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

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.