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

The square root of the positive definite hermitian operator $ D_w^{\dagger} 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^{\dagger} D_w Y_k^{-1}) $ with $ Y_0 = \Id $, where $ Y_k^2 $ converges to $ D_w^{\dagger} 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.