Finding the stable mechanism of ring solitons in two-dimensional Fermi superfluids

TL;DR

The paper investigates the stability of ring solitons in a two-dimensional Fermi superfluid using static and time-dependent Bogoliubov-de Gennes equations. In a uniform system, curvature-induced effective potentials drive ring solitons outward, precluding static solutions, but a harmonic trap can balance this force and create a stable equilibrium at $r_s$ where the free energy $F$ is locally maximal for the ring soliton. The stability landscape is modulated by Friedel oscillations: near the healing length $\xi$, dissipation can cause decay into sound ripples, while larger radii permit persistent, undamped oscillations around $r_s$. The results establish a concrete mechanism for stabilizing ring dark solitons in 2D Fermi superfluids and provide quantitative benchmarks (e.g., $r_s k_F\approx 10$ for certain $E_b$) to guide future experiments and theory on ring solitons in related systems.

Abstract

We theoretically investigate the stable mechanism of a ring soliton in two-dimensional Fermi superfluids by solving the Bogoliubov-de Gennes equations and their time-dependent counterparts. In the uniform situation, we discover that the ring soliton is always driven away from its initial location, and moves towards the edge due to a curvature-induced effective potential. The ring soliton is impossible to remain static at any location in the uniform system. To balance the density difference between the ring soliton's two sides, a harmonic trap is introduced, which can exert an effect to counterbalances the curvature-induced effective potential. This enables the ring dark soliton to become a stable state at a particular equilibrium position r_s, where the free energy of the ring dark soliton just reaches the maximum value. Once ring soliton is slightly deviated from r_s, some stable periodic oscillations of ring soliton around r_s will turn out. Some dissipation will possibly occur to ring soliton once its minimum radius is comparable to the healing length of soliton's Friedel oscillation. This dissipation will increase the oscillation amplitude and finally make the ring soliton decay into sound ripples. Our research lays the groundwork for a more in-depth understanding of the stable mechanism of a ring dark soliton in the future.

Figures (8)

• Figure 1: The density (upper panels) and order parameter (lower panels) distributions of an instantaneous ring dark soliton in both the radial direction (left panels) and $x$-$y$ plane (right panels). The soliton is located at located at $r_0k_F=13.0$. The binding energy $E_b=0.5E_F$.
• Figure 2: The time evolution of ring solitons in a uniform system with different initial locations, namely (a) $r_0k_F=7$, (b) $r_0k_F=15$ and (c) $r_0k_F=23$. The binding energy $E_b=0.5E_F$.
• Figure 3: The time evolution of ring solitons in a harmonic trap with different initial locations, namely (a) $r_0k_F=5.5$, (b) $r_0k_F=9.0$, (c) $r_0k_F=10.1$ and (d) $r_0k_F=12.2$. The binding energy $E_b=0.5E_F$.
• Figure 4: The free energy of instantaneous ring dark solitons at different locations. (a) uniform system with the same parameters as that in Fig. \ref{['Fig:rds']} and Fig. \ref{['Fig:rs_uniform']}; (b) system in harmonic trap with the same parameters as that in Fig. \ref{['Fig:rs_trap']}.
• Figure 5: The stable location of a ring dark soliton in the harmonic trap system at different binding energy $E_b$.