7. Concluding section

comes down to making a discretized version of the vortex structure: f <sup>1</sup>; f <sup>2</sup>;…; f <sup>N</sup>

reached and the structure is not changing any more.

in [38].

28 Superfluids and Superconductors

f <sup>1</sup> ¼ 0 and f <sup>N</sup> ¼ 1 due to the boundary conditions. During the minimization procedure, a program runs through the list of points <sup>f</sup> <sup>n</sup>jn<sup>∈</sup> f g <sup>2</sup>; <sup>3</sup>;…; <sup>N</sup> � <sup>1</sup> , where it suggests a (random) new value; if the new value results in a lower energy, it is accepted as the new value of the vortex structure. The minimization program continues to run until a certain tolerance is

Figure 6. The vortex variational healing length (90) throughout the BEC-BCS crossover for the case β ¼ 100 and ζ ¼ 0. The dotted lines yield the exact solutions in the deep BEC [36] and BCS [37] limits. This plot made using the same data as

Once the exact structure is obtained, it can be analyzed and compared to the variational vortex structure. As an example, we can look at the relative difference in the free energy throughout the BEC-BCS crossover for different temperatures and polarizations. From the plots shown in Figure 7, it can be seen that the difference in free energy is around the order of 1%; this seems

Figure 7. The relative energy difference between the exact and variational solutions throughout the BEC-BCS crossover for different values of the temperature β and polarization ζ. These results were also shown and discussed in [38].

, where

In this chapter, an effective field theory for the description of dilute fermionic superfluids was derived. The main advantages of an effective field theory are the gain in computational speed and the fact that it allows analytic solutions for dark solitons and the variational healing length of the vortex structure. Both the gain in computational speed and the availability of an analytic starting point contribute to the possibility to study several soliton/vortex phenomena throughout the entire BEC-BCS crossover at finite temperatures β for a given polarization ζ within a reasonable computational time span.

On the subject of soliton dynamics, we specifically looked at 1D dark solitons, for which an exact analytical solution was derived. Using this solution, the effect of spin-imbalance on the soliton properties was studied, revealing that the unpaired particles of the excess component mainly reside inside the soliton core. Additionally, the EFT has also been employed in the study of the snake instability of dark solitons [25] and the dynamics of dark soliton collisions [27] in imbalanced superfluid Fermi gases.

For vortices, the structure of a vortex was studied, for which unfortunately no analytical solution is available at the moment. Using a variational model, an analytical solution for the vortex healing length was derived. The variational model was compared with the exact solution. From this analysis, the variational model was found to be a good fit for the exact vortex structure. Other EFT research on vortices includes the behavior of vortices in multiband systems [39] and the study of the "vortex charge" [40].
