Symmetry relation for multifractal spectra at random critical points
Abstract
Random critical points are generically characterized by multifractal properties. In the field of Anderson localization, Mirlin, Fyodorov, Mildenberger and Evers [Phys. Rev. Lett 97, 046803 (2006)] have proposed that the singularity spectrum f(α) of eigenfunctions satisfies the exact symmetry f(2d-α)=f(α)+d-α at any Anderson transition. In the present paper, we analyse the physical origin of this symmetry in relation with the Gallavotti-Cohen fluctuation relations of large deviation functions that are well-known in the field of non-equilibrium dynamics: the multifractal spectrum of the disordered model corresponds to the large deviation function of the rescaling exponent γ=(α-d) along a renormalization trajectory in the effective time t= L. We conclude that the symmetry discovered on the specific example of Anderson transitions should actually be satisfied at many other random critical points after an appropriate translation. For many-body random phase transitions, where the critical properties are usually analyzed in terms of the multifractal spectrum H(a) and of the moments exponents X(N) of two-point correlation function [A. Ludwig, Nucl. Phys. B330, 639 (1990)], the symmetry becomes H(2X(1) -a)= H(a) + a-X(1), or equivalently (N)=(1-N) for the anomalous parts (N) X(N)-NX(1). We present numerical tests in favor of this symmetry for the 2D random Q-state Potts model with various Q.