Induced interactions in the BCS-BEC crossover of two-dimensional Fermi gases with Rashba spin-orbit coupling

We investigate the Gorkov–Melik-Barkhudarov (GM) correction to superfluid transition temperature in two-dimensional Fermi gases with Rashba spin-orbit coupling (SOC) across the SOC-driven BCS-BEC crossover. In the calculation of the induced interaction, we find that the spin-component mixing due to SOC can induce both of the conventional screening and additional antiscreening contributions that interplay significantly in the strong SOC regime. While the GM correction generally lowers the estimate of transition temperature, it turns out that at a fixed weak interaction, the correction effect exhibits a crossover behavior where the ratio between the estimates without and with the correction first decreases with SOC and then becomes insensitive to SOC when it goes into the strong SOC regime. We demonstrate the applicability of the GM correction by comparing the zero-temperature condensate fraction with the recent quantum Monte Carlo results.

Figure 1

Fermi surfaces of two-dimensional Fermi gases with Rashba SOC in noninteracting limit. The solid (dotted) line presents the $\Uparrow (\Downarrow )$ helicity branch of the energy dispersion. The horizontal dashed lines indicate the Fermi energies for the SOC strengths $v_R=1$ (a) and 1.7 (b) that belong to the weak and strong SOC regimes, respectively.

Figure 2

Second-order diagrams of the medium-induced interaction between the $\uparrow$- and $\downarrow$-spin components. The solid lines are propagators, and the single dotted line indicates the bare two-body interaction. The diagram in (a) is conventional in usual two-component systems, and the one in (b) is mediated by the spin-component mixing due to SOC. (c) The schematic diagram of the effective interaction (double dotted line) with the induced interaction correction.

Figure 3

Induced interaction as a function of the SOC strength. The values are scaled with the density of states ${\mathcal{N}}_{2\mathrm{D}}$ of two-dimensional Fermi gases without SOC. (a) Total induced interaction ${\overline{g}}_{\mathrm{ind}}/(-g^2)\equiv {\overline{\mathrm{\Pi }}}_1+{\overline{\mathrm{\Pi }}}_2$, and the screening and antiscreening contributions ${\overline{\mathrm{\Pi }}}_1$ and ${\overline{\mathrm{\Pi }}}_2$ from diagrams 1 and 2 given in Fig. 2, respectively. The inter- and intra-helicity-branch components ${\left(\overline{\mathrm{\Pi }}\right)}_{ab}$ are presented for diagrams 1 (b) and 2 (c). The vertical dotted lines indicate $v_R=\sqrt{2}$ where the character of the Fermi sea abruptly changes.

Figure 4

Induced-interaction correction to the mean-field transition temperature as a function of the interaction strength. The ratio of the transition temperature without the correction ($T_c^{\left(0\right)}$) and with the correction ($T_c^{\left(\mathrm{GM}\right)}$) is compared between the weak and strong SOC regimes at (a) $v_R=0.5$ and (b) $v_R=2.0$, respectively.

Figure 5

Superfluid transition temperature with the GM correction and comparison with the QMC data of condensate fraction. The mean-field transition temperature $T_c$ (a) and (b), and the BKT transition temperature $T_{\mathrm{BKT}}$ (c) and (d), are calculated as a function of the SOC strength $v_R$ at weak attractions of ${\epsilon }_B=0.001$ and ${\epsilon }_B=0.01$. The superscripts $\left(\mathrm{GM}\right)$ and $\left(0\right)$ indicate ones with and without the GM correction, respectively. The condensate fraction at zero temperature is compared with the recent QMC results [53] for the weak and strong SOC regimes at (e) $\alpha \equiv 2v_R^2/{\epsilon }_F=1$ and (f) $\alpha =7$. For direct comparison, the binding energy ${\epsilon }_B$ is converted into the scattering length $a$ through the relation ${\epsilon }_B\equiv 4{\hslash }^2/m{a}^2{e}^{2\gamma }$ where $\gamma$ is the Euler's constant. The lines marked by $\mathrm{MF}$ indicate the mean-field values without the GM correction.