Fermi gas interacting with topological solitons is analyzed at low temperatures. It is shown that at high enough densities fermions become unstable and decay into the pairs of solitons with half-integer fermion number.
We study the soliton modes carrying fractional quantum numbers in one-dimensional fermionic systems. In particular, we consider the solitons in the one-dimensional fermionic superfluids. For the s -wave order parameter with phase twisted by an angle φ , the complex Z 2 soliton may occur carrying a localized fractional quantum number φ /(2 π ). If the system is finite with length L , we show the existence of a uniform background − φ /(2 πL ) which, though vanishing in the thermodynamic limit, is essential to maintain the conservation of the total integer quantum number. This analysis is also applicable to other systems with fractional quantum numbers, thus providing a mechanism to understand the compatibility of the emergence of fractional quantum number in the thermodynamic limit of a finite system with only integer quantum numbers. For the p -wave pairing case, the Majorana zero mode may occur associated with a real Z 2 soliton, and the fractionalized quantum number is the dimension of the single particle Hilbert space, which turns out to be 1/2. Again for a finite system with length L , there is an accompanying uniform dimension density −1/(2 L ) to maintain the dimension of the Hilbert space invariant. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We will discuss recent results on the theoretical and experimental investigations on solitons in a superfluid Fermi gas along the BCS-BEC crossover. On the theory side, by solving the Bogoliubov de Gennes equations and looking for real and odd solutions for the order parameter, it has been shown that a dark soliton at unitarity posses a large density contrast and fermionic bound states. The superfluid gap is found to be significantly quenched by the presence of the soliton due to the occurrence of Andreev fermionic bound states localized near the nodal plane of the order parameter. By solving the time dependent Bogoliubov de Gennes equations, also the decay and collisions of dark solitons has been investigated. Recently, a cascade of Solitonic Excitations in a Superfluid Fermi gas has been experimentally observed.
A mechanism of both formation of peaks in the density of states near the Fermi surface and phase instabilities of nearly ideal degenerate Fermi gas in low-dimensional optical lattices is proposed. According to this mechanism, peak formation is caused by the quasi-classical quantization of the one- and two-dimensional fermionic spectrum in the neighborhood of its extremal points under interaction with an long-wave periodical perturbation. The new spectra result in the instabilities with respect to spontaneous formation of an equilibrium superstructure. In the one-dimensional case this happens for low enough numbers of fermionic atoms. As a result of such transition, fermions become localized (a transition of the metal-insulator type). In the two-dimensional system the transition is possible for a nearly half-filled band. In this case fermions are localized in the wave direction only. It is briefly discussed the possible influence of the results obtained in the paper on the superfluid transition temperature in high anisotropic lattices possessing quasi-(one,two)-dimensional subsystems of fermionic atoms.