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

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. 1 The effects of magnetic fields on two-dimensional systems have attracted continual interests from condensed matter to elementary particle physics. Magnetic fields not only bundle energy spectrum but also rearrange the eigenvalues as they continuously changes, giving rise to rich band structure. The most impressive is the energy spectrum of tightly bounded electron in a uniform magnetic field. It is well-known that the energy spectrum of this system has a fractal structure known as the butterfly diagram [1]. The key ingredient that underlies such variety is the topological nature of magnetic fields. Relativistic analogue of the Hamiltonian of tightly bounded electron is the hermitian Wilson-Dirac operator (HWDO), which attracts renewed interests in the context of overlap Dirac operator (ODO) describing chiral gauge theories on the lattice [2]. Magnetic field also affects the spectrum of HWDO, giving rise to interesting phenomena such as chiral anomaly [3]. We also expect the spectrum has a fractal structure similar to the butterfly diagram. In this paper we investigate the spectrum of HWDO for a uniform magnetic field in two dimensions. We show that the two dimensional system can be converted to one dimensional problem for an arbitrary uniform magnetic field and, as the consequence, the spectrum can be characterized by a set of polynomials, which enables us not only to understand the fractal structure of the spectrum but also to compute the exact index of the ODO. The spectral flow of HWDO for the one-parameter family of link variables was investigated in Ref. [4] and the connection to chiral anomaly was elucidated. Recently, one of the present author reanalyzed the same system [5] and gave some exact results on the spectrum for a particular set of uniform magnetic fields, for which the index of the ODO can be obtained rigorously. The purpose of this paper is to extend them for an arbitrary uniform magnetic field. We consider a two-dimensional square lattice of unit lattice spacing and of sides L. The HWDO H is defined by