Dirac spectrum of one-flavor QCD at $\theta =0$ and continuity of the chiral condensate

We derive exact analytical expressions for the spectral density of the Dirac operator at fixed $\theta$-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism results in a chiral condensate that remains constant when the quark mass changes sign.

Figure 1

Three-dimensional plot of the quenched (top) and dynamical (bottom) parts of the rescaled spectral density as a function of $\widehat{x}$ and $\widehat{m}$. The asymptotic value of the rescaled spectral density is equal to $1/\pi$.

Figure 2

Three-dimensional plot of ${\widehat{\rho }}^q(\widehat{x},\widehat{m})+{\widehat{\rho }}^d(\widehat{x},\widehat{m})$.

Figure 3

Comparison of the exact result (25) for the quenched part of the spectral density for $\widehat{m}=40$ (solid red curve) with the asymptotic result (28) (dashed black curve).

Figure 4

Comparison of the rescaled exact result (25) for the quenched part of the spectral density for $\widehat{m}=-40$ (solid red curve) with the asymptotic result (30) (black dashed curve).

Figure 5

The rescaled chiral condensate (black solid curve) for one-flavor QCD as a function of $\widehat{m}$ is the sum of a quenched contribution, which includes the contribution of the zero modes (red curve), and a contribution due to the dynamical quarks (blue curve). In the thermodynamic limit both the red and blue curves develop a discontinuity at $m=\widehat{m}/V\mathrm{\Sigma }=0$.