Symmetry relations for multifractal spectra at random critical points

Abstract

Random critical points are generically characterized by multifractal properties. In the field of Andersonlocalization, Mirlin et al (2006 Phys. Rev. Lett. 97 046803) have proposed that the singularity spectrumf(α) of eigenfunctions satisfies the exact symmetryf(2d−α) = f(α)+d−α. In the present paper, we analyze the physical origin of this symmetry in relation to theGallavotti–Cohen fluctuation relations of large deviation functions that are wellknown in the field of non-equilibrium dynamics: the multifractal spectrum of thedisordered model corresponds to the large deviation function of the rescaling exponentγ = (α−d) along a renormalization trajectory in the effective timet = lnL. We conclude that the symmetry discovered for the specific example of Andersontransitions should actually be satisfied at many other random critical points after anappropriate translation. For many-body random phase transitions, where thecritical properties are usually analyzed in terms of the multifractal spectrumH(a) and of the momentexponents X(N) of the two-point correlation function (Ludwig 1990 Nucl. Phys. B 330 639), the symmetry becomesH(2X(1)−a) = H(a)+a−X(1), orequivalently Δ(N) = Δ(1−N) for the anomalous parts . We present numerical tests favoring this symmetry for the 2D randomQ-state Potts modelwith varying Q.