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

Motivated by the recent realization of the Haldane model in a shaking optical lattice, we investigate the effects of attractive interaction and the 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 the phase space for intermediate strength of the attractive interaction on the interaction-density plane. When inversion symmetry is broken, a gapless weak 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. Near the topological phase transition, the damping of the Leggett mode reaches its maximum.

Figure 1

Panel (a): the honeycomb lattice in the Haldane model. Taking the distance of nearest-neighbor sites as the length unit, here two-dimensional vectors ${\mathbf{a}}_1=[\sqrt{3}/2,-1/2],{\mathbf{a}}_2=[-\sqrt{3}/2,-1/2]$, and ${\mathbf{a}}_3=[0,1].{\mathbf{b}}_1=[\sqrt{3},0],{\mathbf{b}}_2=[-\sqrt{3}/2,-3/2]$, and ${\mathbf{b}}_3=[-\sqrt{3}/2,3/2]$. The red (black) sites denote sublattices $\mathbf{A}\left(\mathbf{B}\right)$. The phase factor $e^{i\varphi }$ associated with next-nearest-neighbor anticlock hopping also is labeled. Panel (b) is Haldane's topological phase diagram on the $M\text{−}\varphi$ plane. Panel (c) shows two Fermi pockets around Dirac points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ in the Brillouin zone (BZ).

Figure 2

(a) Phase diagram for $M=0$ on the $n\text{−}U$ plane. The region within the black line (except the red dashed line) is in a topological superfluidity phase with Chern number $C=2$. The exterior region of the black line is topologically trivial ($C=0$). The red dashed line denotes the anomalous quantum Hall insulator (${\mathrm{\Delta }}_{\mathrm{A}\left(\mathrm{B}\right)}\equiv 0$) at half-filling ($n=2$). With increasing interaction, e.g., from point $a$ to point $c$, the system crosses the phase boundary. The inset shows chemical $\mu$ potentials and the pairing gaps (${\mathrm{\Delta }}_{\mathrm{A}}={\mathrm{\Delta }}_{\mathrm{B}}=\mathrm{\Delta }$) as functions of interaction strength $U$ for filling factor $n=2.1$.

Figure 3

(a) $E_{\mathrm{gap}}$ and $\delta$ as a function of $M$ for $U=3.0$ (filling $n=2.1$). There exist two critical values $M_{\mathrm{cr}1}=0.1135$ and $M_{\mathrm{cr}2}=0.5718$. When $M<M_{\mathrm{cr}1}$, the bulk gap $E_{\mathrm{gap}}$ remains finite, and the system is in the gapped topological phase. Further increasing $M$ until $M=M_{\mathrm{cr}1}$, $E_{\mathrm{gap}}$ closes, and the system is in the gapless weak topological phase. For $M>M_{\mathrm{cr}2}$, the system becomes a topologically trivial superfluid. For $M_{\mathrm{cr}1}<M<M_{\mathrm{cr}2}$, the system is in the gapless weak topological superfluid state. The Chern number is shown in the inset. (b) Edge states in the boundary of strip geometry. (c) The quasiparticle spectrum (blue) and edge states [red dashed line (black solid line) for left (right) going] in the gapped topological phase for $M=0$. (d) The quasiparticle spectrum (blue) and edge states [red dashed line (black solid line) for left (right) going] in the gapless weak topological phase for $M=0.3$.

Figure 4

The collective excitation spectrum for various values of interaction strength ($M=0$): (a) $U=3.0$, (b) $U=3.4$, and (c) $U=3.8$, corresponding to the three blue points $a,b$, and $c$, respectively, in Fig. 2. The momentum $\mathbf{q}$ is chosen along the $\widehat{x}$ direction $\mathbf{q}=q\widehat{x}$. The shaded region denotes the quasiparticle continuum with its lowest boundary (red) given by ${\min }_k[E_k^{-}+E_{k+q}^{-}]$. There are two types of collective excitations. The lower black line corresponds to the gapless Goldstone mode, whereas the upper blue line is the Leggett mode. (d) The spectral function of the Leggett mode for various values of interaction strength $U=3.0,3.2,3.4,3.6,3.8$ (the corresponding peaks are from right to left). Note that the maximal damping occurs at the transition point ($U=3.4$). The inset shows the variations of the half-width of the spectral function with the interaction strength.

Figure 5

The variations of order parameters with $M$ at $U=3$. For our chosen parameters, the order parameters for the density wave are always very small.