Groupe de Physique Statistique

Equipe 106, Institut Jean Lamour

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Articles dans des revues à comité de lecture

Symmetry relation for multifractal spectra at random critical points
Monthus C., Berche B. and Chatelain C.
J. Stat. Mech. (2009) P12002
DOI : 10.1088/1742-5468/2009/12/P12002
ArXiv : arxiv:0909.0584 [PDF]

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 $Q$.

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