Synthetic gravitational horizons in low-dimensional quantum matter

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 position-dependent hopping integrals $t\left(x\right)\propto x^{\gamma }$ when $\gamma \ge 1$, signaling the formation of black hole event horizons. For $\gamma <1$ 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.

Figure 1

Sketch of the lattice-gravity correspondence. (a) A lattice model with position-dependent hopping which mimics a curved space-time and can possess event horizons. (b) Schematics of the 2D black hole anti-de Sitter space-time considered here. The velocity in space-time diminishes at the vicinity of a static horizon $x_H$. The black curved line indicates a lightlike geodesic (worldline of free-falling massless particles) which exponentially approaches the horizons at $\tau \to \infty$. The local light cones attached to the worldlines are also shown at some points in space-time. Moving from right to left the light cones always shrink, as dictated by $f_{\gamma =1}\left(x\right)=\alpha x$ in Eq. (2).

Figure 2

Time evolution of a Gaussian wave packet in the lattice model, with $\gamma =1, n_0=800, p_0=-\pi /2, w=50$, and $N=1001$. The wave packet slows down and localizes at the origin of the lattice, where it disintegrates.

Figure 3

Position as a function of time of a Gaussian wave packet in the lattice model for five values of $\gamma$, with $x_0=800, p_0=-\pi /2, w=50$, and $N=1001$. The solid lines represent numerical calculations, while the dashed lines represent Eq. (7). For $\gamma <1$, the wave packet bounces on the origin of the lattice, while for $\gamma \ge 1$, it reaches the end of the lattice asymptotically.

Figure 4

Overlap of ${\psi }_G\left(\tilde{\tau }\right)$ and $\psi \left(\tilde{\tau }\right)$ as a function of time. It stays almost constant up to the point where the wave packet reaches the proximity of the origin of the lattice.

Figure 5

DOS $D\left(E\right)$ of the lattice model for four values of $\gamma$. While the uniform hopping model $t_n=1$ has van Hove singularities at the two ends of the band, for $\gamma \ge 1$ a divergent DOS appears at $E=0$.