Finite-temperature effective field theory for dark solitons in superfluid Fermi gases

We use a finite-temperature effective field theory recently developed for superfluid Fermi gases to investigate the properties of dark solitons in these superfluids. Our approach provides an analytic solution for the dip in the order parameter and the phase profile across the soliton, which can be compared with results obtained in the framework of the Bogoliubov–de Gennes equations. We present results in the whole range of the BCS-BEC crossover, for arbitrary temperatures and taking into account Gaussian fluctuations about the saddle point. The obtained analytic solutions yield an exact energy-momentum relation for a dark soliton showing that the soliton in a Fermi gas behaves like a classical particle even at nonzero temperatures. The spatial profile of the pair field and for the parameters of state for the soliton are analytically studied.

Figure 1

Amplitude modulation function at the soliton center dependent on the soliton velocity $v_S$ for different scattering lengths and temperatures. Left panels: Mean-field calculation. Right panels: Results obtained accounting for Gaussian fluctuations. Symbols (filled circles) show the results of the BdG theory from Ref. [21].

Figure 2

Relative fermion density at the soliton center $n_0/n_{\infty }$ as a function of the soliton velocity $v_S$ for different scattering lengths and temperatures. Thick curves: Density calculated accounting for gradient terms. Thin curves: Density calculated within the LDA. Filled circles show the results of the BdG theory from Ref. [21].

Figure 3

The fermion density profile $n_x/n_{\infty }$ of the dark soliton as a function of the distance to the soliton center ($x=0$) is shown for several soliton velocities at $1/\left(k_F{a}_s\right)=1$ and compared with BdG results (filled circles).

Figure 4

Phase difference $\delta \theta$ as a function of the soliton velocity $v_S$. Notation is the same as in Figs. 1 and 2.

Figure 5

Soliton energy ${\mathcal{E}}_S\left(v_S\right)$ as a function of the soliton velocity $v_S$. Filled circles represent BdG data [21].

Figure 6

Soliton momentum ${\mathcal{P}}_S\left(v_S\right)$ as a function of the soliton velocity $v_S$.

Figure 7

Number of fermions in the soliton $N_S\left(v_S\right)$ as a function of the soliton velocity $v_S$. Filled circles represent BdG results from Ref. [21].

Figure 8

Effective mass of the soliton $M_S\left(v_S\right)$ as a function of the soliton velocity $v_S$.