Topological charge and the spectrum of exactly massless fermions on the lattice

Abstract

The square root of the positive definite Hermitian operator ${D}_{w}^{\ifmmode\dagger\else\textdagger\fi{}}{D}_{w}$ in Neuberger's proposal of exactly massless quarks on the lattice is implemented by the recursion formula ${Y}_{k+1}=\frac{1}{2}{(Y}_{k}{+D}_{w}^{\ifmmode\dagger\else\textdagger\fi{}}{D}_{w}{Y}_{k}^{\ensuremath{-}1})$ with ${Y}_{0}=1,$ where ${Y}_{k}^{2}$ converges to ${D}_{w}^{\ifmmode\dagger\else\textdagger\fi{}}{D}_{w}$ quadratically. The spectrum of the lattice Dirac operator for single massless fermion in two dimensional background U(1) gauge fields is investigated. For smooth background gauge fields with nonzero topological charge, the exact zero modes with definite chirality are reproduced to a very high precision on a finite lattice and the index theorem is satisfied exactly. The fermionic determinants are also computed and they are in good agreement with the continuum exact solution.