Abstract
The Laplace operator encodes the behavior of physical systems at vastly different scales, describing heat flow, fluids, as well as electric, gravitational, and quantum fields. A key input for the Laplace equation is the curvature of space. Here we discuss and experimentally demonstrate that the spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved) and flat twodimensional spaces has a universally different structure. We use a lattice regularization of hyperbolic space in an electriccircuit network to measure the eigenstates of a ‘hyperbolic drum’, and in a timeresolved experiment we verify signal propagation along the curved geodesics. Our experiments showcase both a versatile platform to emulate hyperbolic lattices in tabletop experiments, and a set of methods to verify the effective hyperbolic metric in this and other platforms. The presented techniques can be utilized to explore novel aspects of both classical and quantum dynamics in negatively curved spaces, and to realise the emerging models of topological hyperbolic matter.
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Introduction
Curved spaces, traditionally studied in highenergy physics and cosmology, have recently been elevated to paramount importance in condensed matter physics for two reasons. First, the discovery of holographic principles^{1,2} revealed a fundamental hidden structure underlying certain interacting quantum manybody systems that allows to compute their properties from a theory in hyperbolic space of negative curvature. Remarkably, these insights have been applied successfully to analyze strongly correlated electronic systems with tools from holography and to gain insight into the nature of quantum entanglement in condensed matter systems^{3,4,5,6,7,8,9,10,11}. Second, major advancements in the mathematical characterization of classical and quantum states in negatively curved spaces^{12,13,14,15} sparked a resurgence of interest of the condensed matter and metamaterials communities in hyperbolic lattices^{16,17,18}, ushering the research of hyperbolic topological matter^{19,20,21}. These rapid developments call for new experimental platforms to implement tabletop simulations of hyperbolic toymodels.
However, systems that furnish negatively curved space^{22,23} are hard to realize experimentally. The mathematical reason for this is encompassed in Hilbert’s theorem: even the lowest dimensional model of a hyperbolic space, the hyperbolic plane, cannot be embedded in threedimensional Euclidean (flat) laboratory space. We cannot build a hyperbolic drum. This is in sharp contrast to the case of positive curvature: a sphere can be embedded in threedimensional space, and we can study the standing waves (hereafter called eigenmodes) of a spherical membrane, which directly relate to quantum numbers of atomic orbitals. Despite such obstacles, hyperbolic space can be emulated experimentally. For instance, it has been suggested^{24} that a nontrivial metric can be implemented in metamaterials via spatial variations of the electromagnetic permittivity of continuous media. However, it is very challenging to induce these variations in a controlled manner, which limits the applicability of such approaches.
Electric circuits^{25,26,27,28,29,30,31,32} and similar systems, e.g., coplanar waveguide resonators^{16}, overcome these experimental limitations by relying on a discretization of space. In electric circuit networks, the physical distances between the nodes are fundamentally decoupled from the metric that enters the longwavelength description of its degrees of freedom, namely the voltages and currents that pass through the circuit nodes. The latter depend merely on the circuit elements that connect the nodes. Compared to other experimental platforms, electric circuits significantly excel in their flexibility of design, ease of fabrication, and high accessibility to measurements.
In this work we present a strategy for verifying that electric circuits can emulate the physics of negatively curved spaces and we demonstrate that electric circuits can do so efficiently. For concreteness, we consider the most fundamental differential operator on curved spaces, the LaplaceBeltrami operator, which generalizes the notion of the Laplace operator on flat space. The first key result of our work is the experimental observation of negative curvature in the spectral ordering of the eigenmodes of the LaplaceBeltrami operator in hyperbolic space. To paraphrase the words of ref. 33, our measurements confirm that a hyperbolic drum has a sound distinct from a Euclidean drum. Second, since electric circuits allow for timeresolved measurements, we can study not only static, but also dynamic properties. Our measurements confirm that signals in the present realization travel along hyperbolic geodesics, a smoking gun signature for the negative curvature of space. Based on our results, we infer that electric circuit networks could be readily utilized to implement and to experimentally verify the predicted features of the recently studied hyperbolic models of refs. 13,14,15,16,17,18,19,20. We expect the presented methodology for extracting fingerprints of negative curvature to be generalizable to other platforms, in particular to superconducting waveguide resonators that may allow for exciting future incorporation of quantum phenomena^{16}.
Results
Spectra of Euclidean and hyperbolic drums
We start by comparing the eigenmodes of Euclidean and hyperbolic drums in the continuum. The hyperbolic plane, characterized by a constant negative Gaussian curvature K < 0, is naturally embedded in (2 + 1)dimensional Minkowski space as a hyperboloid sheet with fixed timelike distance from the origin, see Fig. 1a. To solve for the eigenmodes of the wave equation, it is convenient to set K = −4 and to employ the stereographic projection Fig. 1a, which maps the hyperbolic plane onto the Poincaré disk, i.e., the unit disk with length element \({{{\mathrm{d}}}}{s}^{2}={(1{x}^{2}{y}^{2})}^{2}({{{\mathrm{d}}}}{x}^{2}+{{{\mathrm{d}}}}{y}^{2})\).
The eigenmodes of the hyperbolic drum with \({x}^{2}+{y}^{2}\le {r}_{0}^{2}\, < \, 1\) correspond to the spectrum^{17,34,35} of the LaplaceBeltrami operator:
where Δ_{E} = (∂^{2}/∂x^{2} + ∂^{2}/∂y^{2}) is the usual Laplace operator in the Euclidean plane. Adopting Dirichlet boundary conditions, which yield vanishing amplitude on the disk boundary, the spectrum of the drum is given by solutions to:
where g ∈ {E, H} indicates the geometry, and \({\lambda }_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) is the frequency of the mode with angular momentum ℓ and with n radial zeroes. Solutions to Eq. (2) are superpositions of Bessel functions (associated Legendre functions) in the Euclidean (hyperbolic) case, cf. Methods.
We plot in Fig. 1b the first few solutions to Eq. (2) on the Euclidean vs. Poincaré disk for r_{0} = 0.94, which corresponds to our experimental realization discussed below. We observe a significant reordering of the eigenmodes characterized by (n, ℓ): while in the Euclidean case the first eigenmode with n = 1 is the fourth (not counting degenerate eigenmodes separately), in the hyperbolic case, it is only the sixth mode. This reordering becomes even more apparent when considering the angular momentum dispersion \({\lambda }_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) vs. ℓ displayed in Fig. 2a. In both the Euclidean and the hyperbolic case, several branches (corresponding to different values of n, indicated by red numbers) are discernible. The spectral reordering manifests as a reduced slope of the branches (relative to their spacing) compared to their behavior for the Euclidean drum. Consequently, eigenmodes with large ℓ and small n appear much earlier in the spectrum in hyperbolic compared to Euclidean space. The spectral reordering is stronger for larger radii r_{0}. This is intuitively understood from the fact that the circumference of a hyperbolic drum grows superlinearly with its radius, such that oscillations in the angular direction stretch over larger distances. This makes them energetically favorable over oscillations in the radial direction, resulting in the observed reordering.
Lattice regularization of the hyperbolic plane
To experimentally realize a hyperbolic drum in an electric circuit network, we discretize the continuous space formed by the hyperbolic plane. This is achieved by tessellating the hyperbolic plane with regular polygons; a regular tessellation with q copies of psided polygons meeting at each vertex is conventionally denoted by the Schläfli symbol {p, q}. The curvature of the continuous space constrains the possible choices of p and q: for vanishing curvature (Euclidean plane) they need to satisfy (p − 2)(q − 2) = 4, while negative curvature (hyperbolic plane) requires (p − 2)(q − 2) > 4. A given regular hyperbolic tessellation uniquely fixes the distance between neighboring sites (cf. Supplementary Note 3), in contrast to the Euclidean case where the distance can be scaled arbitrarily.
Interpreting the vertices as sites of a lattice and the edges as connections between nearest neighbors, we obtain a hyperbolic lattice. The sites and nearestneighbor connections form a graph whose Laplacian matrix gives the lattice regularization of the continuum LaplaceBeltrami operator^{17}, which is fully determined by the topology of the lattice. The metric of the underlying continuous space is manifested in the connectivity of the lattice sites and therefore in the graph without reference to the positions of the vertices. However, the positions of the graph nodes (i.e., lattice sites) are relevant for the interpretation of the graph as a lattice when explaining the effective physics.
Different tessellations of the hyperbolic plane are possible, and they generally differ in their symmetries and in how densely their vertices cover the disk. For our experiments, three different aspects of the modeled lattice are important: (i) the lattice should provide a good approximation of the continuum, (ii) a large fraction of the Poincaré disk should be covered to obtain strong signatures of the negative curvature, and (iii) ℓ = 0 modes should be easy to excite and distinguish from ℓ ≠ 0 modes. While aspects (i) and (ii) can both be satisfied by having a sufficiently large number of vertices, in practice, there will be a trade off between the two aspects: for a fixed number of vertices, tessellations with larger area per vertex cover a larger fraction r_{0} of the Poincaré disk, while for fixed coverage r_{0} a good approximation of the continuum is naturally achieved by tessellations that feature small area per vertex (i.e., which tile the hyperbolic plane densely)^{17}. Finally, (iii) depends on the symmetry properties of the lattice: a vertex at the origin of the disk allows for easy excitation and identification of ℓ = 0 modes and a high order of rotation symmetry prevents ℓ ≠ 0 modes to have nonvanishing weight at the origin of the disk, which would impede the identification of ℓ = 0 modes. We analyze and compare several different tessellations with respect to these three aspects in the Supplementary Note 3. These considerations favor the {3, 7} tessellation, which exhibits a sevenfold rotation symmetry with respect to a site at the center, and which covers a disk with radius r_{0} = 0.94 with only 85 sites, see Fig. 3a.
In the longwavelengthregime, eigenvectors of the Laplacian matrix can be associated with eigenmodes of the LaplaceBeltrami operator in the continuum. We match them by systematically determining the absolute value of the angular momentum ℓ of the eigenvectors by a Fourier transform of their components on the outermost sites. Due to the discreteness of the lattice, this analysis is only reliable for modes with sufficiently small ℓ and n, i.e., in the longwavelength limit. Note that while the Laplacian matrix is defined purely on the graph, to define angular momentum we need to interpret the graph as a regular lattice, i.e., identify the vertices with lattice sites. But, as mentioned above, the (relative) positions of those sites are uniquely defined by the graph via the values of p and q. We extract the angular momentum dispersion for the chosen tessellation, and in Fig. 2b compare it to the corresponding Euclidean {3, 6} tessellation with the same number of sites. As in the continuum, a strong spectral reordering is observed. This reordering is a universal feature of the spatial curvature and does, therefore, not rely on the details of the tessellation, as long as it adequately approximates the continuum.
Implementation in an electric circuit
In our experiments, the tessellation is realized as an electric circuit network (right half of Fig. 3a) with a node at each site. Nodes are coupled capacitively among each other and inductively to ground. The boundary conditions are implemented by additional capacitive coupling of the nodes in the outermost shell to ground. Effectively, this corresponds to adding one more shell with all nodes shorted to ground, i.e., it represents the lattice equivalent of the Dirichlet boundary conditions. A generic electric circuit network is described by Kirchoff’s law:
where I_{a} and V_{a} are the input current and voltage amplitude (for angular frequency ω) at node a, respectively. The matrix J(ω) is called^{29} the grounded circuit Laplacian, and generally depends on ω. In the continuum limit, the input current I at some position is related to the divergence of the current density j via I = ∇ ⋅ j, with j = σE = σ∇V, σ the conductivity, E the electric field due to an applied voltage V, and ∇ the del operator (for brevity, we dropped the subscript g indicating the geometry). Hence, \(I={{{{{{{\boldsymbol{\nabla }}}}}}}}\cdot \left(\sigma {{{{{{{\boldsymbol{\nabla }}}}}}}}V\right)=\sigma {{\Delta }}V\), establishing the interpretation of J as the restriction of the continuum Laplace operator to the grounded circuit. The impedance to ground of node a, Z_{a}(ω) = V_{a}/I_{a}, is fully determined by J and its resonances correspond to eigenmodes of J with eigenvalues λ ∝ 1/ω^{2} (see Methods). Note that this relationship could be changed to λ ∝ ω^{2} by exchanging the roles of capacitors and inductors in implementing the connections between the nodes resp. to the ground.
Three types of experiments are performed. First, an impedance analyzer is used to measure Z_{a} as a function of frequency f = ω/2π for each node a. The data for three input nodes are shown in Fig. 3b. Second, these eigenmodes are resonantly excited and their voltage profile is measured using lockin amplifiers. For the modes at the highest six frequencies, both magnitude (relative to the voltage at the input node) and phase (relative to a reference signal) are shown in Fig. 3c. In the final experiment, the circuit is stimulated by the broadband voltage pulse shown in Fig. 4b fed into the circuit as a current pulse at a node close to the boundary. Subsequently, the voltage is measured as a function of time at each node. We observe the pulse to propagate in the Poincaré disk (the full time dependence is shown in Supplementary Movie 1 and discussed in Supplementary Note 6). A snapshot of the instantaneous phase profile (obtained via a Hilbert transform) is shown in Fig. 4d, which visualizes the propagation of the pulse.
Evaluation of the experimental data
We proceed with discussing the results of these three measurements. Comparing the impedance of input node 1 (blue curve) to nodes 14 and 18, see Fig. 3b, we clearly observe the spectral reordering discussed in the previous section: there are four additional peaks for input node 14 and 18 located between the two highestfrequency peaks for input node 1. This implies that the second ℓ = 0 mode (i.e., the first mode with n > 0) is the sixth eigenmode. The explicit values of ℓ and n for specific modes can be deduced from the voltage profiles of the eigenmodes obtained in the second experiment, see Fig. 3c.
We further plot (orange squares in Fig. 2b) the extracted dispersion of the Laplacian frequencies \({\lambda }_{{{{{{{{\rm{H}}}}}}}}}^{n \ell }\) with the angular momentum \(\left\ell \right\), obtained by a circular Fourier transform of the measured signal. We observe an almost perfect match with the theoretically predicted values (blue dots in Fig. 2b) for the first few measured modes. However, higher modes are increasingly difficult to excite and detect, due to the finite resolution in frequency and space. We remark that the boundary sites of the present experimental realization of a hyperbolic lattice could be used to probe holographic dualities. For each eigenmode of the system, only its angular distribution on the boundary is important (cf. the angular momentum dispersion in Fig. 2b), yielding a novel and universal onedimensional physical system on the boundary. We leave a detailed examination of these intriguing edge modes to future studies.
Finally, we discuss the timeresolved measurements. We excite the densest region of the frequency spectrum (Fig. 3b) using a current pulse (Fig. 4b) of mean frequency 500 kHz (Fig. 4c). By exciting a large number of modes, we approximate the continuum response. The propagation of the pulse through the circuit network leads to the profile of instantaneous phases depicted in Fig. 4d, where the phase fronts can be easily identified by the positions of equal instantaneous phase. Since the connectivity of the nodes implements the metric of the Poincaré disk, these phase fronts form concentric hyperbolic circles, highlighted by black circles in Fig. 4d. This agrees with the theoretical expectation that the signal emanates from the excited node along geodesics, which are the generalization of straight lines in curved space (red lines in Fig. 4a).
Wave fronts are perpendicular to these geodesics and thus constitute concentric circles (black circles in Fig. 4a) up to corrections due to boundary reflections. In Fig. 4d–f, we have chosen an early time during the excitation such that contributions from such reflections do not have a significant impact on the measured phases. Finally, when plotting the phase vs. hyperbolic (d_{H}) and Euclidean (d_{E}) distance in Fig. 4e, f, respectively, we observe that the correlation of the phase with d_{H} is stronger than with d_{E}. This manifests that the propagation of the signal indeed follows hyperbolic rather than Euclidean geodesics, thereby verifying that the system realizes the hyperbolic rather than Euclidean metric.
Discussion
We have experimentally simulated the negatively curved hyperbolic plane, as evidenced both in the spectral ordering of the Laplace operator and in the signal propagation along curved geodesics. With an implementation encompassing only 85 lattice sites, we have readily observed an excellent approximation of the hyperbolic plane; at the same time, no technical constraint hinders significantly enlarging the number of sites in future applications. In particular, using existing chip manufacturing technology and commercially available components, electric circuits representing lattices with ~10^{4} sites should be within reach. In combination with the presented results, the efficient fabricability and high scalability of electric circuits elevates them into a versatile platform for emulating classical hyperbolic models, with several advantages over the previously considered methods^{16,24}.
First, electric circuits provide easy means for embedding hyperbolic lattices on a flat physical geometry, while allowing for unconnected wire crossings. Such flexibility could be utilized to include coupling beyond nearest neighbors and to implement the plethora of other hyperbolic tessellations^{22,23}. In particular, going beyond the presented emulation of the Laplace operator in a negatively curved space, the platform allows to emulate much more complex tightbinding models. These could, for example, be used to test the recently emerging concepts of hyperbolic band theory^{12,13,14}, hyperbolic crystallography^{15}, and hyperbolic topological insulators^{19,20}. Electric circuits also excel at providing time and spatially resolved access to the individual degrees of freedom.
Furthermore, including nonlinear and nonreciprocal elements in the network, such as transistors and diodes, is trivial^{36}. This enables experimental investigation of how phenomena like topological insulators^{19,20,27,28}, the nonHermitian skin effect^{37,38}, further nonHermitian topological systems^{39} or nonlinear topological systems^{40,41,42} interact with with the negative curvature underlying hyperbolic lattices. Given their large scalability, electric circuits could be manufactured with the goal to experimentally study nonlinear dynamics of systems with sizes that are unwieldy for numerical simulations. Staying instead within the linear regime, there is a relationship between particles moving freely on geodesics of negatively curved space and deterministic chaos, as illustrated by the Hadamard system^{43}. In combination with our experimental verification of the signal propagation along the geodesics, this relationship designates electric circuits a promising experimental platform to investigate classical models of chaos.
Crucially, our work demonstrates the experimental viability of two methods for verifying the hyperbolic nature, i.e., the negative curvature, of the simulated model, which is an important step toward realizing more complicated models. The two methods rely on approximating the LaplaceBeltrami operator using a simple nearestneighbor tightbinding model and then observing (1) a reordering of eigenmodes compared to flat space, or (2) the propagation of a pulse along hyperbolic geodesics. These methods are, at least in principle, transferable to other platforms, even though it may generally be more challenging to experimentally access the necessary (spatially or timeresolved) information. However, the first method can be applied in a minimal fashion that requires access to significantly less experimental data. As we show in Fig. 3b, it is sufficient to measure the response (here the impedance to ground) at two vertices, one at the origin and one away from it, in order to distinguish ℓ = 0 from ℓ ≠ 0 modes and observe the predicted mode reordering. In this respect, note that waveguide resonator circuits, were previously proposed as a platform for realizing hyperbolic models as well^{16}. However, no substantial experimental verification of the curvature has been performed so far. Our methods could be used to perform a similar analysis on that platform.
Let us finally remark that while coplanar waveguide resonators have been proposed as a promising platform for implementing quantum hyperbolic matter, it is also conceivable^{36} that superconducting qubits could potentially be combined with electric circuits in the future. This suggests another route toward exciting future generalizations of our work to quantum models. We expect such generalizations to inspire a new paradigm for designing and measuring holographic toymodels and topological or conformal boundary field theories in discrete geometries. In this context, it is worth reminding that theoretical models of hyperbolic quantum systems were proposed^{44}, which still await experimental implementation, including MERA tensor networks^{5,8} and topological quantum memories^{45,46}. These efforts have the potential to fundamentally alter our understanding of physics in curved spaces and imply novel views on problems in condensed matter theory, quantum gravity, cosmology, and holography.
Methods
Eigenmodes of the LaplaceBeltrami operator
The solutions to Eq. (2) on the disk \({{{{{{{{\mathcal{D}}}}}}}}}_{{r}_{0}}\) of radius r_{0} < 1 correspond to the eigenmodes of a drum of radius r_{0} in the corresponding geometry. They can be conveniently expressed using special functions. Going to polar coordinates (r, θ), one finds (cf. Supplementary Note 1) for the Euclidean metric:
where \({{{{{{{{\mathcal{J}}}}}}}}}_{\ell }(z)\) are the Bessel functions of the first kind and k_{nℓ} is the (n + 1)th zero of \(k\,\mapsto\, {{{{{{{{\mathcal{J}}}}}}}}}_{\ell }(k{r}_{0})\). From the angular part of the solution it follows that ℓ can be interpreted as the angular momentum. Furthermore, \({k}_{n\ell }=\frac{{z}_{\ell,n+1}}{{r}_{0}}\), where z_{ℓ,n} is the nth zero of \({{{{{{{{\mathcal{J}}}}}}}}}_{\ell }(z)\). The radial zeroes r_{m} of \({u}_{{{{{{{{\rm{E}}}}}}}}}^{n\ell }(r,\, \theta )\) are then given by:
such that m = 1, 2, …, n for the nontrivial zeroes r_{m} < r_{0}. Thus, \({u}_{{{{{{{{\rm{E}}}}}}}}}^{n\ell }\) has exactly n nontrivial radial zeroes.
For the hyperbolic metric, on the other hand, one finds (cf. Supplementary Note 1):
with \({P}_{\lambda }^{\ell }(z)\) the associated Legendre functions and k_{nℓ} the (n + 1)th zero of \(k\,\mapsto\, {P}_{\frac{1}{2}\left(1+{{{{{{{\rm{i}}}}}}}}k\right)}^{\ell }\left(\frac{1+{r}_{0}^{2}}{1{r}_{0}^{2}}\right)\). Again we can interpret ℓ as the angular momentum and n as the number of radial zeroes of \({u}_{{{{{{{{\rm{H}}}}}}}}}^{n\ell }\).
Lattice regularization
The graph Laplacian of a simple (i.e., undirected) graph is given by:
where D is the degree matrix (the diagonal matrix containing the number of adjacent sites for each site as entries) and A the adjacency matrix (A_{ab} = 1 if sites a and b are adjacent and zero otherwise). Assuming the graph represents a lattice regularization of a continuum, then any function u(x, y) induces a function on the lattice, via a ↦ u(x_{a}, y_{a}) =: u_{a}, and the action of the Laplacian matrix, ∑_{b} Q_{ab}u_{b}, can be expressed in terms of the continuum LaplaceBeltrami operator, e.g., following the steps outlined in ref. 17.
Tessellations of the Euclidean or hyperbolic plane constitute a lattice regularization of the continuum^{17}, and the boundaries of the tiles (i.e., vertices and edges) can be interpreted as forming a graph. If only a finite segment of the plane is tiled, the tessellation has a boundary, which corresponds to vertices of the graph that are attached to fewer edges than the bulk vertices. Naturally, this is reflected both in the adjacency matrix A as well as in the degree matrix D. However, if we impose Dirichlet boundary conditions for u(x, y) as we do in the main text, then u vanishes on the boundary sites, which allows us to drop them from the matrix description. Consequently, we are left only with the bulk part of Q. For a Euclidean {3, 6} tessellation, we find (cf. Supplementary Note 2):
where d is the distance between sites. For the hyperbolic tessellation {3, 7}, on the other hand, we find (cf. Supplementary Note 2):
where \(h=\tanh ({d}_{0})=0.496\,970\), and d_{0} = 0.545 275 is the hyperbolic distance between two neighboring sites in the Poincaré disk representation. For both tessellations, the leading contribution is the LaplaceBeltrami operator for the appropriate metric, such that eigenstates of Q correspond to \({u}_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) from Eq. (2) and the eigenvalues are proportional to \({\lambda }_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) (up to higherorder corrections).
Extraction of angular momentum dispersion
The angular momentum dispersion, \({\lambda }_{{{{{{{{\rm{g}}}}}}}}}^{n\ell }\) vs. \(\left\ell \right\), shown in Fig. 2b is extracted from the spectrum and eigenstates of the graph Laplacian using Fourier analysis on shells of the graph, i.e., sites that have approximately the same distance from the disk center and form a circle. A shell can therefore be considered as a onedimensional system with periodic boundary conditions with the polar angle taking the role of position. For each eigenvector u, its components on one of the shells, therefore, define a periodic function u_{shell}(θ) defined at discrete θ. By first interpolating u_{shell}(θ) and then performing a discrete Fourier transform on regular samples, we determine the dominant Fourier component which is interpreted as the angular momentum \(\left\ell \right\) of u. For the eigenstates shown in Fig. 2b it is sufficient to consider the outermost shell, but for higher eigenstates, considering additional shells can improve the results.
Theoretical description of electric circuit
In our circuit network, nodes are coupled with capacitance C, each node is coupled to ground via an inductance L and the boundary conditions are implemented by adding additional capacitive couplings to ground such that each node is capacitively coupled to seven other components. The grounded circuit Laplacian is then given by the graph Laplacian Q of the underlying (bulk) lattice and a contribution from the inductive grounding (neglecting resistances and other parasitic effects):
The spectral decomposition is therefore given by the eigenstates ψ^{β} and eigenvalues q^{β} of the Laplacian matrix, −Qψ^{β} = q^{β}ψ^{β}, with eigenvalues:
The eigenmode index can be decomposed into the principal and orbital index, β = (n, ℓ), to match the analytic solution in the continuum.
The inverse of J is called the circuit Green function and can be obtained by expanding J into eigenmodes (here we assume that J is Hermitian and the circuit grounded, as is the case for our circuit) \(J(\omega )={\sum }_{\beta }{j}^{\beta }(\omega ){\psi }^{\beta }{{\psi }^{\beta }}^{{{{\dagger}}} }\); then:
Assuming current fed into node a, i.e., I_{a} = ∑_{c}Iδ_{ca}, the impedance of that node to ground can be written in terms of the eigenmodes of J:
and the stationary voltage response, i.e., after equilibration, at some other node b is given by:
We observe that in both cases the result is a superposition of eigenmodes of J with the weight proportional to 1/j^{β}(ω), which has a resonance at:
By combining this result with Eq. (9) for the bulktolattice correspondence, it follows that a resonance of Z_{a} at frequency f^{β} = ω^{β}/(2π) corresponds to an eigenmode of the hyperbolic drum with eigenvalue:
This results in a spectral reversal where the lowestfrequency (small λ) eigenmodes of the LaplaceBeltrami operator correspond to the highestfrequency (large f) oscillations of the electric circuit. Equation (16) is used to plot the experimental data in Fig. 2. If the circuit is probed at one of the resonance frequencies, ω^{β}, then the dominant contribution to V_{b} is:
where j^{β} does not diverge in practice due to the presence of small resistive terms (see Supplementary Note 4 for a discussion of the impact of parasitic resistances and Supplementary Note 5 for an extended analysis of measured eigenmodes). This implies that the voltage profile encodes the eigenvectors \({\psi }_{b}^{\beta }\).
Electric circuit parameters
The capacitances of the electric circuit are implemented by ceramic capacitors with C = 1 nF and 1% tolerance, the inductances as power inductors with L = 10 μH, 5% tolerance and a minimal quality factor of 40 at 1 MHz. Nodes on the boundary have additional capacitors C to ground such that each node is connected to seven identical capacitors in total. Finally, each node is made accessible for in and ouput via SMB connectors.
Measurement details
The impedance measurements were performed in a twoterminal measurement configuration using a Zurich Instruments MFIA 5 MHz impedance analyzer. A short/open compensation routine was used to remove the residual impedance and stray capacitance of the test fixture. The impedance of all 85 circuit nodes has been recorded for frequencies in the range from 250 KHz to 1.75 MHz. To exclude transmission line effects in the measurement, the maximum cable length was restricted to be below 1.8 m.
For the measurement of the voltage profiles of the eigenmodes, a reference voltage signal and phase sensitive detection is needed. This was achieved by using three Zurich Instrument MFIA 5 MHz impedance analyzers as lockin amplifiers synchronized in frequency and phase. Each mode was excited by a current signal of the corresponding frequency fed into the node with the highest impedance peak at that frequency. The current signal was obtained by applying the sinusoidal reference voltage signal with fixed peaktopeak voltage of 1 V produced by one of the lockin amplifiers to a shunt resistor of 12 Ω. The other two lockin amplifiers were used to measure the voltages of the different nodes. All voltage signals demodulated with the reference signal were filtered with a digital lowpass filter of eighth order and a cutoff frequency of f_{−3 dB} = 0.7829 Hz. The readout of the real and imaginary part of the voltage took place after at least 16 filter time constants which corresponds to at least 99% settling of the lowpass filters in a step response.
The timeresolved measurements were carried out by seven Picoscope 4824, which are eight channel USB oscilloscopes with 20 MHz bandwidth and 12 bit resolution. In the experiment, the circuit was stimulated at node 31 by the broadband pulse:
with V_{0} = 4.3 V and f = 500 kHz. The pulse is generated by a 50 Ω function generator and the output current was fed directly into the input node. Since the oscilloscopes do not provide a separate trigger channel, one channel of each instrument was fed with a rectangular pulse synchronized with the excitation pulse to trigger the instruments. They used an edge trigger at 1 V in rapid trigger mode and sampled with 40 MS/s, i.e., every 25 ns. Assuming equal behavior of the circuit under repeated stimulation, which was verified during the measurement process by repeating the process described below ten times, the measurement was performed in two steps. First, the seven oscilloscopes were used to measure the voltage at nodes 1 through 49 (including the input node 31), then, in the second run, the input node and nodes 38 through 85 were measured. Finally, the measured realvalued signals V(t) were transformed into complexvalued ones using the Hilbert transform, therefore giving access to the instantaneous phase as the argument of the complexvalued signal:
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/10.3929/ethzb000503548.
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/10.3929/ethzb000503548.
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Acknowledgements
P.M.L. and T.B. were supported by the Ambizione grant No. 185806 by the Swiss National Science Foundation. T.N. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERCStGNeupert757867PARATOP). 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 Studienstiftung des deutschen Volkes. I.B. acknowledges support from the University of Alberta startup fund UOFAB Startup Boettcher and Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grants RGPIN202102534 and DGECR202100043.
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R.T. initiated the project, and together with T.N. and T.B. led the collaboration. P.M.L., A.S., L.K.U., T.Ho., T.He. and T.B. performed the theoretical analysis of the hyperbolic tessellations. P.M.L., A.S. and A.V. designed the electric circuit. S.I., H.B. and T.K. carried out the measurements, and together with P.M.L. and A.S. analyzed the collected data. P.M.L., A.S., I.B., T.N. and T.B. wrote the manuscript. P.M.L., A.S., L.K.U., T.Ho., T.He., A.V., M.G., C.H.L., S.I., H.B., T.K., I.B., T.N., R.T., and T.B. discussed together and commented on the manuscript.
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Lenggenhager, P.M., Stegmaier, A., Upreti, L.K. et al. Simulating hyperbolic space on a circuit board. Nat Commun 13, 4373 (2022). https://doi.org/10.1038/s41467022320424
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DOI: https://doi.org/10.1038/s41467022320424
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