The Chiral Condensate of One-Flavor QCD and the Dirac Spectrum at \theta=0

TL;DR

This paper investigates the behavior of the chiral condensate in one-flavor QCD at heta=0, revealing how spectral density oscillations reconcile fixed-topology discontinuities with constant condensate at heta, and emphasizing the importance of zero modes.

Contribution

It demonstrates how spectral density oscillations explain the continuous chiral condensate in one-flavor QCD at heta=0, resolving apparent contradictions in previous observations.

Findings

Spectral density becomes oscillatory for negative quark mass.

Oscillation period scales inversely with space-time volume.

Zero modes are crucial for finite condensate in the thermodynamic limit.

Abstract

In a sector of fixed topological charge, the chiral condensate has a discontinuity given by the Banks-Casher formula also in the case of one-flavor QCD. However, at fixed \theta-angle, the chiral condensate remains constant when the quark mass crosses zero. To reconcile these contradictory observations, we have evaluated the spectral density of one-flavor QCD at \theta=0. For negative quark mass, it becomes a strongly oscillating function with a period that scales as the inverse space-time volume and an amplitude that increases exponentially with the space-time volume. As we have learned from QCD at nonzero chemical potential, if this is the case, an alternative to the Banks-Casher formula applies, and as we will demonstrate in this talk, for one-flavor QCD this results in a continuous chiral condensate. A special role is played by the topological zero modes which have to be taken into account exactly in order to get a finite chiral condensate in the thermodynamic limit.