Charge fluctuations of solitons with arbitrary fractional fermion number in 1 + 1 dimensions are calculated, generalizing the result of Kivelson and Schrieffer for solitons with fermion number \textonehalf{}. The soliton charge is measured by a sampling function
f(x) such that $f(x)\ensuremath{\simeq}1$ over a region of width
L around the soliton and then falls to zero in a distance
l. It is shown that vacuum fluctuations vanish as ${l}^{\ensuremath{-}1}$ for large
l while the additional fluctuations due to the presence of a soliton vanish as either $\mathrm{exp}(\ensuremath{-}\frac{L}{\ensuremath{\xi}})$ or $\mathrm{exp}(\ensuremath{-}2{\ensuremath{\Delta}}_{0}L)$; $\ensuremath{\xi}$ is the soliton width and ${\ensuremath{\Delta}}_{0}$ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.
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.
Quantum electrodynamics in
1+1 dimensions (Schwinger model) on an interval admits lattice discretization with a finite-dimensional Hilbert space and is often used as a testbed for quantum and tensor network simulations. In this work we clarify the precise mapping between the boundary conditions in the continuum and lattice theories. In particular we show that the conventional Gauss law constraint commonly used in simulations induces a strong boundary effect on the charge density, reflecting the appearance of fractionalized charges. Further, we obtain by bosonization a number of exact analytic results for local observables in the massless Schwinger model. We compare these analytic results with the simulation results obtained by the density matrix renormalization group method and find excellent agreements.
We study the soliton modes carrying fractional quantum numbers in one-dimensional superfluids. In the
s-wave pairing superfluid with the phase of the order parameter twisted by opposite angles
±φ/2 at the two ends there is an emergent complex
Z2 soliton mode carrying fractional spin number
φ/(2π) if there is only one pairing branch. We demonstrate that in finite systems of length
L, the spin density for one pairing branch in the presence of a single soliton mode consists of two terms, a localized spin density profile carrying fractional quantum number
φ/(2π), and a uniform background
−φ/(2πL). The latter one vanishes in the thermodynamic limit leaving a single soliton mode carrying fractional excitation, however it is essential to keep the total quantum number conserved in finite systems. This analysis is also applicable to other systems with fractional quantum numbers, thus provides a mechanism to understand the compatibility of the emergence of fractional charges with the integral quantization of charges in a finite system. For the
p-wave pairing superfluid with the chemical potential interpolating between the strong and weak pairing phases, the
Z2 soliton is associated with a Majorana zero mode. By introducing the dimension density, we argue that the Majorana zero mode may be understood as an object with 1/2 dimension of the single particle Hilbert space. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We examine charge fractionalization by chiral separation in a one-dimensional fermion system described by Luttinger liquid theory. The focus is on the question of whether the fractional charges are quantum-mechanically sharp, and in the analysis we make a distinction between the global charge, which is restricted by boundary conditions, and the local charge where a background contribution is subtracted. We show, by way of examples, that fractional charges of arbitrary values, all which are quantum-mechanically sharp, can be introduced by different initial conditions. Since the system is gapless, excitations of arbitrary low frequency contribute to the fluctuations, it is important to make a precise definition of sharp charges, and this we we do by subtraction of the ground-state contribution. We very briefly comment on the relevance of our analysis for proposed experiments.