Normalization sum rule and spontaneous breaking of U(N) invariance in random matrix ensembles

It is shown that the two-level correlation function ${\mathit{R}}_2^{\mathrm{\infty }}$(s,${\mathit{s}}^{\mathcal{’}}$) in the invariant random matrix ensembles (RME’s) with soft confinement exhibits unexpected long-range correlations in the form of a sharp peak at s≊-s’. The presence of such a ‘‘ghost’’ correlation peak lifts the sum rule prohibition for the level number variance to have a Poisson-like term var(n)=ηn that is typical of RME’s with broken U(N) symmetry. Thus we conclude that the U(N) invariance is broken spontaneously in the RME’s with soft confinement, η playing the role of an order parameter.