Large Chern-number topological superfluids in a coupled-layer system

Large Chern-number topological phase is always an important topic in modern physics. Here we investigate the topological superfluids in a coupled-layer system, in which transitions between different topological superfluids can be realized by controlling the binding energy, interlayer tunneling, and layer asymmetry, etc. These topological transitions are characterized by energy gap closing and reopening at the critical points at zero momentum, where the Chern number and sign of Pfaffian undergo a discontinuous change. Topological protected edge modes at the boundaries are ensured by the bulk-edge correspondence. In a trapped potential the edge modes are spatially localized at the interfaces between distinct topological superfluids, where the number of edge modes is equal to the Chern-number difference between the left and right superfluids. These topological transitions can be detected by spin texture at or near zero momentum, which changes discretely across the critical points due to band inversion. The model can be generalized to a multilayer system in which the Chern number can be equal to any positive integer. These large Chern-number topological superfluids provide fertile grounds for exploring exotic quantum matters in the context of ultracold atoms.

Figure 1

The evolution of order parameters ${\Delta }_i$ and chemical potential $\mu$ as a function of binding energy (a), interlayer tunneling (b), and asymmetry (c). In (a) $t=0.5E_{\text{F}},\delta \mu =0.5E_{\text{F}}$; (b) ${\varepsilon }_{\text{b}}=1.0E_{\text{F}},\delta \mu =0.5E_{\text{F}}$; and (c) $t=0.5E_{\text{F}},{\varepsilon }_{\text{b}}=1.0E_{\text{F}}$. The effect of asymmetry on the population occupation projected to each layer is plotted in (d) with parameters from (c). In all subfigures $\Gamma =1.0E_{\text{F}}$.

Figure 2

Phase diagrams defined by Chern number $\mathcal{C}$ and sign of Pfaffian $\nu$. In (a), $t=\delta \mu =0.5E_{\text{F}}$. In (b), ${\varepsilon }_{\text{b}}=0.6E_{\text{F}}$ and $\Gamma =0.97E_{\text{F}}$.

Figure 3

Edge states in a strip geometry with Chern number $\mathcal{C}=2$ (a) and $\mathcal{C}=1$ (b). (c) The total wave function of the edge states in each layer at momentum $k_y=\pm 0.25k_{\text{F}}$ (inset shows the two states with the same momentum propagate at the same edge). (d) Edge state velocities as a function of asymmetry. Parameters are $\Gamma =1.28E_{\text{F}}$ in (b) and $\Gamma =1.43E_{\text{F}}$ in (a) and (c), and the other parameters in (a)–(c) are ${\varepsilon }_{\text{b}}=1.2E_{\text{F}}$ and $t=\delta \mu =0.5E_{\text{F}}$. (d) The velocities of the edge modes as a function of asymmetry under parameters $t=0.2E_{\text{F}},{\varepsilon }_{\text{b}}=0.6E_{\text{F}}$, and $\Gamma =0.97E_{\text{F}}$. (e) Generalized bulk-edge correspondence in a trapped potential. The edge modes are spatially localized at the interface between two topological superfluids, and the number of edge modes $\mathcal{N}=|{\mathcal{C}}_{\text{L}}-{\mathcal{C}}_{\text{R}}|$, where ${\mathcal{C}}_{\text{L}}$ and ${\mathcal{C}}_{\text{R}}$ are the Chern number of the superfluids at the left and right interfaces, and all the other edge modes are gapped out by direct coupling due to their opposite chiralities. In general, $\mathcal{N}=1$.

Figure 4

(a) Band inversion and topological phase transitions. Across the critical points the spin polarization $S_{\text{z}}\left(0\right)=\langle c_{0\uparrow }^{†}{c}_{0\uparrow }-c_{0\downarrow }^{†}{c}_{0\downarrow }\rangle$ (marked by arrows) changes discretely due to interchange of symmetry between the occupied and unoccupied bands. The band inversions (see the third subfigure) among the occupied bands—a typical feature in a coupled-layer system—do not change the ground state topology, thus $S_{\text{z}}\left(0\right)$ is invariant. (b) Evolution of spin polarization at zero momentum $S_{\text{z}}\left(0\right)$ and total population imbalance $S_{\text{z}}=(n_{\uparrow }-n_{\downarrow })/n$ as a function of tunneling. $S_{\text{z}}\left(0.05\right)$ is the spin polarization at momentum $\left|\mathbf{k}\right|=0.05k_{\text{F}}$. Parameters are ${\varepsilon }_{\text{b}}=0.6E_{\text{F}},\Gamma =0.97E_{\text{F}},\delta \mu =0.2E_{\text{F}}$.