Equation of state and phase transition in spin-orbit-coupled Bose gases at finite temperature: A perturbation approach

We study two-component Bose gases with Raman induced spin-orbit coupling via a perturbation approach at finite temperature. For weak coupling, free energy is expanded in terms of Raman coupling strength up to the second order, where the coefficient (referred to as Raman susceptibility) is determined according to linear response theory. The equation of state for the stripe phase and the plane-wave phase are obtained in Popov approximation, and the first order transition between these two phases is investigated. As temperature increases, we find the phase boundary bends toward the stripe phase side in most temperature regions, which implies the ferromagnetic order is more robust than the crystalline order in presence of thermal fluctuations. Our results qualitatively agree with the recent experimental observation in rubidium atomic gases. A method to measure Raman susceptibility through the two-photon Bragg scattering experiment is also discussed.

Figure 1

(a) Comparison of the free energy of the spin-balanced phase and the spin-polarized phase at finite temperature for $g_{-}/g=0.002$ (solid line) and $g_{-}/g=-0.002$ (dashed line). The threshold temperatures for the spin-balanced phase and the spin-polarized phase are indicated by $▹$ and $▸$, respectively (see text). Parameters: $n=0.5k_{\mathrm{r}}^3$, and $g=0.828E_{\mathrm{r}}/k_{\mathrm{r}}^3$. $T_{\mathrm{c}}=2\pi {\hslash }^2{[n/\zeta \left(\frac{3}{2}\right)]}^{2/3}/\left(m{k}_{\mathrm{B}}\right)$ is the transition temperature in the noninteracting case. $\zeta (·)$ is the Riemann zeta function. Lower panels show two ruled-out scenarios of the phase diagram with Raman coupling: (b) a ruled out scenario for $g_{-}>0$, and (c) a ruled out scenario for $g_{-}<0$.

Figure 2

Raman susceptibility $\chi$ of the spin-balanced phase and the spin-polarized phase at finite temperature for (a) $g_{-}/g=0.002$ and (b) $g_{-}/g=-0.002$. The threshold temperatures for the spin-balanced phase and the spin-polarized phase are indicated by $▹$ and $▸$, respectively. Other parameters are the same as in Fig. 1.

Figure 3

The transition lines between the STR phase and the PW phase for various densities. $g_{-}/g=0.002$; other parameters are the same as in Figs. 1 and 2. The threshold temperature for the spin-polarized phase is indicated by $▴$. When the temperature is close to $T_{\mathrm{c}}$ (for instance, in the shadow region above 0.9 $T_{\mathrm{c}}$), mean-field theory is usually inaccurate. For a qualitative comparison, the experimental data ($\circ$ with error bar) measured in a harmonic trap from Ref. [7] are also shown. At low temperature, the typical atomic density at the center of the trap is around $0.5k_{\mathrm{r}}^3$.

Figure 4

Comparison of the actual free energy with the perturbation value $F_{\mathrm{perp}}$ in a noninteracting Bose gas at various temperatures. Inset: noninteracting Raman susceptibility as a function of temperature ($n=0.5k_{\mathrm{r}}^3$).

Figure 5

The transition lines between the STR phase and the PW phase with $F$ and $\chi$ calculated in the Hartree-Fock approximation. Parameters and notations are the same as in Fig. 3.