Mimicking Dirac fields in curved spacetime with fermions in lattices with non-unitary tunneling amplitudes

Abstract

In this paper we show that a Dirac Hamiltonian in a curved background spacetime can be interpreted, when discretized, as a tight-binding Hamiltonian with non-unitary tunneling amplitudes. We find the form of the non-unitary tunneling matrices in terms of the metric tensor. The main motivation behind this exercise is the feasibility of such Hamiltonians by means of laser-assisted tunnelings in cold atomic experiments. The mapping thus provides a physical interpretation of such Hamiltonians. We demonstrate the use of the mapping on the example of a time-dependent metric in 2+1 dimensions. Studying the spin dynamics, we find qualitative agreement with known theoretical predictions, namely particle pair creation in an expanding Universe.