Charge fluctuations and fractional charge of fermions in 1 + 1 dimensions

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 ½. The soliton charge is measured by a sampling function $f\left(x\right)\mathrm{}$ such that $f\left(x\right)\mathrm{}\simeq 1$ 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 $l^{-1\mathrm{}}$ for large $l$ while the additional fluctuations due to the presence of a soliton vanish as either $\exp (-\frac{L}{\xi })$ or $\exp (-2\mathrm{}{\Delta }_{0}L)$; $\xi$ is the soliton width and ${\Delta }_0$ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.