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 ≈ −s′. This lifts the sum rule prohibition for the level number variance to have a Poisson-like term var(n) = ηn 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, η playing the role of an order-parameter. Typeset using REVTEX 1 The statistical description of complex systems by ensembles of random matrices turned out to be a powerful general approach that was successively applied to a great variety of systems in different fields from nuclear physics [1] to mesoscopics [2] and quantum chaos [3]. The classical random matrix theory (RMT) by Wigner, Dyson and Mehta [1] describes the statistics of eigenvalues for a Gaussian ensemble of random Hermitian N ×N -matrices H with the probability distribution P (H) ∝ exp[−TrH]. By definition, the statistical properties of this ensemble are invariant under unitary transformations U(N) of matrices H and thus there is no basis preference in the RMT. This means that the classical RMT can be applied only to quantum systems where all (normalized) linear combinations of eigenfunctions have similar properties. For disordered electronic systems, it implies that all eigenstates must be extended. In other words, the classical RMT is applicable only for describing the energy level statistics in the metal phase [4,5] that exists in the dimensionality d > 2 at a relatively weak disorder. With disorder increasing the system goes through the Anderson transition to an insulating phase in which all eigenstates are localized. The level statistics in this phase obviously cannot be described by the U(N)-invariant RME, since one can construct an extended state by a linear combination of localized states randomly positioned throughout the sample. Thus the proper probability distribution P (H) must contain a basis preference in order to exclude unitary transformations which would lead to formation of such extended states. The ensemble of random banded matrices (RBME) [6,7] is an example of such a noninvariant RMT. It describes properties of systems belonging to so-called quasi-1d universality class which includes quasi-1d disordered electronic systems with localization [6] and certain quantum chaotic systems [7]. The corresponding eigenvalue statistics are Poissonic in the N → ∞ limit (at a fixed bandwidth b) and reduce to the Wigner-Dyson form at b → ∞. Thus changing the parameter b/N , one can describe the crossover from Wigner-Dyson to Poisson level statistics which occurs in quasi-1d disordered systems with increasing the ratio L/ξ of the sample length L and the localization radius ξ. While the localization in quasi-1d systems seems to be well described in terms of the 2 RBME, the problem of the random matrix description of the critical region near the Anderson transition and the Anderson insulator phase for d > 2 remains open. The recent works [8-10] where the existence of the universal critical level statistics has been demonstrated, resulted in an intensive search for the proper RME description. In this connection, two different generalizations of the classical RMT have been recently proposed [11,12]. The generalized RME studied in Ref.[11] was obtained from the Gaussian invariant ensemble by introducing a symmetry-breaking term: P (H) ∝ e 2 e 2N2Tr([Λ,H][Λ,H]†). (1) The h-dependent term breaks the U(N) invariance and tends to align H with a symmetry breaking unitary matrix Λ thus setting the basis preference. It turned out [11] that even after averaging over Λ the resulting ensemble leads to the eigenvalue statistics that deviate from the Wigner-Dyson form. The difference between the Wigner-Dyson statistics that correspond to h = 0 and the level statistics for any non zero h turns out to be dramatic in the thermodynamic (TD) limit N → ∞. Namely, for h 6= 0 the variance var(n) of the number of levels in an energy window that contains n levels on the average, grows linearly with n at n ≫ 1: var(n) = 〈(δn)〉 = η(h)n ∼ hn, η(0) = 0. (2) For the classical RMT [1], var(n) ∝ lnn that is negligible as compared to Eq.(2) for any nonzero 0 < η(h) < 1 in the limit n → ∞. The Poisson-like behavior described by Eq.(2) is valid also for RBME in the TD limit. In contrast to Eq.(1), the probability distribution suggested in Ref.[12] is explicitly U(N)invariant: