Different models of gravitating Dirac fermions in optical lattices
ICFO-Institut de Ciències Fotòniques,The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
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Received: 9 December 2016
Revised: 11 January 2017
Published online: 26 July 2017
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 et al., New J. Phys. 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 π-flux square lattices (where one of the dimension is implemented by using internal spin states of atoms as we originally proposed in [Boada et al., Phys. Rev. Lett. 108, 133001 (2012)] and [Celi et al., Phys. Rev. Lett. 112, 043001 (2014)]).
© EDP Sciences, Springer-Verlag, 2017