The n-level spectral correlations for chaotic systems

TL;DR

This paper demonstrates that the $n$-level spectral correlation functions of chaotic quantum systems without time-reversal symmetry match the predictions of the Circular Unitary Ensemble, confirming a key universality conjecture.

Contribution

The authors derive the $n$-level correlation functions for chaotic systems using a semiclassical approach, extending the diagonal approximation to match CUE predictions.

Findings

Correlation functions agree with CUE predictions

Extended diagonal approximation is effective

Supports universality conjecture for chaotic systems

Abstract

We study the $n$-level spectral correlation functions of classically chaotic quantum systems without time-reversal symmetry. According to Bohigas, Giannoni and Schmit's universality conjecture, it is expected that the correlation functions are in agreement with the prediction of the Circular Unitary Ensemble (CUE) of random matrices. A semiclassical resummation formalism allows us to express the correlation functions as sums over pseudo-orbits. Using an extended version of the diagonal approximation on the pseudo-orbit sums, we derive the $n$-level correlation functions identical to the $n \times n$ determinantal correlation functions of the CUE.