Chiral $d$-Wave Superfluid in Periodically Driven Lattices

A chiral $d$-wave superfluid is a preliminary example of interacting topological matter. However, unlike $s$-wave superfluids prevalent in nature, its existence requires a strong $d$-wave interaction, a criterion that is difficult to access in ordinary systems. There is no experimental observation of such unconventional superfluid at the moment. Here, we present a new principle for creating a two-dimensional (2D) chiral $d$-wave superfluid using periodically driven lattices. Because of an imprinted 2D pseudospin-orbit coupling, where the sublattice index serves as the pseudospin, the $s$-wave interaction between two hyperfine spin states naturally creates a chiral $d$-wave superfluid. This scheme can be directly implemented in current experiments.

Figure 1

(a) Applying circularly polarized light (red helix) or circular shaking (red circle), the intersublattice tunneling becomes $\pm \mathrm{\Omega }$ and $\pm i\mathrm{\Omega }$ along the $x$ and the $y$ directions, respectively. $\mathrm{\Omega }$ is the strength of the photon-assisted intersublattice tunneling. (b) Red and blue clouds represent the Wannier wave functions in the $A$ and $B$ sublattices, respectively. $\mathrm{\Delta }$ is the energy offset and $\delta =\mathrm{\Delta }-\hslash \omega$ is the one-photon detuning.

Figure 2

(a) Purple arrows are pseudospins $\mathit{\eta }$ and blue arrows represent their projections on the plane. At the $\mathrm{\Gamma }$ and $M$ points in the BZ, $\mathit{\eta }$ is perpendicular to the plane due to the absence of interband hybridization. Red and green spheres represent hyperfine spin up and spin down fermions, respectively. The shadowed region represents occupied states. (b) Top view of (a). The projection of $\mathit{\eta }$ rotates $2\pi$ on the Fermi surface. (c) Schematic of ${\beta }_{\mathbf{k}}^2/{\alpha }_{\mathbf{k}}^2$ in the BZ. ${\beta }_{\mathbf{k}}=0$ at the $\mathrm{\Gamma }$ and $M$ points.

Figure 3

(a) Phase diagram. The black curve represents the topological transition from 2 to 0. For the numerical parameters we chose, it is very close to the boundary between the chiral $d$-wave and the $s+d$ superfluids. (b) ${\mathrm{\Delta }}_{\mathit{s}}$ and ${\mathrm{\Delta }}_{\mathit{d}}$ as a function of $U_A/U_B$ at half filling $n=1$ where $E_R={\hslash }^2{\pi }^2/(2M{d}^2)$. In terms of intersublattice tunneling, ${\mathrm{\Delta }}_{\mathit{d}}\sim 10\mathrm{\Omega }$. Parameters used here are $V=6E_R$, $\zeta =0.06$, and $f=0.1d$. The inset shows the region near the topological phase transition point (the dashed line). (c) Subpanels I–III represent Fermi surfaces of the $s$-wave, the mixed, and the chiral $d$-wave superfluids, respectively, before turning on the interaction.

Figure 4

(a) For small $U_A/U_B$, the topologically nontrivial ground state has two chiral edge states on each edge (red and green solid lines). (b) When $U_A/U_B$ increases to a critical value, the gap in the excitation spectrum vanishes. A topological phase transition occurs, and the edge states disappear. (c) With a further increase of $U_A/U_B$, the gap reopens in the bulk and no edge state exists in the topologically trivial state. Parameters used here are $V=6E_R$, $\zeta =0.06$, $f=0.1d$, $n=1$, and $U_A/U_B=0.035,0.037,0.04$.