Wannier functions in one-dimensional disordered systems: Application to fractionally charged solitons

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Abstract
Wannier functions can be defined as the eigenstates of the position operator projected onto a given band. Though this definition is equivalent to the usual definition of Wannier functions in a crystal, the present definition is useful in disordered systems as well. It is shown that if a band is separated from all other bands by a finite energy gap, the Wannier functions are spatially localized. Although the Wannier functions are typically too complicated to compute explicitly, they are a useful conceptual tool. As an example of their usefulness, they are here used to study the charge fluctuations associated with a fractionally charged topological defect or soliton. It is shown that fractional charge is a sharp quantum observable, thus confirming the results of previous continuum-model calculations S. Kivelson and J. R. Schrieffer, [Phys. Rev. B 25, 6447 (1982)].