Nonperturbative results for level correlations from the replica nonlinearσmodel

We show that for all three standard symmetry classes (unitary, orthogonal, and symplectic), the conventional replica nonlinear $\sigma$ model gives the correct nonperturbative result for the two-level correlation functions $R_2\left(\omega \right)$ of electrons in disordered metals in the limit of large $\omega .$ In this limit, nonperturbative oscillatory contributions arise from a degenerate saddle-point manifold within this $\sigma$ model that correspond to the replica-symmetry breaking. Moreover, we demonstrate that in the unitary case the very same results can be extracted from the exact integral representation for $R_2\left(\omega \right)$ used by Verbaarschot and Zirnbauer [J. Phys. A 17, 1093 (1985)] to criticize the replica trick.