Normalization Sum Rule and Spontaneous Breaking of U(N) Invariance in Random Matrix Ensembles

TL;DR

This paper demonstrates that in certain invariant random matrix ensembles with soft confinement, the U(N) symmetry is spontaneously broken, leading to a Poisson-like variance in level number fluctuations, characterized by a new sum rule.

Contribution

It introduces a normalization sum rule and shows how soft confinement causes spontaneous U(N) symmetry breaking in invariant RME, affecting level statistics.

Findings

Identification of a ghost peak in the two-level correlation function.

Demonstration of a Poisson-like variance term in level number fluctuations.

Establishment of the order-parameter role of $ta$ for symmetry breaking.

Abstract

It is shown that the two-level correlation function $R(s,s')$ in the invariant random matrix ensembles (RME) with soft confinement exhibits a "ghost peak" at $s\approx -s'$. This lifts the sum rule prohibition for the level number variance to have a Poisson-like term ${\rm var}(n)=\eta n$ that is typical of RME with broken U(N) symmetry. Thus we conclude that the U(N) invariance is broken spontaneously in the RME with soft confinement, $\eta$ playing the role of an order-parameter.