Field-induced topological pair-density wave states in a multilayer optical lattice

We study the superfluid phases of a Fermi gas in a multilayer optical lattice system in the presence of out-of-plane Zeeman field, as well as spin-orbit (SO) coupling. We show that the Zeeman field combined with the SO coupling leads to exotic topological pair-density wave (PDW) phases in which different layers possess different superfluid order parameters, even though each layer experiences the same Zeeman field and the SO coupling. We elucidate the mechanism of the emerging PDW phases, and characterize their topological properties by calculating the associated Chern numbers.

Figure 1

(a) Phase diagram of the bilayer system in the $t_{\perp }-h_z$ plane by setting the SO coupling strength $\alpha =0$. (b) Phase diagram of the bilayer system in the $\alpha -h_z$ plane by setting the interlayer hopping strength $t_{\perp }=1.5t$. Other parameters are $\mu =0$ and $U=-4t$. The order parameter for the BCS and the PDW phases are given by $(\mathrm{\Delta },\mathrm{\Delta })$ and $(\mathrm{\Delta },-\mathrm{\Delta })$, respectively.

Figure 2

The single-particle dispersion in the absence of the SO coupling. Without the Zeeman field (left panel), pairing between opposite spins occur within the same energy band, as shown by the red dashed circles. When $h_z$ is comparable to $t_{\perp }$ (right panel), the interband pairing (shown by the blue solid circle) becomes favored.

Figure 3

(a) Phase diagram in $h_z-t_{\perp }$ plane at fixed SO coupling strength $\alpha =0.5t$, chemical potential $\mu =0$, and interaction strength $U=-4t$. The color scheme represents the magnitude of the order parameter $|{\mathrm{\Delta }}_{1,2}|=\mathrm{\Delta }$. (b) Wave function of the topological edge states localized near the two edges of the system supported by the topological PDW state whose spectrum is shown in (d). The amplitude of the spin-resolved wave function in the two layers are the same. (c) The spectrum of the nontopological BCS state at $t_{\perp }=1.5t$ and $h_z=0.5t$. (d) The spectrum of the topological PDW state at $t_{\perp }=1.5t$ and $h_z=1.5t$. The blue solid lines in the gap represent a pair of chiral edge states. In (b)–(d), a hard-wall potential is added in the $x$ axis.

Figure 4

(a) and (b) Phase diagram for a trilayer optical lattice system in the $t_{\perp }-h_z$ with the SO coupling strength $\alpha =0$. The color describes the amplitude of (a) ${\mathrm{\Delta }}_0$ and (b) ${\mathrm{\Delta }}_0^{\prime }$. (c) and (d) Phase diagram in the $\alpha -h_z$ plane at the fixed interlayer hopping strength $t_{\perp }=1.2t$. The color describes the magnitude of (c) ${\mathrm{\Delta }}_0$ and (d) ${\mathrm{\Delta }}_0^{\prime }$. Other parameters are $\mu =0$ and $U=-4.0t$. The order parameter for the BCS, the ${\mathrm{PDW}}_1$, and the ${\mathrm{PDW}}_2$ phases are given by $({\mathrm{\Delta }}_0,{\mathrm{\Delta }}_0^{\prime },{\mathrm{\Delta }}_0), ({\mathrm{\Delta }}_0,0,-{\mathrm{\Delta }}_0)$, and $({\mathrm{\Delta }}_0,-{\mathrm{\Delta }}_0^{\prime },{\mathrm{\Delta }}_0)$, respectively.

Figure 5

The single-particle dispersion of the trilayer system in the absence of the SO coupling. Without the Zeeman field (left panel), the pairing between opposite spins occurs within the same band (red dashed circles). When the Zeeman field strength is increased such that $h_z\approx t_{\perp }/\sqrt{2}$, the pairing between two adjacent bands is favored (blue solid circle), which leads to the ${\mathrm{PDW}}_1$ phase. With further increase in the Zeeman field such that $h_z\approx \sqrt{2}{t}_{\perp }$, the pairing is favored between the top and the bottom band (red solid circle), and the ${\mathrm{PDW}}_2$ phase emerges.

Figure 6

Phase diagram in the $t_{\perp }-h_z$ plane at fixed SO coupling $\alpha =0.25t$. The Chern numbers for various superfluid phases are indicated. The Chern numbers for BCS phase are 0, 2 for the ${\mathrm{PDW}}_1$ phase and 4 for the ${\mathrm{PDW}}_2$ phase. The other parameters are $\mu =0$ and $U=-4t$.