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 $f(\alpha )$ of eigenfunctions satisfies the exact symmetry $f(2d-\alpha )=f(\alpha )+d-\alpha$ 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 $\gamma =(\alpha -d)$ along a renormalization trajectory in the effective time $t=\ln L$. 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 $\Delta (N)=\Delta (1-N)$ for the anomalous parts $\Delta (N)\equiv X(N)-NX(1)$. We present numerical tests in favor of this symmetry for the 2D random $Q-$state Potts model with various $Q$.

15 pages, 3 figures, v2=final version

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