Localization in Lattice QCD

TL;DR

This paper investigates the phase diagram of quenched QCD with Wilson fermions, revealing the nature of near-zero modes and their impact on chiral symmetry, and discusses implications for domain-wall and overlap fermions.

Contribution

It provides a microscopic picture of the phase diagram, characterizes near-zero modes in different regions, and analyzes their effects on fermion formulations in lattice QCD.

Findings

Localized near-zero modes cause divergence in pion two-point functions outside the Aoki phase.

Quenched QCD is well-defined only with a non-zero twisted mass.

Domain-wall and overlap fermions work outside but not inside the Aoki phase.

Abstract

In this paper, we examine the phase diagram of quenched QCD with two flavors of Wilson fermions, proposing the following microscopic picture. The super-critical regions inside and outside the Aoki phase are characterized by the existence of a density of near-zero modes of the (hermitian) Wilson-Dirac operator, and thus by a non-vanishing pion condensate. Inside the Aoki phase, this density is built up from extended near-zero modes, while outside the Aoki phase, there is a non-vanishing density of exponentially localized near-zero modes, which occur in "exceptional" gauge-field configurations. Nevertheless, no Goldstone excitations appear outside the Aoki phase, and the existence of Goldstone excitations may therefore be used to define the Aoki phase in both the quenched and unquenched theories. We show that the density of localized near-zero modes gives rise to a divergent pion two-point function, thus providing an alternative mechanism for satisfying the relevant Ward identity in the presence of a non-zero order parameter. This divergence occurs when we take a "twisted" quark mass to zero, and we conclude that quenched QCD with Wilson fermions is well-defined only with a non-vanishing twisted mass. We show that this peculiar behavior of the near-zero-mode density is special to the quenched theory by demonstrating that this density vanishes in the unquenched theory outside the Aoki phase. We discuss the implications for domain-wall and overlap fermions constructed from a Wilson-Dirac kernel. We argue that both methods work outside the Aoki phase, but fail inside because of problems with locality and/or chiral symmetry, in both the quenched and unquenched theories.