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arXiv:2104.03875 · Physical Review Research

Synthetic gravitational horizons in low-dimensional quantum matter

Corentin Morice, Ali G. Moghaddam, Dmitry Chernyavsky, Jasper van Wezel, Jeroen van den Brink

Published Jun 9, 2021• 35 citations
arXivHTMLPDF

Abstract

We propose a class of lattice models realizable in a wide range of setups whose low-energy dynamics exactly reduces to Dirac fields subjected to (1+1)-dimensional [(1+1)D] gravitational backgrounds, including (anti-)de Sitter space-time.Wave packets propagating on the lattice exhibit an eternal slowdown for power-law positiondependent hopping integrals t (x) ∝ x γ when γ 1, signaling the formation of black hole event horizons.For γ < 1 instead the wave packets behave radically different and bounce off the horizon.We show that the eternal slowdown relates to a zero-energy spectral singularity of the lattice model and that the semiclassical wave packets trajectories coincide with the geodesics on (1+1)D dilaton gravity, paving the way for new and experimentally feasible routes to mimic black hole horizons and realize (1+1)D space-times as they appear in certain gravity theories.

DOI
10.1103/physrevresearch.3.l022022

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We propose a class of lattice models realizable in a wide range of setups whose low-energy dynamics exactly reduces to Dirac fields subjected to (1+1)-dimensional [(1+1)D] gravitational backgrounds, including (anti-)de Sitter space-time.Wave packets propagating on the lattice exhibit an eternal slowdown for power-law positiondependent hopping integrals t (x) ∝ x γ when γ 1, signaling the formation of black hole event horizons.For γ < 1 instead the wave packets behave radically different and bounce off the horizon.We show that the eternal slowdown relates to a zero-energy spectral singularity of the lattice model and that the semiclassical wave packets trajectories coincide with the geodesics on (1+1)D dilaton gravity, paving the way for new and experimentally feasible routes to mimic black hole horizons and realize (1+1)D space-times as they appear in certain gravity theories.
We investigate the wave packet dynamics and eigenstate localization in recently proposed generalized lattice models whose low-energy dynamics mimics a quantum field theory in (1+1)D curved spacetime with the aim of creating systems analogous to black holes. We identify a critical slowdown of zero-energy wave packets in a family of 1D tight-binding models with power-law variation of the hopping parameter, indicating the presence of a horizon. Remarkably, wave packets with non-zero energies bounce back and reverse direction before reaching the horizon. We additionally observe a power-law localization of all eigenstates, each bordering a region of exponential suppression. These forbidden regions dictate the closest possible approach to the horizon of states with any given energy. These numerical findings are supported by a semiclassical description of the wave packet trajectories, which are shown to coincide with the geodesics expected for the effective metric emerging from the considered lattice models in the continuum limit.
We investigate the wave packet dynamics and eigenstate localization in recently proposed generalized lattice models whose low-energy dynamics mimics a quantum field theory in (1+1)D curved spacetime with the aim of creating systems analogous to black holes. We identify a critical slowdown of zero-energy wave packets in a family of 1D tight-binding models with power-law variation of the hopping parameter, indicating the presence of a horizon. Remarkably, wave packets with non-zero energies bounce back and reverse direction before reaching the horizon. We additionally observe a power-law localization of all eigenstates, each bordering a region of exponential suppression. These forbidden regions dictate the closest possible approach to the horizon of states with any given energy. These numerical findings are supported by a semiclassical description of the wave packet trajectories, which are shown to coincide with the geodesics expected for the effective metric emerging from the considered lattice models in the continuum limit.
We investigate the wave packet dynamics and eigenstate localization in recently proposed generalized lattice models whose low-energy dynamics mimics a quantum field theory in (1+1)D curved spacetime with the aim of creating systems analogous to black holes. We identify a critical slowdown of zero-energy wave packets in a family of 1D tight-binding models with power-law variation of the hopping parameter, indicating the presence of a horizon. Remarkably, wave packets with non-zero energies bounce back and reverse direction before reaching the horizon. We additionally observe a power-law localization of all eigenstates, each bordering a region of exponential suppression. These forbidden regions dictate the closest possible approach to the horizon of states with any given energy. These numerical findings are supported by a semiclassical description of the wave packet trajectories, which are shown to coincide with the geodesics expected for the effective metric emerging from the considered lattice models in the continuum limit.
We investigate the sine model, a one-dimensional tight-binding Hamiltonian with position-dependent sinusoidal hopping, and demonstrate the formation of synthetic horizons where electronic wave packets exhibit exponential slowdown. Interestingly, using the exact transformation between this model and the Harper equation, we find that analogous semiclassical horizons can emerge in a quantum Hall setup within the Harper-equation representation and at half-filling for specific values of the magnetic flux. The Harper equation describes the eigenstates of a square-lattice tight-binding model subjected to a perpendicular magnetic field, as a family of one-dimensional problems parametrized by a conserved transverse quasi-momentum. We demonstrate the correspondence between the sine and Harper models by comparing their semiclassical phase-space portraits and by numerically studying wave-packet dynamics in both models. Furthermore, by applying sudden quenches to the sine model's hopping profile, we observe the emergence of states with an effective thermal Fermi-Dirac distribution characterized by a corresponding Unruh temperature. Our numerical calculations reveal a non-universal behavior of this temperature, suggesting the involvement of mechanisms beyond a simple low-energy description.
Synthetic horizons in models for quantum matter provide an alternative route to explore fundamental questions of modern gravitational theory. Here we apply these concepts to the problem of emergence of thermal quantum states in the presence of a horizon, by studying ground-state thermalization due to instantaneous horizon creation in a gravitational setting and its condensed matter analog. By a sudden quench to position-dependent hopping amplitudes in a one-dimensional lattice model, we establish the emergence of a thermal state accompanying the formation of a synthetic horizon. The resulting temperature for long chains is shown to be identical to the corresponding Unruh temperature, provided that the postquench Hamiltonian matches the entanglement Hamiltonian of the prequench system. Based on detailed analysis of the outgoing radiation we formulate the conditions required for the synthetic horizon to behave as a purely thermal source, paving a way to explore this interplay of quantum-mechanical and gravitational aspects experimentally.
Black holes are usually described either as classical geometries on which quantum fields propagate or as objects whose complete quantum description lies beyond semiclassical gravity. Hawking radiation already shows, however, that a black hole is not an exactly isolated system: the horizon carries a thermal scale and, after coarse graining, may be associated with fluctuations, dissipation and environmental correlations. Whether this horizon-scale openness can be encoded directly in the gravitational background, before any stochastic treatment of the metric, remains an open question. Here we answer it by introducing, for the first time, a conserved Hawking-scale anisotropic source representing the coarse-grained near-horizon atmosphere and solving the Einstein equations for a static black hole at fixed ADM mass. The dressed geometry has a renormalized mass function, a self-consistently shifted horizon and a modified surface gravity. Because the same lapse governs the horizon and timelike geodesics, the thermal response is accompanied by a correlated orbital restructuring: the effective potential is renormalized, the ISCO moves inward and the near-horizon residence time increases. The construction adds no Langevin force to a fixed Schwarzschild background; rather, it defines a deterministic dressed mean geometry, providing a controlled background on which stochastic metric fluctuations may subsequently be treated in the Einstein-Langevin framework. This establishes the deterministic mean-field sector as the necessary first step for future stochastic gravity programs addressing horizon-scale openness.