Critical spectral statistics as the Luttinger liquid of energy levels at a finite temperature

Abstract

We review the recent results on the energy level statistics in disordered conductors near the Anderson localization transition and other critical systems with multifractal eigenstates. We focus on the critical random matrix ensemble (CRMT) suggested by Mirlin and Fyodorov as a possible generic description of the critical parametric spectral statistics (CPSS). We show that in the region of large energy and/or parameter separations and weak multifractality this CRMT is equivalent to the Luttinger liquid of energy levels at a finite temperature with the interaction between the fictitious 1D fermions (energy levels) being dependent on the Dyson symmetry parameter β = 1, 2, 4 and the temperature being proportional to the multifractality exponent η = 1 — D2. We show that the CRMT is the simplest extension of the classical Wigner-Dyson random matrix ensemble for the case of finite dimensionless conductance g ≫ 1 and suggest the form of the two-level correlation functions for all the symmetry classes and all energy separations. We argue that these correlation functions coincide with the density correlation functions for the Calogero-Sutherland model at finite temperatures.