Energy level dynamics in systems with weakly multifractal eigenstates: Equivalence to one-dimensional correlated fermions at low temperatures

Abstract

It is shown that the parametric spectral statistics in the critical random matrix ensemble with multifractal eigenvector statistics are identical to the statistics of correlated one-dimensional (1D) fermions at finite temperatures. For weak multifractality the effective temperature of fictitious 1D fermions is proportional to Teff α (1 - dn)/n < 1, where dn is the fractal dimension found from the nth moment of the inverse participation ratio. For large energy and parameter separations the fictitious fermions are described by the Luttinger liquid model which follows from the Calogero-Sutherland model. The low-temperature asymptotic form of the two-point equal-parameter spectral correlation function is found for all energy separations and its relevance for the low-temperature equal-time density correlations in the Calogero-Sutherland model is conjectured.