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Progress of Theoretical Physics

Local Induced Charge and Its Fluctuation in 2 + 1 Dimensions

Katsuya Hasebe, Hiroaki Nakatani, T. Nonoyama

Published Mar 1, 1992• 0 citations
Publisher

Abstract

The Induced Charge of the masive fermion in 2 + 1 dimensions is studied with the local charge operator, defined by the local area integral of charge density. Numerical computations show that the fractional Induced Charge has non-vanishing fluctuation, which means that no eigenstate of the fractional Induced Charge exists. Analysis is given to the work which claims that there exists an eigenstate of fractional charge.

DOI
10.1143/ptp.87.693

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The Induced Charge of the masive fermion in 2 + 1 dimensions is studied with the local charge operator, defined by the local area integral of charge density. Numerical computations show that the fractional Induced Charge has non-vanishing fluctuation, which means that no eigenstate of the fractional Induced Charge exists. Analysis is given to the work which claims that there exists an eigenstate of fractional charge.
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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)f(x)f(x) such that $f(x)\ensuremath{\simeq}1$ over a region of width LLL around the soliton and then falls to zero in a distance lll. It is shown that vacuum fluctuations vanish as ${l}^{\ensuremath{-}1}$ for large lll 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.
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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.
Noise-driven noninteracting fermions have an interacting effective action. This makes it exceedingly difficult to compute their noise-averaged density matrix. Nevertheless, here we show that the exact time dependence of noise-averaged quadratic observables (such as energy and the fermionic Green's function) can be obtained from a Heisenberg-picture approach. Our results provide access to a long-time limit, which is not amenable to numerical simulations. As a simple example, we examine the fate of fractional charge in a noisy dimerized lattice (relevant to cold-atom emulations of polyacetylene). We find that the fractional charge remains robust against noisy nearest-neighbor hopping processes, while it becomes unstable to fluctuations in the second-neighbor hopping.