Different models of gravitating Dirac fermions in optical lattices

TL;DR

This paper constructs lattice Dirac Hamiltonians for fermions in 2D optical metrics, exploring various lattice geometries and gauge transformations, including Rindler spacetime, with potential applications in quantum simulations.

Contribution

It introduces a method to realize lattice Dirac Hamiltonians for Rindler spacetime in different optical lattice geometries, expanding the toolkit for quantum simulation of relativistic effects.

Findings

Lattice Dirac Hamiltonians for Rindler spacetime can be realized with position-dependent tunneling.

Gauge transformations can modify hopping phases to simulate different spacetime metrics.

The approach applies to honeycomb, brickwall, semi-synthetic, and $ ext{pi}$-flux square lattices.

Abstract

In this paper I construct the naive lattice Dirac Hamiltonian describing the propagation of fermions in a generic 2D optical metric for different lattice and flux-lattice geometries. First, I apply a top-down constructive approach that we first proposed in [Boada {\it et al.,New J. Phys.} {\bf 13} 035002 (2011)] to the honeycomb and to the brickwall lattices. I carefully discuss how gauge transformations that generalize momentum (and Dirac cone) shifts in the Brillouin zone in the Minkowski homogeneous case can be used in order to change the phases of the hopping. In particular, I show that lattice Dirac Hamiltonian for Rindler spacetime in the honeycomb and brickwall lattices can be realized by considering real and isotropic (but properly position dependent) tunneling terms. For completeness, I also discuss a suitable formulation of Rindler Dirac Hamiltonian in semi-synthetic brickwall and $\pi$-flux square lattices (where one of the dimension is implemented by using internal spin states of atoms as we originally proposed in [Boada {\it et al.,Phys. Rev. Lett. } {\bf 108} 133001 (2012)] and [Celi {\it et al.,Phys. Rev. Lett. } {\bf 112} 043001 (2012)]).