Thermodynamic properties of Rashba spin-orbit-coupled Fermi gas

We investigate the thermodynamic properties of a superfluid Fermi gas subject to Rashba spin-orbit coupling and effective Zeeman field. We adopt a $T$-matrix scheme that takes beyond-mean-field effects, which are important for strongly interacting systems, into account. We focus on the calculation of two important quantities: the superfluid transition temperature and the isothermal compressibility. Our calculation shows very distinct influences of the out-of-plane and the in-plane Zeeman fields on the Fermi gas. We also confirm that the in-plane Zeeman field induces a Fulde-Ferrell superfluid below the critical temperature and an exotic finite-momentum pseudogap phase above the critical temperature.

Figure 1

Thermodynamic quantities at zero temperature, (a) order parameter and (b) pairing momentum, as functions of interaction strength characterized by $1/a_s{K}_F$ for different ${\theta }_h$. Here the SO-coupling strength is $\alpha K_F=2.0E_F$, and the effective Zeeman field strength is $h=0.5E_F$. ${\theta }_h$ is the angle between the effective Zeeman field and the $z$ axis, such that $h_z=hcos{\theta }_h$ and $h_x=hsin{\theta }_h$. Therefore, ${\theta }_h=0$ represents a pure out-of-plane Zeeman field, while ${\theta }_h=\pi /2$ represents a pure in-plane Zeeman field. $K_F$ and $E_F$ are the Fermi wave number and Fermi energy of the ideal Fermi gas, respectively.

Figure 2

Superfluid gap at $T=0$ as a function of the Zeeman field strength for (a) $1/a_s{K}_F=-0.5$, (b) 0, and (c) 0.5. The solid, dashed, and dot-dashed lines correspond to ${\theta }_h=0$, $\pi /4$, and $\pi /2$, respectively. The SO-coupling strength is $\alpha K_F=2.0E_F$. The vertical arrows indicate the critical Zeeman field strength at which the system becomes gapless.

Figure 3

(a) The critical magnetic field that separates the gapped region and the gapless region. The region below the lines is gapped. Here the SO-coupling strength is $\alpha K_F=2.0E_F$. (b) The nodal surface in the gapless region for $h_z=0.0$, $h_x=1.0E_F$, $\alpha K_F=2.0E_F$, and $1/a_s{K}_F=0.0$. The range of the plot is from $-2K_F$ to $2K_F$ along each direction.

Figure 4

(a) The total gap $\Delta$, pseudogap ${\Delta }_{\mathrm{pg}}$, and superfluid gap ${\Delta }_{\mathrm{sc}}$ as functions of the temperature. (b) The corresponding pairing momentum $Q$ as a function of the temperature. The parameters used are $1/a_s{K}_F=0.0$, $\alpha K_F=2.0E_F$, $h_z=0.0$, and $h_x=0.5E_F$. In this example, the green vertical dashed line indicates the critical temperature $T_c=0.210E_F$.

Figure 5

Critical temperature $T_c$ in the BEC-BCS crossover for (a) different spin-orbit couplings (the values of the SO-coupling strength $\alpha$, in units of $E_F/K_F$, are shown in the legends) with no Zeeman field and for (b) different ($h$, ${\theta }_h$) with $\alpha K_F=2.0E_F$. $h$ is in units of $E_F$.

Figure 6

Critical temperature $T_c$ as functions of the Zeeman field strength for (a) $1/a_s{K}_F=-0.5$, (b) 0, and (c) 0.5. The solid, dashed, and dot-dashed lines correspond to ${\theta }_h=0$, $\pi /4$, and $\pi /2$, respectively. The SO-coupling strength is $\alpha K_F=2.0E_F$. The vertical arrows indicate the value of the Zeeman field strength at which the system become gapless.

Figure 7

Compressibility in the superfluid regime as a function of temperature for different Zeeman field strengths and scattering lengths. The scattering lengths are $1/K_F{a}_s=-0.5$ for panels (a1) and (b1), $1/K_F{a}_s=0$ for panels (a2) and (b2), and $1/K_F{a}_s=0.5$ for panels (a3) and (b3). For panels (a1), (a2), and (a3), $h_x=0$; for panels (b1), (b2), and (b3), $h_z=0$. The SO-coupling strength is $\alpha K_F=2.0E_F$. The compressibility is normalized by ${\kappa }_0=\frac{3}{2}\frac{1}{N{E}_F}$, the compressibility of an ideal Fermi gas, and the temperature is normalized to the superfluid transition temperature $T_c$ for the given set of parameters. In the legends, we also indicate whether the quasiparticle excitations of the corresponding system are gapped or gapless. In (a2), we show $\kappa$ in the absence of SO couplings and Zeeman fields (the dotted line) by taking $\kappa =h=0$. In this limit, our calculation recovers the result reported in Ref. [41].

Figure 8

Compressibility at $T=0$ as a function of the Zeeman field strength. The dashed line corresponds to the in-plane Zeeman field (${\theta }_h=\pi /2$), and the solid line corresponds to the out-of-plane Zeeman field (${\theta }_h=0$). The dashed and solid arrows indicate the critical field strength at which the system becomes gapless in the presence of an in-plane Zeeman field and an out-of-plane Zeeman field, respectively. The SO-coupling strength is $\alpha K_F=2.0E_F$, and the scatter length is set at $1/K_F{a}_s=0$.

Figure 9

Feynman diagrams in a $T$-matrix scheme.