Black Hole on a Chip: Proposal for a Physical Realization of the Sachdev-Ye-Kitaev model in a Solid-State System

A system of Majorana zero modes with random infinite-range interactions—the Sachdev-Ye-Kitaev (SYK) model—is thought to exhibit an intriguing relation to the horizons of extremal black holes in two-dimensional anti–de Sitter space. This connection provides a rare example of holographic duality between a solvable quantum-mechanical model and dilaton gravity. Here, we propose a physical realization of the SYK model in a solid-state system. The proposed setup employs the Fu-Kane superconductor realized at the interface between a three-dimensional topological insulator and an ordinary superconductor. The requisite $N$ Majorana zero modes are bound to a nanoscale hole fabricated in the superconductor that is threaded by $N$ quanta of magnetic flux. We show that when the system is tuned to the surface neutrality point (i.e., chemical potential coincident with the Dirac point of the topological insulator surface state) and the hole has sufficiently irregular shape, the Majorana zero modes are described by the SYK Hamiltonian. We perform extensive numerical simulations to demonstrate that the system indeed exhibits physical properties expected of the SYK model, including thermodynamic quantities and two-point as well as four-point correlators, and discuss ways in which these can be observed experimentally.

Popular Summary

Black holes are of significant interest to physicists because they constitute one place where two fundamental theories that underlie our understanding of the Universe—general relativity and quantum mechanics—come into a stark conflict. One well-known example of such a conflict is Hawking’s black hole information paradox: General relativity demands that all information encoded in a piece of matter cast into a black hole is irretrievably lost, which contradicts the principle of unitary evolution that is inherent to quantum mechanics. Reconciling general relativity with quantum mechanics is perhaps the most important challenge facing modern physics. However, there is a fundamental difficulty to extracting experimental evidence: Astrophysical black holes can be observed from a great distance but cannot be actively probed by any existing or imaginable technique. Here, we engineer a solid-state system that behaves in many important ways precisely like a black hole but is amenable to experimental explorations using standard techniques.

Some strongly interacting electron systems—occurring in, for instance, certain types of solids and cold atomic gases—can behave in a very real and quantifiable sense as black hole horizons. These structures are often called “holographic quantum matter.” We propose a specific way to experimentally realize and probe such a model in a solid-state system. Our proposed setup employs special properties of the interface between a three-dimensional topological insulator containing a single massless Dirac fermion and an ordinary superconductor (e.g., lead or niobium). We perform extensive numerical simulations to demonstrate that the system indeed exhibits the physical properties expected of black hole horizons and discuss ways in which these properties can be probed experimentally.

We expect that our findings will pave the way for future studies of information theory and quantum chaos.

Figure 1

The proposed setup for a solid-state realization of the SYK model.

Figure 2

(a) Spectral functions, measurable in a tunneling experiment, in the conformal (strongly interacting) limit (red lines) and free-fermion limit (blue lines). (b) Numerically evaluated large-$N$ Matsubara Green’s functions for $J=1.0$, $T=0.001$, and different values of $K$. Red dashed line shows the conformal limit behavior Eq. (3.7) while the thick green and brown lines correspond to free-fermion result Eq. (3.4) with $K=0.1$ and 0.5, respectively.

Figure 3

Band structure (4.2) of the lattice model Eq. (4.1) for $\lambda =1$ and $m=0$ (blue dashed) and $m=0.5$ (red solid line). $X$ and $M$ denote the $(0,\pi )$ and $(\pi ,\pi )$ points of the Brillouin zone, respectively.

Figure 4

Numerical simulations of the BdG Hamiltonian Eq. (4.5). (a) Stadium-shaped hole geometry employed in the simulations. $R$ parametrizes the hole size whereas $R_B$ denotes the radius inside which the magnetic field is nonzero. (b) Energy levels $E_n$ of the BdG Hamiltonian Eq. (4.5) calculated for $N=0$ and $N=24$. Energies have been sorted in ascending order and plotted as a function of their integer index $n$. The shaded band represents the SC gap region. (c)–(f) Density plots of the typical zero-mode wave function amplitudes for $N=24$. The dashed circle in (c) has radius $R_B$. The following parameters are used to obtain these results: $\lambda =1$, $m=0.5$, ${\mathrm{\Delta }}_0=0.3$, $\mu =w_{\mu }=0$, $w_{\lambda }=w_{\mathrm{\Delta }}=0.1$, $L=42$, $R=10$, and $R_B=15$.

Figure 5

Thermodynamic properties of the many-body Hamiltonian Eq. (2.15). (a) Thermal entropy per particle and (b) heat capacity per particle. Dashed lines show the expected behavior for the SYK model with random independent couplings, solid lines show results for the couplings obtained from the giant vortex system. In all panels the same parameters are used as in Fig. 4 with $V_0$ adjusted so that $J=1$.

Figure 6

Level statistics analysis. Top row: Histograms of $\ln r_n$ obtained from the energy levels of the SYK Hamiltonian Eq. (2.15) with coupling constants taken from the giant vortex model with $V_0$ and $w_{\mu }$ adjusted so that $J=1$ and $K=0$. Solid lines indicate the expected distributions GOE (green), GUE (red), and GSE (orange) specified in Eq. (5.5). Bottom row: Results for the noninteracting case $J=0$, $K=1$. Black solid line represents the Poisson distribution Eq. (5.6). Histograms in all panels are averaged over eight independent realizations of disorder except for $N=24$, where 16 realizations are employed to obtain satisfactory statistics, and $N=32$, for which a single realization is used.

Figure 7

Spectral function $A(\omega )$ computed at zero temperature for coupling constants $J_{ijkl}$ obtained from the giant vortex calculation (top) and taken from the Gaussian distribution (bottom). Thin gray lines represent individual disorder realizations corresponding to a physical measurement in a system with quenched disorder. Thick lines reflect the average over 25 independent disorder realizations. Dashed lines represent the expected low-frequency behavior in the large-$N$ conformal limit Eq. (3.7). All parameters are as in Fig. 4 with $J=1$, $K=0$ and broadening $\delta =0.04$ in Eq. (5.8).

Figure 8

Out-of-time-order correlators for our giant vortex system (top) and random Gaussian couplings (bottom). Nonzero temperature $T=1$ has been taken and other parameters as in Fig. 4. Correlators are displayed for a single disorder realization (unaveraged), but different realizations give very similar results for all interacting cases. Detailed shape of the oscillations apparent in the $J=0$ curves depends sensitively on the specific disorder realization, but all realizations show qualitatively similar behavior.

Figure 9

Spectral function $A(\omega )$ calculated for the giant vortex geometry with $N=28$ and coupling constants $(J,K)=J_0(\cos \theta ,\sin \theta )$ taken to interpolate between the fully interacting and noninteracting limits.