Sum rules for the Dirac spectrum of the Schwinger model

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 also show that the sum rules can be obtained from the clustering property of the scalar correlation functions. This argument 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 superconductors.