Hidden nonsymmorphic symmetry in optical lattices with one-dimensional spin-orbit coupling

We uncover the nonsymmorphic symmetry and investigate its effects on the noncollinear band structures of a quasi-two-dimensional optical lattice with synthetic one-dimensional spin-orbit coupling and a tunable Zeeman field. The perpendicular Zeeman field breaks time-reversal symmetry and lifts the Kramers degeneracy which is protected by time-reversal and generalized inversion symmetries. Interestingly, we find that the degeneracy of Bloch bands on the border of the Brillouin zone is immune to the Zeeman field. This degeneracy, reminiscent of that in nonsymmorphic crystals, is protected by the hidden glide-plane symmetry that comprises a physical reflection involving both spatial and spin degrees of freedom followed by a nonprimitive lattice translation. Furthermore, we show that the band degeneracy can be lifted by the glide-plane-symmetry-breaking lattice potential. Finally, we propose to detect these effects by measuring a dynamical structure factor with optical Bragg spectroscopy.

Figure 1

Spatial pattern of the optical lattice potential (a) ${\mathcal{V}}_0\left(\mathit{r}\right)$ and (b) ${\mathcal{V}}_y\left(\mathit{r}\right)$ in the $z=0$ plane with $\{V_0,V_{\text{T}},V_y\}=\{3,12,1\}{E}_{\text{R}}$. $E_{\text{R}}={\pi }^2{\hslash }^2/2m{d}^2$ is recoil energy. Open circles indicate positions of minima in ${\mathcal{V}}_0\left(x,y,z=0\right)$.

Figure 2

Band structures along the high symmetry line of first Brillouin zone under a perpendicular Zeeman field $V_z/E_{\text{R}}=0,0.1,0.2$ with $\{V_0,V_{\text{T}},V_y\}=\{3,12,1\}{E}_{\text{R}}$ in units of recoil energy $E_{\text{R}}={\pi }^2{\hslash }^2/2m{d}^2$. The lowest four bands are projected into glide parity $+$ (open triangle) and $-$ (open square) branches.

Figure 3

(a) The in-plane spatial pattern of the glide-plane-symmetry-breaking potential ${\mathcal{V}}_{\text{G}}\left(\mathit{r}\right)$ with $V_{\text{G}}=0.2E_{\text{R}}$. (b) Band structures along the high symmetry line in Brillouin zone with $V_{\text{G}}/E_R=0\text{,}0.2$ and $\{V_0,V_{\text{T}},V_y,V_z\}=\{3,12,1,0.2\}{E}_{\text{R}}$ in units of recoil energy $E_{\text{R}}={\pi }^2{\hslash }^2/2m{d}^2$.

Figure 4

Color map of dynamical structure factor $S\left(\mathit{q},\omega \right)$ along high symmetry line $\mathit{q}=\left(q_x,0\right)$ with integer band filling $\nu =1$ and $\{V_z,V_{\text{G}}\}=\{0,0\}{E}_R$ (a), $\{V_z,V_{\text{G}}\}=\{0.2,0\}{E}_R$ (b), $\{V_z,V_{\text{G}}\}=\{0.2,0.2\}{E}_R$ (c). (d) Color map of single-particle transition probability $P_{2,1}\left(\mathit{k}\right)={\sum }_{{\mathit{k}}^{\prime }}{\left|A_{2\mathit{k},1{\mathit{k}}^{\prime }}\left(\mathit{q}\right)\right|}^2$ between the lowest and second lowest energy bands at $\mathit{q}=\left(\pi ,0\right)$ as the dominating excitation channel indicated by the white circle in (c).