Fractional excitations in one-dimensional fermionic systems

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 phi, the complex Z(2) soliton may occur carrying a localized fractional quantum number phi/(2 pi). If the system is finite with length L, we show the existence of a uniform background -phi/(2 pi L) 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/(2L) 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.