenerate in comparison to their separation ω from all the other bands, i.e. |ε(k)| ω.In this limit we can apply degenerate perturbation theory.Since the middle two bands (n = 0) are the best approximation of the quasi-energy spectrum (the other bands are more distorted copies [10]), we will focus on the middle two bands and compute the energy-splitting at the nodal region.The effective Hamiltonian describing the energy splitting is [13] H eff = P H 0 P + P H where E 0 = 0 is the energy at the nodal region and P is the projection operator onto the middle two bands given by Plugging in H 0 and H 1 yields Due to nonlinear behaviour of the multipliers for applied voltages at the upper boundary of the allowed input range, i.e. voltages near 10 V, the magnitude of the impedances where cut in half by using two identical inductors in parallel and reducing the resistance with R = ωL still holding.Therefore the control voltages V a and V b can be operated in half of the input range, i.e. ±5 V, leading to the same behaviour of the phase element with doubled diagonal entries.The full Laplacian of four phase elements connected to the unit cell is the sum of the above Laplacians, Choosing ω 2 = 9/LC reduces the real part of the diagonal elements to zero in Eq. (S63).The remaining imaginary part on the diagonal only induces a constant shift of the spectrum and therefore does not alter the band structure under consideration in a qualitative sense.

SUPPLEMENTARY SECTION S VIII EXPERIMENTAL REALIZATION OF THE PHASE ELEMENT
A detailed circuit diagram is presented in Fig. S9, with the actual circuit board shown in Fig. S10.The phase element consists of four analog multipliers of type AD633 by Analog Devices Inc.The outputs W of the upper two multipliers are connected to two parallel inductors of type SRR7045-471M, with a nominal inductance of 470 µH at 1 kHz, forming inductance L. The outputs of the lower two multipliers are connected to a 50 Ω PTF6550R000BYBF resistor with a 50 Ω Bourns 3296W500 potentiometer in series.This combination of equally sized resistances and reactances allows for counter-rotating phase-variable impedances between V 1 and V 2 , as desired.The Z inputs are used for manual DC offset compensation.To set the offset a high ohmic potentiometer is inserted between the positive and negative supply lines and its output range is down scaled by a voltage divider for precise adjustability.Therefore the divider consists of a 50 kΩ Bourns 3299W503 potentiometer (R Z,2 ) between the supply voltages in combination with a 300 kΩ Yageo MFR-25FTF52-300K resistor (R Z,3 ) and a 1 kΩ Bourns 3296W102 resistor (R Z,1 ) to ground is used.The inputs for the supply voltages of the multipliers are buffered with 1 µF Murata GRM55DR72D105KW01L capacitors

FIG. 1 .
FIG.1.Unit-cell circuits.a Euclidean {6, 3} honeycomb lattice with two sites in the unit cell (full orange circles).Each site has 3 neighbors, some of them in adjacent unit cells (empty orange circles).b The wave function of particles hopping between unit cells picks up a complex Bloch phase, see Eq. (1).c The associated unit-cell circuit diagram encodes the Bloch-wave Hamiltonian H(k), Eq. (2), and the energy bands.Momentum k = (k1, k2) is an external parameter.d In topolectrical circuits, a complex-phase element imprints tunable Bloch phases along edges connecting neighboring sites.The circuit element is directed, with e iφ imprinted in one direction, and e −iφ in the other.This leads to Hermitian matrices H(k).e, f Unit-cell circuits for the {8, 4} (e) and {8, 3} (f ) hyperbolic lattices.The Bravais lattice is the {8, 8} lattice in either case, with 4 and 16 sites in the unit cell, respectively.In these lattices, Bloch waves carry a four-dimensional momentum k = (k1, k2, k3, k4).

FIG. 3 .
FIG. 3.Density of states.Integrated DOS computed from finite {p, 3} lattices vs. predictions from hyperbolic band theory (HBT) realized in unit-cell circuits.a DOS of a {10, 3} flake with 2880 sites.b Bulk-DOS of the same lattice as in a.With the boundary contribution removed, it agrees well with band theory.c Bulk-DOS of a {7, 3} flake with 847 sites vs. band theory.d The averaged DOS of five {8, 3} regular maps (each with ∼2000 sites) reveals excellent agreement with band theory.

4 µ=1
cos(k µ ) and d y (k) = − 4 µ=1 sin(k µ ) with hopping amplitude J set to 1.The energy bands are ε ± (k) = ± d x (k) 2 + d y (k) 2 , so the band-touching region is determined by the two equations d x (k) = 0 and d y (k) = 0.With four k-components, these two equations define the two-dimensional nodal surface S visualized in Fig. 4(b).Near every node Q ∈ S, H {10,5} (k) is approximated by the Dirac Hamiltonian

u 3 ,
and T 2 T −1 3 u 3 .The neighbours of u 3 are u 2 , T 3 T −1 2 u 2 , u 4 , and T 3 T −1 4 u 4 .The neighbours of u 4 are u 3 , T −1 1 u 1 , T 4 u 1 , and T 4 T −1 3 u 3 .Therefore the Bloch-wave Hamiltonian is FIG.S2.HBT vs. Hyperbolic Flakes.The top panels compare the normalized integrated DOS obtained from Bloch-wave Hamiltonians and finite lattices in flake geometry.In the bottom panels, the boundary effect of the latter is effectively removed by using the bulk-DOS defined in Eq. (S21).For all lattices in consideration, both total DOS and bulk-DOS are independent of the system size, indicated by the number of shells used in the lattice construction.(Note that the 2-shell {10, 5} lattice has no bulk sites, so its bulk-DOS is undefined.)The efficacy of the boundary removal is most apparent in the low-energy region, where the linear growth in the integrated DOS is restored as dictated by Weyl's law[7].For lattices {7, 3}, {8, 3}, and {10, 3}, the bulk-DOS of finite lattices agrees very well with the band-theoretical prediction.On the other hand, lattices {8, 4} and {10, 5} demonstrate a stronger discrepancy.
FIG.S6.Bulk-boundary correspondence.Left.We compute the bulk-DOS and edge-DOS on a finite {6, 3}-flake with 2400 sites.While there is a reduced bulk-DOS at zero energy E = 0, due to the system being semi-metallic, a sharp peak of edge-DOS is visible at E = 0, corresponding to a topological boundary mode.Right.A similar analysis of a {10, 5}-flake, i.e. hyperbolic graphene, with 7040 sites does not yield the same pattern.While pronounced peaks of the edge-DOS are visible, they do not appear in energy regions of reduced bulk-DOS, and so their topological nature cannot be inferred from this analysis.
FIG. S8.Complex-phase element (schematic).Circuit diagram of the implementation of a boundary phase element using analog multipliers to enable continuous gain tuning.
FIG. S9.Complex-phase element (detailed).We show the detailed circuit diagram of the implementation of a boundary phase element using analog multipliers to enable continuous phase tuning.Four analog multipliers built the core of this element.The voltages Va and V b which cause the phase tuning are fed into the X inputs of the upper and lower multipliers respectively.The outputs W of the upper two multipliers are connected inductors, whereas the outputs of the lower two multipliers are each connected to a tunable resistor, allowing for tunable phases as described in the derivation of the element's Laplacian.The resistors RZ,1, RZ,3 and the potentiometer RZ,2 are used for DC offset compensation.The supply voltage to the multipliers is connected via the connectors V+ and V−.The lines of the supply voltages are connected to capacitors to ground, to avoid high frequency signals to couple into the multipliers via these inputs.Via the connectors V1 and V2 the phase tuned signal is fed into an attached unit cell, which is not depicted.