Topological superfluids and the BEC-BCS crossover in the attractive Haldane-Hubbard model

Abstract

Motivated by the recent realization of the Haldane model in shaking optical lattice, we investigate the effects of attractive interaction and BEC-BCS crossover in this model at and away from half filling. We show that, contrary to the usual s-wave BEC-BCS crossover in the lattice, a topological superfluid with Chern number C = 2 appears in an extended region of phase space for intermediate strength of the attractive interaction in the interaction-density plane. When inversion symmetry is broken, a new gapless topological state is realized. We also investigate the fluctuations in these superfluid phases and show that the Anderson-Bogoliubov mode is quadratic due to time-reversal symmetry breaking and the existence of an undamped Leggett mode in the strong coupling limit.Introduction. Cold atoms in optical lattices can be used to simulate important models in condensed matter physics due to their high controllability and versatility [1,2]. This was demonstrated beautifully by the recent realizations of Haldane [3][4][5] and Hofstadter models using shaking lattice and Raman laser techniques, respectively. In these experiments, the existence of topological bands is verified using Bloch oscillations with either a Bose condensate [6] or free fermions [3]. With tunable interactions in the optical lattices, these advances open new avenues for the controlled study of interaction effects in topological system and pave the way to the possibility of realizing fractional Chern insulators.Perhaps by far the best studied interacting topological state is the fractional quantum Hall state, in which the strong Coulomb repulsion between electrons generate emergent fractional quasi-particles that obey abelian or non-abelian statistics [7][8][9][10][11][12][13]. Naturally, with the discovery of topological insulators [14,15], a great deal of efforts has been to investigate its interacting counterparts [16][17][18][19]. In the case of Haldane model, several recent studies have focused on the interplay between magnetic instabilities and the possible topological ground states in the case of repulsive interactions [20][21][22][23][24], as well as possible superconducting states with attractive interactions at half filling [25].In this Letter, we consider the analogous of BEC-BCS crossover [26] in the attractive Haldane-Hubbard model. In contrast to usual BEC-BCS crossover, we found that away from half filling, there are extended regions of parameter space (interaction-density) for which a topological superfluid is the ground state. With increasing breaking of inversion symmetry, a gapless topological state intervenes before the system enters a trivial superfluid. We also consider the fluctuation effects on the ground states and show that the usual Anderson-Bogoliubov mode becomes quadratic in the absence of time-reversal symme-

Band geometry, Berry curvature, and superfluid weight

“…The interplay between band topology and Hubbard interactions, both repulsive and attractive, has been studied extensively and various phases have been predicted [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]. Investigation of possible exotic phases goes beyond the scope of this work.…”

Topological Phase Transitions in the Repulsively Interacting Haldane-Hubbard Model

“…The above symmetries then imply that the phase diagram is symmetric under the reflection ∆ AB → −∆ AB and that the chemical potential is zero. We do not consider large values of t for which chiral spin liquid phases, topological Mott insulators and exotic kinds of magnetic order have been proposed to appear [14,[25][26][27][28][29][30], or attractive interactions which may support topological superfluids [17,31].…”

Superfluid Density and Collective Modes of Fermion Superfluid in Dice Lattice

“…It is worthy noting that the important roles played by particle-hole symmetry in this model. Similarly as Haldane-Hubbard model 39 , applying particle-hole transformation to the system,…”

“…In this section, we would present Gaussian fluctuation method [52][53][54] to calculate the collective modes and spectral functions 39 .…”