Unconventional Fulde-Ferrell-Larkin-Ovchinnikov pairing states in a Fermi gas with spin-orbit coupling

We study the phase diagram of a two-dimensional Fermi gas with the synthetic spin-orbit coupling that has recently been realized experimentally. In particular, we characterize in detail the properties and the stability region of the unconventional Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states in such a system, which are induced by spin-orbit coupling and Fermi surface asymmetry. We identify several distinct nodal FFLO states by studying the topology of their respective gapless contours in momentum space. We then examine the phase structure and the number density distributions in a typical harmonic trapping potential under the local density approximation. Our studies provide detailed information on the FFLO pairing states with spin-orbit coupling and Fermi surface asymmetry, and will facilitate experimental detection of these interesting pairing states in the future.

Figure 1

Typical structure of the thermodynamic potential $\Omega$ as a function of $\Delta$ for $\mathbf{Q}=0$. The parameters for the curves are $\mu /h=1.374$ (dash-dotted), $\mu /h=1.474$ (solid), and $\mu /h=1.624$ (dashed). For all curves, $E_b/h=0.5$, $h_x/h=0.2$, and $\alpha k_h/h=0.7$. Here, the effective Rabi frequency $h$ is taken to be the energy unit, while the unit of momentum $k_h$ is defined as ${\hslash }^2{k}_h^2/\left(2m\right)=h$.

Figure 2

Contour plot of typical thermodynamic potential landscape in the $\Delta$–$Q_x$ plane for $E_b/h=0.5$, $h_x/h=0.2$, $\alpha k_h/h=0.7$, $\mu /h=1.524$.

Figure 3

Typical phase diagrams in the $\alpha$–$\mu$ plane for (a) $E_b/h=0.5$, $h_x/h=0.2$ and (b) $E_b/h=0.2$, $h_x/h=0.2$. Solid curves represent the first-order phase boundaries, the dash-dotted curves represent the continuous phase boundary between the fully gapped FFLO state (gFFLO${}_x$) and various nodal FFLO${}_x$ states, and the dashed curves represent the continuous phase boundary between different nodal FFLO${}_x$ states or the normal state.

Figure 4

(a–c) Gapless contours in momentum space for different nodal FFLO${}_x$ states with: (a) $\alpha k_h/h=1$, $\mu /h=-0.2$ (ns1); (b) $\alpha k_h/h=0.58$, $\mu /h=1.2$ (ns2); and (c) $\alpha k_h/h=0.83$, $\mu /h=1.15$ (mixed). (d–f) Quasiparticle (hole) dispersion spectra along the $k_x=Q_x/2$ axis for the nodal ${\text{FFLO}}_x$ states that correspond to panels (a–c), respectively.

Figure 5

Typical phase diagram in the $\alpha$–$n$ plane for $E_b/h=0.5$ and $h_x/h=0.2$. Solid curves represent first-order phase boundaries, dash-dotted curves represent continuous phase boundary between the fully gapped FFLO state (gFFLO${}_x$) and various nodal FFLO${}_x$ states, and dashed curves represent continuous phase boundaries between different nodal FFLO${}_x$ states or the normal state. We take the unit of number density as $n_0=k_h^2$.

Figure 6

(a) Number density distribution in a typical trapping potential. The solid curve is the total number density distribution, and the dashed (dash-dotted) curve represents the density profile of the spin-up (spin-down) component. The units for the number density $n_0$ and for the distance from the trap center $R$ are defined in the text. (b) Variation of the pairing order parameter ${\Delta }_Q$ as a function of the distance from the trap center. (c) Variation of the CoM momentum $Q_x$ of the pairing state as a function of the distance from the trap center.