  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 gravitational backgrounds, including (anti-)de Sitter
spacetime. Wave-packets propagating on the lattice exhibit an eternal slowdown
for power-law position-dependent hopping integrals $t(x)\propto x^\gamma$ when
$\gamma\geq 1$, signalling 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 spacetimes as they appear in certain gravity
theories.
