Spectrum of the Hermitian Wilson-Dirac Operator for a Uniform Magnetic Field in Two Dimensions

TL;DR

This paper analyzes the spectrum of the Hermitian Wilson-Dirac operator under a uniform magnetic field in two dimensions, revealing a fractal structure and deriving an exact index theorem for the overlap Dirac operator.

Contribution

It reduces the eigenvalue problem to a one-dimensional relativistic Harper equation and provides explicit secular equations and an exact index theorem.

Findings

Spectrum exhibits fractal structure in the infinite volume limit

Explicit secular equations are derived using polynomials

An exact index theorem for the overlap Dirac operator is established

Abstract

It is shown that the eigenvalue problem for the hermitian Wilson-Dirac operator of for a uniform magnetic field in two dimensions can be reduced to one-dimensional problem described by a relativistic analog of the Harper equation. An explicit formula for the secular equations is given in term of a set of polynomials. The spectrum exhibits a fractal structure in the infinite volume limit. An exact result concerning the index theorem for the overlap Dirac operator is obtained.