Abstract
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 complexphase 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 complexphase element developed here can be a key ingredient for future experimental simulation of various Hamiltonians with topological ground states.
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Introduction
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 farreaching applications from quantum manybody 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 groundbreaking experimental implementation of hyperbolic lattices^{6,7,8} 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^{16,17,18,19}. Conceptually, recent mathematical insights into hyperbolic lattices from algebraic geometry promise to inspire a fresh quantitative perspective onto curved space physics in general^{20,21,22}.
Hyperbolic lattices emulate particle dynamics that are equivalent to those in negatively curved space. They are twodimensional lattices made from regular pgons such that q lines meet at each vertex, denoted {p, q} for short, with (p − 2)(q − 2) > 4^{6}. 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 \({{{{{{{\mathcal{H}}}}}}}}=J{\sum }_{\langle i,j\rangle }({c}_{i}^{{{{\dagger}}} }{c}_{j}+{c}_{j}^{{{{\dagger}}} }{c}_{i})\), with \({c}_{i}^{{{{\dagger}}} }\) the creation operator of particles at site i, J the hopping amplitude, and the sum extending over all nearest neighbors.
In all previous experiments^{6,7,8}, 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 complexphase 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 twodimensional hyperbolic matter is four, six or higherdimensional, 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 work paves the way for theoretical studies of more complex hyperbolic matter systems and their experimental realization.
Results
Infinite hyperbolic lattices as unitcell 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 \({e}^{{{{{{{{\rm{i}}}}}}}}{k}_{\mu }}\) 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 tightbinding limit, ε_{n}(k), are the eigenvalues of the N × N Blochwave Hamiltonian matrix H(k). In the latter, the matrix entry at position \((n,{n}^{{\prime} })\) is the sum of all Bloch phases for hopping between neighboring sites z_{n} and \({z}_{{n}^{{\prime} }}\) after endowing the unit cell with periodic boundaries. (See Methods for an explicit construction algorithm of H(k).) The approach is visualized in Fig. 1a and b for the {6, 3} honeycomb lattice with N = 2 unit cell sites. The associated 2 × 2 Blochwave Hamiltonian is
with eigenvalues \({\varepsilon }_{\pm }({{{{{{{\bf{k}}}}}}}})=\pm J1+{e}^{{{{{{{{\rm{i}}}}}}}}{k}_{1}}+{e}^{{{{{{{{\rm{i}}}}}}}}{k}_{2}}\). This models the band structure of graphene in the noninteracting limit^{23,24}.
Recent theoretical insights into hyperbolic band theory (HBT) and nonEuclidean crystallography revealed that this construction also applies to hyperbolic lattices, as many of them split into Bravais lattices and unit cells^{20,25}. There are two crucial differences between twodimensional 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, \({T}_{\mu }{T}_{{\mu }^{{\prime} }}\ne {T}_{{\mu }^{{\prime} }}{T}_{\mu }\). Nonetheless, Bloch waves transforming as in Eq. (1) can be eigenfunctions of the Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}\) on the infinite lattice. These solutions are labelled by 2g momentum components k = (k_{1}, … ,k_{2g}) from a higherdimensional 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 Blochwave 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 unitcell 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 Fig. 1c, e, f. The infinite extent of space is implemented through distinct momenta k. Due to the noncommutative nature of hyperbolic translations, other eigenfunctions of \({{{{{{{\mathcal{H}}}}}}}}\) in higherdimensional 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^{21,22}, 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 unitcell circuits. In topolectrics, tightbinding 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 realvalued edges in unitcell circuits can be implemented using existing technology^{14}, we had to develop a tunable complexphase element to imprint the nonvanishing 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 offdiagonal entries are controlled by external voltages V_{a} and V_{b} according to V_{b}/V_{a}=\(\tan \phi\), so ϕ is tunable, with resolution limited only by the resolution of the sources that provide those voltages. Equation (3) therefore realizes a Blochwave term with ϕ = ϕ(k).
Validity of Blochwave assumption
Unitcell circuits of hyperbolic lattices only capture the Blochwave 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 higherdimensional 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^{6,18}, the boundary effect on the DOS can be partly eliminated by considering the bulkDOS^{17,27,28}, defined as the sum of local DOS over all bulk sites (see Methods). To implement periodic boundary conditions, we utilize finite graphs known as regular maps^{29,30,31,32}, which are {p, q} tessellations of closed hyperbolic surfaces with constant coordination number q that preserve all local pointgroup 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 Blochwave Hamiltonian H_{{p, q}}(k)^{25}. Our extensive numerical analysis, presented in Supplementary Info. Secs. I–III, shows that both bulkDOS 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^{21}, HBT is exact and all singleparticle energies on the graph read ε_{n}(k_{i}) with certain quantized momenta k_{i}. We explore their connection to higherdimensional Euclidean lattices in Supplementary Info. Sec. S III.
For the {8, 4} and {10, 5} lattices, we find that the bulkDOS on hyperbolic flakes deviates more significantly from the predictions of HBT. This may originate from (i) the omission of higherdimensional representations or (ii) enhanced residual boundary contributions to the approximate bulkDOS. The latter is due to the larger boundary ratio for {8, 4} and {10, 5} lattices (see Supplementary 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 higherdimensional representations mix with Bloch waves (selection rules) will shed light on their role in manybody or interacting hyperbolic matter in the future.
Note that the unitcell circuits can be adapted to simulate nonAbelian Bloch states. One such option is to use a specific irreducible representation as an ansatz for constructing the corresponding nonAbelian eigenstates^{22,33}. If the representation is ddimensional, then the nonAbelian Bloch Hamiltonian can be emulated by building a circuit with d degrees of freedom on each node, giving a total of Nd 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 unitcell circuit depicted in Fig. 4a. The {10, 5} lattice has two sites in its unit cell and four independent translation generators, resulting in the Blochwave Hamiltonian
with crystal momentum k = (k_{1}, k_{2}, k_{3}, k_{4}) (see Supplementary 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 noninteracting with only nearestneighbor hopping). Both systems belong to a larger family of {2(2g + 1), 2g + 1} Bravais lattices with twosite unit cells and 2g translation generators^{25}. 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 k_{3} = k_{4} + π in h(k).
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 \({{{{{{{\mathcal{S}}}}}}}}\) in momentum space, determined by the condition h(k) = 0. This is a complex equation and thus results in a manifold of real codimension two. Whereas this implies isolated Dirac points in graphene, the nodal surface of Dirac excitations in hyperbolic graphene is twodimensional because momentum space is fourdimensional, see Fig. 4b. The associated Dirac Hamiltonian is derived in Supplementary Info. Sec. S IV. At each Dirac point \({{{{{{{{\bf{k}}}}}}}}}_{0}\in {{{{{{{\mathcal{S}}}}}}}}\), 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 timereversal and inversion symmetries^{34,35}. Therefore, hyperbolic graphene is a synthetic topological semimetal and a platform to study topological states of matter. Its momentumspace topology is the natural fourdimensional analogue of twodimensional graphene and threedimensional nodalline semimetals^{36}.
We experimentally realized the unitcell circuit for hyperbolic graphene in topolectrics with four tunable complexphase 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 twodimensional 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 Fig. 4d, we measured the band structure along lines connecting representative points in the Brillouin zone. This further highlights both the tunability of the experimental setup and the extended bandtouching 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 \(\left{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\right\rangle=(1,\pm {e}^{{{{{{{{\rm{i}}}}}}}}{\alpha }_{{{{{{{{\bf{k}}}}}}}}}})\). The phase α_{k} changes by 2π upon encircling a Dirac node in the normal plane, creating a momentumspace vortex, and \(\left{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\right\rangle\) picks up a Berry phase of π (see Methods). We numerically compute the lowerenergy eigenstates \(\left{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{}\right\rangle\) in the twodimensional plane defined by k = (k_{1}, k_{2}, 0, π) and observe a vortexantivortex pair, see Fig. 4e. While the nontrivial Berry phase in graphene implies zeroenergy boundary modes, the bulkboundary correspondence in hyperbolic graphene is complicated by the mismatch of position and momentumspace dimensions, see Supplementary Info. Sec. S V.
By periodic tuning of the complexphase 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 \({{{{{{{\bf{a}}}}}}}}(t)={a}_{0}(\sin (\omega t),\cos (\omega t))\), where ω is the frequency of light, realizes a Floquet system with topologically nontrivial band gaps^{37,38}. In the unitcell 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 \({k}_{\mu }(t)={k}_{\mu }A\sin (\omega t+{\varphi }_{\mu })\), 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 complexphase 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^{17,32}. 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 twodimensional lattices to create effectively higherdimensional Euclidean or hyperbolic models. Electric circuits also admit measurements of the timeresolved evolution of states, thus giving access to various nonequilibrium phenomena beyond the Floquet experiment discussed in the text. Additionally, together with nonlinear, nonHermitian 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 fourdimensional momentum space. By joining multiple unitcell circuits, multilayer settings can be emulated: for instance, using two realvalued connections to join the same sublattice sites of hyperbolic graphene realizes AAstacked 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 momentumspace dimensions requires to reevaluate many properties of Dirac particles in the context of hyperbolic graphene such as the bulkboundary correspondence discussed earlier, or Klein tunneling and Zitterbewegung, which have been observed in onedimensional Euclidean condensed matter systems^{44,45,46} and discussed for graphene^{23,47}.
Methods
Blochwave Hamiltonian matrix
One can construct the Blochwave 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 \({\{{z}_{n}\}}_{n=1,\ldots,N}\). 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 Supplementary Info Sec. S I. A list of known hyperbolic lattices with their corresponding Bravais lattices and unit cells is given in ref. ^{25}.
Hamiltonian of realspace hyperbolic lattices
The Hamiltonians of hyperbolic lattices with open boundary conditions (flakes) were generated by the shellconstruction method used in refs. ^{6,25}. One obtains the Poincaré coordinates of the lattice sites and the adjacency matrix \({{{{{{{\mathcal{A}}}}}}}}\), where \({{{{{{{{\mathcal{A}}}}}}}}}_{ij}\) is 1 if sites i and j are nearest neighbours and 0 otherwise. The tightbinding Hamiltonian in firstquantized form is then \({{{{{{{\mathcal{H}}}}}}}}=J{{{{{{{\mathcal{A}}}}}}}}\), where J is the hopping amplitude. The adjacency matrices of hyperbolic lattices with periodic boundary condition (regular maps) were identified from mathematical literature^{30} and are listed in Supplementary Info Table S3. A larger set of hyperbolic regular maps has been identified in ref. ^{32}.
BulkDOS of hyperbolic flakes
To effectively remove the boundary contribution to the total DOS of a hyperbolic flake, we define the bulkDOS 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 \({{{{{{{{\mathcal{N}}}}}}}}}_{\epsilon }\) is the set of eigenstates with energies between ϵ and ϵ + δϵ. 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 Supplementary 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 Blochwave Hamiltonian of hyperbolic graphene can be written as
where \({d}_{x}({{{{{{{\bf{k}}}}}}}})=1\mathop{\sum }\nolimits_{\mu=1}^{4}\cos ({k}_{\mu })\) and \({d}_{y}({{{{{{{\bf{k}}}}}}}})=\mathop{\sum }\nolimits_{\mu=1}^{4}\sin ({k}_{\mu })\) with hopping amplitude J set to 1. The energy bands are \({\varepsilon }_{\pm }({{{{{{{\bf{k}}}}}}}})=\pm \sqrt{{d}_{x}{({{{{{{{\bf{k}}}}}}}})}^{2}+{d}_{y}{({{{{{{{\bf{k}}}}}}}})}^{2}}\), so the bandtouching region is determined by the two equations d_{x}(k) = 0 and d_{y}(k) = 0. With four kcomponents, these two equations define the twodimensional nodal surface \({{{{{{{\mathcal{S}}}}}}}}\) visualized in Fig. 4(b). Near every node \({{{{{{{\bf{Q}}}}}}}}\in {{{{{{{\mathcal{S}}}}}}}}\), H_{{10, 5}}(k) is approximated by the Dirac Hamiltonian
where \({{{{{{{\bf{u}}}}}}}}({{{{{{{\bf{Q}}}}}}}})=\mathop{\sum }\nolimits_{\mu=1}^{4}\sin ({Q}_{\mu }){{{{{{{{\bf{e}}}}}}}}}_{\mu }\,{{\mbox{and}}}\,{{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{Q}}}}}}}})=\mathop{\sum }\nolimits_{\mu=1}^{4}\cos ({Q}_{\mu }){{{{{{{{\bf{e}}}}}}}}}_{\mu }\). Here e_{μ} is the unit vector in the direction of k_{μ}. For the detailed derivation, see Supplementary Info. Sec. S IV.
Berry phase in hyperbolic graphene
We write Eq. (8) as
with \({r}_{{{{{{{{\bf{k}}}}}}}}}=\sqrt{{d}_{x}{({{{{{{{\bf{k}}}}}}}})}^{2}+{d}_{y}{({{{{{{{\bf{k}}}}}}}})}^{2}}\) and \({\alpha }_{{{{{{{{\bf{k}}}}}}}}}=\arctan ({d}_{y}({{{{{{{\bf{k}}}}}}}})/{d}_{x}({{{{{{{\bf{k}}}}}}}}))\). The eigenstates are \(\left{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\right\rangle=(1,\pm {e}^{{{{{{{{\rm{i}}}}}}}}{\alpha }_{{{{{{{{\bf{k}}}}}}}}}})\). The relative phase α_{k} undergoes a 2π rotation around any given node \({{{{{{{\bf{Q}}}}}}}}\in {{{{{{{\mathcal{S}}}}}}}}\), implying a π Berry phase. One can verify this numerically by taking a chain of momenta {k_{1}, k_{2},…, k_{n}} on the closed loop \({{{{{{{\bf{k}}}}}}}}(s)={{{{{{{\bf{Q}}}}}}}}+{{{{{{{\bf{u}}}}}}}}({{{{{{{\bf{Q}}}}}}}})\cos (s)+{{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{Q}}}}}}}})\sin (s)\), s ∈ [0, 2π], and then using the lowerenergy state to compute the Berry phase, given by \(\gamma=\,{{\mbox{Im ln}}}\,(\langle {\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{1}}^{}{\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{2}}^{}\rangle \langle {\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{2}}^{}{\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{3}}^{}\rangle \cdots \langle {\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{n}}^{}{\psi }_{{{{{{{{{\bf{k}}}}}}}}}_{1}}^{}\rangle )\) in the discrete formulation^{48}.
Floquet band gaps in hyperbolic graphene
With tunable complexphase elements, it is possible to drive individual momentum components of hyperbolic graphene periodically, realizing the timedependent Hamiltonian
where A is the driving amplitude, ω is the frequency, and φ_{μ} are offsets in the periodic drive. Applying Floquet theory^{49} and degenerate perturbation theory^{50} near a Dirac node \({{{{{{{\bf{k}}}}}}}}\in {{{{{{{\mathcal{S}}}}}}}}\), we determine the effective Hamiltonian in the limit A ≪ 1 and ω ≫ J, to order \({{{{{{{\mathcal{O}}}}}}}}({A}^{4})\), to be
Here \({{{{{{{{\mathcal{J}}}}}}}}}_{0}(A)\) is the zeroth Bessel function of the first kind and
The factor of \({{{{{{{{\mathcal{J}}}}}}}}}_{0}(A)\) in the first term of Eq. (12) slightly shifts the location of the node while the second term opens up a kdependent gap Δ(k). Clearly, if the phases φ_{μ} are identical, Δ(k) is trivial. For a generic set of phases φ_{μ}, however, there exists a onedimensional subspace of \({{{{{{{\mathcal{S}}}}}}}}\) where Δ(k) = 0, implying that the nodes remain gapless up to \({{{{{{{\mathcal{O}}}}}}}}({A}^{4})\). See Supplementary Info. Sec. VI for a more detailed derivation and discussion of the Floquet equations relevant for this work.
Tunable complexphase 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 Supplementary Info. Secs. VII and VIII.
The complexphase element as shown in Fig. 2 features four AD633 analog multipliers by Analog Devices Inc. The transfer function of these multipliers is given by \(W=\frac{({X}_{1}{X}_{2})\cdot ({Y}_{1}{Y}_{2})}{10\,\,{{\mbox{V}}}}+Z\), 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 SRR7045471M 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 complexphase 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 \(\frac{{V}_{a}\,{V}_{1}}{10\,\,{{\mbox{V}}}}{V}_{2}\) and \(\frac{{V}_{b}\,{V}_{1}}{10\,\,{{\mbox{V}}}}{V}_{2}\) 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: \({I}_{2}=\left(\frac{1}{{{{{{{{\rm{i}}}}}}}}\omega L}\left(\frac{{V}_{a}\,{V}_{1}}{10\,\,{{\mbox{V}}}}{V}_{2}\right)+\frac{1}{R}\left(\frac{{V}_{b}\,{V}_{1}}{10\,\,{{\mbox{V}}}}{V}_{2}\right)\right)\). The relationship between the current I_{1} and the applied voltages can be derived in the same fashion, yielding \({I}_{1}=\left(\frac{1}{{{{{{{{\rm{i}}}}}}}}\omega L}\left(\frac{{V}_{a}\,{V}_{2}}{10\,\,{{\mbox{V}}}}{V}_{1}\right)+\frac{1}{R}\left(\frac{{V}_{b}\,{V}_{2}}{10\,\,{{\mbox{V}}}}{V}_{1}\right)\right)\). One then obtains Eq. (3) by further choosing R = ωL and applying voltage signals of \(10\,\,{{\mbox{V}}}\,\,\sin (\phi )\) and \(10\,\,{{\mbox{V}}}\,\,\cos (\phi )\) to V_{a} and V_{b} respectively.
Data availability
All the data (both experimental data and data obtained numerically) used to arrive at the conclusions presented in this work are publicly available in the following data repository: https://doi.org/10.5683/SP3/EG9931.
Code availability
All the Wolfram Language code used to generate and/or analyze the data and arrive at the conclusions presented in this work is publicly available in the form of annotated Mathematica notebooks in the following data repository: https://doi.org/10.5683/SP3/EG9931.
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Acknowledgements
We thank A. Fahimniya, A. Gorshkov, A. Kollár, P. Lenggenhager, and J. Maciejko for inspiring discussions. A.C. and I.B. acknowledge support from the University of Alberta startup fund UOFAB Startup Boettcher. I.B. acknowledges funding from the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grants RGPIN202102534 and DGECR202100043. The work in Würzburg is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through ProjectID 258499086  SFB 1170 and through the WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter – ct.qmat ProjectID 39085490  EXC 2147. T.He. was supported by a Ph.D. scholarship of the German Academic Scholarship Foundation. T.N. acknowledges funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programm (ERCStGNeupert757867PARATOP). T.B. was supported by the Ambizione grant No. 185806 by the Swiss National Science Foundation.
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I.B. and R.T. initiated the project and led the collaboration. I.B. and A.C. performed the theoretical analysis for this work. H.B., T.He., T.Ho., S.I., A.F., T.K., A.S., L.K.U., M.G., and R.T. developed the tunable complex phase element and carried out the experimental implementation of unitcell circuits. A.C., H.B., T.N., T.B., and I.B. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Chen, A., Brand, H., Helbig, T. et al. Hyperbolic matter in electrical circuits with tunable complex phases. Nat Commun 14, 622 (2023). https://doi.org/10.1038/s41467023363596
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DOI: https://doi.org/10.1038/s41467023363596
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