Abstract
The superfluid properties of attractive Hubbard model in dice lattice are investigated. It is found that three superfluid order parameters increase as the interaction increases. When the filling factor falls into the flat band, due to the infinite large density of states, the resultant superfluid order parameters are proportional to interaction strength, which is in striking contrast with the exponentially small counterparts in usual superfluid (or superconductor). When the interaction is weak, and the filling factor is near the bottom of the lowest band (or the top of highest band), the superfluid density is determined by the effective mass of the lowest (or highest) single-particle band. When the interaction is strong and filling factor is small, the superfluid density is inversely proportional to interaction strength, which is related to effective mass of tightly bound pairs. In the strong interaction limit and finite filling, the asymptotic behaviors of superfluid density can be captured by a parabolic function of filling factor. Furthermore, when the filling is in flat band, the superfluid density shows a logarithmic singularity as the interaction approaches zero. In addition, there exist three undamped collective modes for strong interactions. The lowest excitation is gapless phonon, which is characterized by the total density oscillations. The two others are gapped Leggett modes, which correspond relative density fluctuations between sublattices. The collective modes are also reflected in the two-particle spectral functions by sharp peaks. Furthermore, it is found that the two-particle spectral functions satisfy an exact sum-rule, which is directly related to the filling factor (or density of particle). The sum-rule of the spectral functions may be useful to distinguish between the hole-doped and particle-doped superfluid (superconductor) in experiments.
Highlights
The superfluid properties of attractive Hubbard model in dice lattice are investigated
We investigate the superfluid properties of attractive Hubbard model in dice lattice
We emphasize the important roles of effective masses in the understanding of the asymptotic behaviors of superfluid density, especially for small filling factor
Summary
When the filling factor is in between the lowest band and the middle flat band, the order parameter gets a dip due to the small density of states near the Dirac points [especially for weakly interacting cases in panel (a) of Fig. 2]. When the filling is small, superfluid density diminishes with the increasing of interaction, which is opposite to that of order parameters [see panel (b) of Fig. 2]. ≃ 1/(q2ρs) > 0 is a real number and θq the phase o perator[59,60] In such a two-particle spectral functions near the phonon states [ ω ≃ ωphonon(q) or ω ≃ −ωphonon(−q) ] are proportional to product of superfluid order parameters, i.e., Aij(q, ω) ≃ 4Z+ i ∗j δ[ω − ωphonon(q)] − 4Z− i ∗j δ[ω + ωphonon(−q)], Bi,j(q, ω) ≃ −4Z+ i jδ[ω − ωphonon(q)] + 4Z− i jδ[ω + ωphonon(−q)],. Eq (30) still holds (see the last column of the Table 1)
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