Symmetries of multifractal spectra and field theories of Anderson localization

TL;DR

This paper reveals that symmetries in multifractal spectra at Anderson transitions originate from conformal invariance and Weyl group symmetries in the underlying field theories, providing a unified theoretical framework.

Contribution

It establishes a field-theoretic basis for symmetry relations in multifractal spectra, linking them to conformal invariance and Weyl group symmetries in non-linear sigma models.

Findings

Symmetry relations follow from conformal invariance of the critical theory.

The probability distribution of local density of states exhibits a symmetry from Weyl group invariance.

The results apply broadly to Anderson localization and other disordered critical points.

Abstract

We uncover field-theoretic underpinnings of symmetry relations for multifractal spectra at Anderson transitions and at critical points of other disordered systems. We show that such relations follow from the conformal invariance of the critical theory, which implies their general character. We also demonstrate that for the Anderson localization problem the entire probability distribution for the local density of states possesses a symmetry arising from the invariance of correlation functions of the underlying non-linear $\sigma$-model with respect to the Weyl group of the target space of the model.