Hidden long-range order in a spin-orbit-coupled two-dimensional Bose gas

A spin-orbit-coupled two-dimensional Bose gas is shown to simultaneously possess quasi- and true long-range order in the total and relative phase sectors, respectively. The total phase undergoes a Berenzinskii-Kosterlitz-Thouless transition to a low-temperature phase with quasi-long-range order, as expected for a two-dimensional quantum gas. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin-orbit coupling in combination with spin-dependent particle-particle interactions. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low-temperature regime. Numerical simulations of the stochastic projected Gross-Pitaevskii equation give a good agreement with the analytical predictions.

Figure 1

Relative-phase fluctuations from Bogoliubov theory at two different temperatures. Panels (a) and (d) show the relative-phase fluctuations $〈{\left(\mathrm{\Delta }{\varphi }_{\mathrm{r}}\right)}^2〉$ from Eq. (6) where the axes denote the separations $x=x^{\prime }-x^{\prime \prime }$ and $y=y^{\prime }-y^{\prime \prime }$. A plateau is seen to develop at large separation. The fluctuations $〈{\left(\mathrm{\Delta }{\varphi }_{\mathrm{r}}\right)}^2〉$ are also shown in panels (b) and (e) while panels (c) and (f) depict the correlation function $G_{\mathrm{r}}$ from Eq. (7). The magenta (grey) lines are plotted along the $x$-axis and black lines along the $y$-axis for panels (b), (c), (e), and (f). The temperature is set to $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ for panels (a), (b), (c) and to $T/T_{\mathrm{BKT}}^{\infty }\approx 1.33$ for panels (d), (e), (f), and $\mu =13,{\tilde{g}}_{12}/\tilde{g}=0.9$ and $({\tilde{\kappa }}_x,{\tilde{\kappa }}_y)=(1,0)$.

Figure 2

Total phase from stochastic simulations of Eq. (8). Panels (a) and (b) depict the snapshot of the total-phase profile $\arg \left({\mathrm{\Psi }}_1\right)+\arg \left({\mathrm{\Psi }}_2\right)$ at $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ and 1.33 respectively. The correlation function $G_{\mathrm{t}}\left(\right|{\mathbf{r}}^{\prime }-{\mathbf{r}}^{\prime \prime }\left|\right)$ is shown on a doubly-logarithmic scale in panels (c) and (d). Dots represent numerical data and solid lines are algebraic fits for the lower temperatures in panel (c) and exponential fits in panel (d). The temperatures are $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ (blue, top), 0.67 (orange, middle), 0.78 (black, bottom) in panel (c), and 1.33 (green, top), 1.56 (magenta, middle) and 1.78 (red, bottom) in panel (d).

Figure 3

Relative phase from stochastic simulations. Panels (a) and (b) depict the snapshots of the relative-phase profile $\arg \left({\mathrm{\Psi }}_1\right)-\arg \left({\mathrm{\Psi }}_2\right)$ at $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ and 1.33 respectively. The correlation function $G_{\mathrm{r}}$ is plotted in panels (c) and (d) along the SOC direction. While plateaus are reached in (c) for $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ (blue, top), 0.67 (orange, middle), 0.78 (black, bottom), the correlation function quickly decays to zero for the higher temperatures $T/T_{\mathrm{BKT}}^{\infty }\approx 1.33$ (green, solid line), 1.56 (magenta, dashed line) and 1.78 (red, dotted line) in panel (d). Panel (e) shows the plateau values for the phase correlation $G_{\mathrm{r}}(\left|x^{\prime }-x^{\prime \prime }\right|\to \infty )$ versus temperature.

Figure 4

Total-phase correlation functions based on Eq. (A12). Blue (top) and red (bottom) dots indicate the total-phase correlation functions evaluated at temperatures $T/T_{\mathrm{BKT}}^{\infty }\approx 0.44$ and 1.33 respectively. The linearity of the curves implies a power-law behavior.