On the consistency of the Aoki-phase

TL;DR

This paper defends the standard understanding of the Aoki-phase in lattice QCD with Wilson fermions, showing it is consistent with recent reanalyses and deriving related lattice sum rules.

Contribution

It demonstrates the consistency of the traditional Aoki-phase analysis with new approaches and extends continuum sum rules to lattice formulations.

Findings

The standard Aoki-phase understanding aligns with recent reanalysis methods.

The approach confirms the vanishing of the flavor-singlet pseudoscalar expectation value.

Lattice generalizations of Leutwyler and Smilga sum rules are derived.

Abstract

Lattice QCD with two flavors of Wilson fermions can exhibit spontaneous breaking of flavor and parity, with the resulting "Aoki phase" characterized by the non-zero expectation value $<\bar\psi \gamma_5 \tau_3 \psi>\ne0$. This phenomenon can be understood using the chiral effective theory appropriate to the Symanzik effective action. Within this standard analysis, the flavor-singlet pseudoscalar expectation value vanishes: $<i \bar\psi \gamma_5 \psi>=0$. A recent reanalysis has questioned this understanding, arguing that either the Aoki-phase is unphysical, or that there are additional phases in which $<i \bar\psi \gamma_5 \psi>\ne0$. The reanalysis uses the properties of probability distribution functions for observables built of fermion fields and expansions in terms of the eigenvalues of the hermitian Wilson-Dirac operator. Here I show that the standard understanding of the Aoki-phase is, in fact, consistent with the approach used in the reanalysis. Furthermore, if one assumes that the standard understanding is correct, one can use the methods of the reanalysis to derive lattice generalizations of the continuum sum rules of Leutwyler and Smilga.