Strict parabolicity of the multifractal spectrum at the Anderson transition

TL;DR

This paper demonstrates that the multifractal spectrum at the Anderson transition is strictly parabolic by deriving a functional equation for anomalous dimensions, highlighting differences between sigma-models and disordered systems.

Contribution

It derives a functional equation for anomalous dimensions showing the multifractal spectrum's strict parabolicity at the Anderson transition.

Findings

The anomalous dimensions satisfy elta_q=aq(q-1)

The spectrum is strictly parabolic at the transition

Sigma-models do not exactly correspond to disordered systems

Abstract

Using the well-known "algebra of multifractality", we derive the functional equation for anomalous dimensions \Delta_q, whose solution \Delta_q=aq(q-1) corresponds to strict parabolicity of the multifractal spectrum. This result demonstrates clearly that a correspondence of \sigma-models with the initial disordered systems is not exact.