$\pi$-Flux Dirac Bosons and Topological Edge Excitations in a Bosonic Chiral $p$-Wave Superfluid

We study the topological properties of elementary excitations in a staggered $p_x\pm i{p}_y$ Bose-Einstein condensate realized in recent orbital optical lattice experiments. The condensate wave function may be viewed as a configuration space variant of the famous $p_x+i{p}_y$ momentum space order parameter of strontium ruthenate superconductors. We show that its elementary excitation spectrum possesses Dirac bosons with $\pi$ Berry flux. Remarkably, if we induce a population imbalance between the $p_x+i{p}_y$ and $p_x-i{p}_y$ condensate components, a gap opens up in the excitation spectrum resulting in a nonzero Chern invariant and topologically protected edge excitation modes. We give a detailed description of how our proposal can be implemented with standard experimental technology.

Figure 1

(a) The checkerboard potential $V_H$ comprises two classes of wells denoted $A$ and $B$. (b) Geometry of the added square lattice $V_S$ for the simplest case $\eta =0$. (c) The complete lattice $V=V_H+V_S$ comprises four classes of sites denoted as $A1$, $A2$, $B1$, and $B2$. The gray rectangles in (a)–(c) indicate the unit cells of $V_H$, $V_S$, and $V$, respectively. (d) The potential $V$ is shown along the dashed square in (c). The $s$ and $p$ orbitals accounted for in the tight-binding model are indicated. (e) Laser beams at wavelengths $\lambda$, ${\lambda }_1$, and ${\lambda }_2$ are coupled to the interferometric setup for the formation of $V$. Their linear polarizations (orthogonal or parallel with respect to the $xy$ plane) are indicated by white arrows. The phase $\beta$ in Eq. (1) is servo controlled by a movable mirror (M) and a photodetector (PD), in order to tune $\mathrm{\Delta }{V}_{\text{sp}}$ in Eq. (3). (f), Contour plot of the lattice potential $V=V_H+V_S$. The inner blue and middle black solid squares denote the unit cells of the lattice potential $V_H$ and $V$, respectively. The outer red solid square denotes the unit cell of the predicted order parameter. (g) The first Brillouin zones corresponding to the unit cells shown in (f).

Figure 2

(a) The schematic picture of the order parameter in one unit cell denoted by the outer solid square. The integer number $\nu$ denotes the phase of the orbital wave function as $\nu \pi /2$. Specifically, $p$ orbitals are located at the four corners of the inner square, while other sites are occupied by the $s$ orbitals. (b) Excitation spectra for $V_S=0$ along the high-symmetric lines in the first Brillouin zone marked by the red diamond in the lower inset. Two Dirac points are denoted by $D_{+}$ and $D_{-}$. The parameters used in numerics are $J_{\text{sp}}=0.12E_{\text{rec}}$, $J_{\parallel }=0.07J_{\text{sp}}$, $J_{\perp }=0.15J_{\text{sp}}$, $\mathrm{\Delta }{V}_{\text{sp}}=0.3J_{\text{sp}}$, $U_s=0.24E_{\text{rec}}$, $U_p=0.12E_{\text{rec}}$, and ${\mathrm{\Delta }}_{\mathrm{pp}}=0$. The mean boson density corresponds to one particle per orbital. Here, only the lowest 8 bands are shown. (c) Berry curvature of the first band of elementary excitation. (d) The same as (b) except a bias square optical lattice potential $V_S$ is turned on to induce a population imbalance between $p_x+i{p}_y$ and $p_x-i{p}_y$ components, which opens a gap around the Dirac points leading to topological elementary excitations and nonzero Berry curvature for the phonon mode shown in (c). Here, the only difference in parameters used in numerics is ${\mathrm{\Delta }}_{\mathrm{pp}}=3{\mathrm{\Delta }}_{\text{sp}}$.

Figure 3

Elementary excitation spectra for a finite system with a cylinder geometry. A periodic (open) boundary condition is assumed in the $x+y$ ($x-y$) direction. One unit cell of the finite system contains 40 copies of the unit cell of the order parameter. The lattice potential and interaction parameters are the same as that used in Fig. 2. Solid black and blue lines denote the bulk excitation spectra and the chiral edge states, respectively.