Quantum simulation of hyperbolic space with circuit quantum electrodynamics: From graphs to geometry

We show how quantum many-body systems on hyperbolic lattices with nearest-neighbor hopping and local interactions can be mapped onto quantum field theories in continuous negatively curved space. The underlying lattices have recently been realized experimentally with superconducting resonators and therefore allow for a table-top quantum simulation of quantum physics in curved background. Our mapping provides a computational tool to determine observables of the discrete system even for large lattices, where exact diagonalization fails. As an application and proof of principle we quantitatively reproduce the ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincaré disk, and show how conformal symmetry emerges for large lattices. This sets the stage for studying interactions and disorder on hyperbolic graphs in the future. Importantly, our analysis reveals that even relatively small discrete hyperbolic lattices emulate the continuous geometry of negatively curved space, and thus can be used to experimentally resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity.

Figure 1

(a) We consider the regular tessellation of the hyperbolic plane with heptagons, embedded into the Poincaré disk with a hyperbolic metric. All neighboring lattice sites have equal hyperbolic distance and the unit disk boundary is infinitely far away from each point in the interior. (b) Finite graphs preserving sevenfold rotation invariance can be constructed by considering subsets that are topologically equivalent to $\ell =1,2,3\cdots$ concentric rings, shown here for $\ell =3$.

Figure 2

(a) The adjacency matrix can be approximated by the hyperbolic Laplacian in the continuum limit through Eq. (8). To derive this property, we choose an arbitrary site $z_i$ with coordination number 3 (blue diamond). When applying the automorphism $z\mapsto w\left(z\right)=\frac{z_i-z}{1-z{\overline{z}}_i}$, which maps $z_i$ to the origin, the three neighbors of $z_i$ (red squares) are mapped to an equilateral triangle. This implies Eq. (7), which can be expanded in powers of $h$ to yield the desired relation. (b) Sums over lattice sites are replaced by integrals over hyperbolic space according to Eq. (9). This is achieved by assigning to each site $z_i$ an effective hyperbolic triangle with interior angles $2\pi /7$ and area ${\mathcal{A}}_{▵}=\pi /28$.

Figure 3

Ground state energy $E_0$ (blue, lower data) and first excited state energy $E_1$ (orange, upper data) for the graph (filled circles) and corresponding continuous disk (empty circles). The solid lines are the asymptotic continuum formulas from Eq. (17). Inset: Partition function $Z$ summed over negative energies for the graph with $\ell =4$ (blue) and associated disk (red) as a function of inverse temperature $\beta$. The dashed line is the low-temperature asymptote $lnZ\sim -\beta E_0$.

Figure 4

Quantitative match between graph Green function $G_{ij}$ and continuum Green function $G(z_i,z_j)$. We fix site $z_i$ to be on the second ring, and plot the correlations as a function $F_j$ of site $z_j$. Upper panel: Results for $\omega =-2.95$ just below $E_0$. Left: The two plots are $F_j=G_{ij}$ and $F_j=G(z_i,z_j)$. The size of dots is proportional to $|F_j{|}^{1/2}$, and blue (red) corresponds to positive (negative) sign of $F_j$. Right: Mean correlation function vs hyperbolic distance $d_{ij}=d(z_i,z_j)$, where the red (black) data are the graph (continuum) function. To obtain the curves, we make a list of pairs $(F_j,d_{ij})$ and compute the average $F_j$ as a function of distance, with the error bar being the standard deviation. The quantitative agreement between graph and continuum is remarkable. Emergent conformal symmetry is reflected by the data points collapsing onto a single curve $G_{ij}=f\left(d_{ij}\right)$ with some function $f$ for large $\ell$. The main plots are for $\ell =6$, the insets for $\ell =3$. Lower panel: The same setting for $\omega =-2.5+0.1i$ with $\text{Re}\left(\omega \right)>E_0$. We plot the real part of the correlation function.

Figure 5

The angle ${\delta }_i={\chi }_i-{\varphi }_i$ is heavily oscillating between sites with nearby radius. In most cases, ${\delta }_i$ only depends on the value of the radius $|z_i|=r_i$. Here we plot the averaged angle ${\delta }_i$ vs site radius $r_i$ for the first five rings (comprising 847 sites). The red “error bars” indicate those cases where the mapping ${\delta }_i\leftrightarrow r_i$ is not unique. Still, for any given site $i$ we can unambiguously assign the value of ${\delta }_i$ through Eqs. (B7)–(B9).

Figure 6

Finite-size scaling. We define the $h$-dependent operator ${\widehat{A}}_h$ through Eq. (C1), which allows us to connect the physical value of $h=0.275798$ for the heptagonal lattice to a formal $h\to 0$ limit. For the radially symmetric test function $f\left(r\right)=cos\left[\pi d\right(r,0\left)\right]$ we show here ${\widehat{A}}_{h}f$ (empty black circles), the cubic approximation ${\widehat{A}}_h^{\left(3\right)}f$ from Eq. (C3) (filled blue squares), and the quadratic approximation ${\widehat{A}}_h^{\left(2\right)}=3f+\frac{3}{4}{h}^2{\mathrm{\Delta }}_{g}f$ (gray solid line). We subtract $3f$ for better visibility of the deviations. The three figures correspond to decreasing values of $h$, with the individual values given in the plot labels. As $h\to 0$, the quadratic approximation quantitatively reproduces the full expression.

Figure 7

Counting function $\mathcal{N}\left(r\right)$ for radii within the first $\ell =4$ rings versus the linear curve $14\rho +b$, with $b=-9.096$.