Topological Charge and The Spectrum of Exactly Massless Fermions on the Lattice

Abstract

The square root of the positive definite hermitian operator $D_w^{†}{D}_w$ in Neuberger's proposal of exactly massless quarks on the lattice is implemented by the recursion formula $Y_{k+1}=1/2(Y_k+D_w^{†}{D}_w{Y}_k^{-1})$ with $Y_0=\text{Id}$, where $Y_k^2$ converges to $D_w^{†}{D}_w$ 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 non-zero 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.

Comments: 18 pages (LaTeX), 2 figures (EPS)