Fractional excitation in one-dimensional two species fermionic superfluids

We study one-dimensional two-species fermionic superfluids with order parameter twisted by an angle φ at the two ends. By solving the corresponding Bogoliubov-de-Gennes equation, we obtain the U(1) soliton state which turns out to carry φS/(2π) spin, where S is the spin of single fermion. For Z2 soliton with φ = π, the localized state carries only 1/2 of a single fermion spin. To conserve the total spin of the system as an integal multiple of S, we demonstrate, the localized fractional spin φS/(2π) can be compensated by an opposite total spin −φS/(2π), which distributes uniformly in space and is thus undetectable locally in the thermodynamic limit.