Superfluidity of Dirac Fermions in a Tunable Honeycomb Lattice: Cooper Pairing, Collective Modes, and Critical Currents

Abstract

Motivated by recent experiments on atomic Dirac fermions in a tunable
honeycomb optical lattice, we study the attractive Hubbard model of
superfluidity in the anisotropic honeycomb lattice. At weak-coupling, we find
that the maximum mean field pairing transition temperature, as a function of
density and interaction strength, occurs for the case with isotropic hopping
amplitudes. In this isotropic case, we go beyond mean field theory and study
collective fluctuations, treating both pairing and density fluctuations for
interaction strengths ranging from weak to strong coupling. We find evidence
for a sharp sound mode, together with a well-defined Leggett mode over a wide
region of the phase diagram. We also calculate the superfluid order parameter
and collective modes in the presence of nonzero superfluid flow. The
flow-induced softening of these collective modes leads to dynamical
instabilities involving stripe-like density modulations as well as a
Leggett-mode instability associated with the natural sublattice symmetry
breaking charge-ordered state on the honeycomb lattice. The latter provides a
non-trivial test for the experimental realization of the one-band Hubbard
model. We delineate regimes of the phase diagram where the critical current is
limited by depairing or by such collective instabilities, and discuss
experimental implications of our results.