On the Neuberger overlap operator

We compute Neuberger’s overlap operator by the Lanczos algorithm applied to the Wilson-Dirac operator. Locality of the operator for quenched QCD data and its eigenvalue spectrum in an instanton background are studied. 1. Although brute force calculations of the quenched lattice QCD with Wilson fermions have been able to approach the chiral limit [1], there are increased efforts to make the chiral symmetry exact on the lattice [2, 3]. There are different starting points to formulate lattice actions with exact lattice chiral symmetry, but all of them ought to obey the Ginsparg-Wilson condition [5]: γ5D +Dγ5 = aDγ5D (1) The condition implies an operator D with non-compact support over the lattice. Therefore, meaningful solutions are those with an almost compact D, or with exponentially vanishing tails, so that locality of the operator is observed. A candidate is the overlap operator of Neuberger [2]: 1 D = 1− V, V = γ5sign(H), H = γ5(1− aDW ) (2) where a is the lattice spacing and DW the Wilson-Dirac operator, DW = 1 2 ∑ μ [γμ(∂ ∗ μ + ∂μ)− a∂ ∗ μ∂μ] (3) and ∂μ and ∂ ∗ μ are the nearest-neighbor forward and backward difference operators. Note that −H can be seen as the Hermitian variant of the Wilson-Dirac operator with negative bare mass −1. 2 The free overlap operator is analytic and 2π-periodic in momentum space [6], therefore its Fourier transform is local. The locality can also be shown for smooth background fields and in quenched samples simulated at moderate couplings [7]. We make some trivial changes like in [6] for our convenience. The reader familiar with the notations of the hopping parameter κ and the hopping matrix M , may find useful the relation: H = −3γ5(1 − κM), with κ = 1/6.