Sum rules for the Dirac spectrum of the Schwinger model

Abstract

The inverse eigenvalues of the Dirac operator in the Schwinger model satisfy the same Leutwyler-Smilga sum rules as in the case of QCD with one flavor. In this paper we give a microscopic derivation of these sum rules in the sector of arbitrary topological charge. We show that the sum rules can be obtained from the clustering property of the scalar correlation functions. This argument also holds for other theories with a mass gap and broken chiral symmetry such as QCD with one flavor.For QCD with several flavors a modified clustering property is derived from the low energy chiral Lagrangian. We also derive sum rules for a fixed external gauge field.Both types of sum rules are obtained from the bosonized version of the Schwinger model.In the sector of topological charge ν the sum rules are consistent with a shift of the Dirac spectrum away from zero by ν/2 average level spacings. This shift is also required to obtain a nonzero chiral condensate in the massless limit. Finally, we discuss the Dirac spectrum for a closely related two-dimensional theory with a gauge field action that is quadratic in the gauge field and has applications to the quantum Hall effect and d-wave super-conductors.

Lessons From Random Matrix Theory for QCD at Finite Density

“…Since we compare the ratio of two partition functions, it is essential that they are normalized correctly. This has been studied in a random matrix framework [53] confirming the above results.…”

Dirac spectra of two-dimensional QCD-like theories

“…A two-dimensional model for which Dirac spectra have been studied in great detail, both analytically and numerically, is the Schwinger model. The eigenvalue correlations of the one-flavor Schwinger model are given by random matrix theory as was shown numerically [15,16] and analytically by calculating the Leutwyler-Smilga sum rules [17]. The two flavor Schwinger model was analyzed in great detail in [18], and after rescaling the eigenvalues by the average level spacing excellent agreement with chiral random matrix theory is observed.…”

“…In three dimensions the index is not defined. In two dimensions topology is defined for U (1) and can for example be studied for the Schwinger model [17,20]. However, for higher dimensional gauge groups the index of the Dirac operator is zero [19,21,22] although unstable instantons do exist [23,24].…”

Dirac spectrum and chiral condensate for QCD at fixed θ angle

“…by employing the recurrence relations (C1) and (C2) and the properties of the determinant. In this form the terms involving the upper left 2 × 2 block yield the quenched level density (26). The remaining terms represent the dynamical part and can be simplified to…”