Generalized multifractality at metal-insulator transitions and in metallic phases of 2D disordered systems

TL;DR

This paper investigates generalized multifractality in eigenstates of disordered 2D systems across metallic and insulating phases, revealing symmetry violations at critical points and the role of sigma-model manifold topology.

Contribution

It constructs pure-scaling eigenfunction observables for symmetry classes AII, D, and DIII and verifies their properties through numerical simulations, highlighting symmetry violations at transitions.

Findings

Exponents satisfy Weyl symmetry and parabolicity in metallic phases.

Parabolicity is violated at metal-insulator transition critical points.

Weyl symmetry breaks down at critical points in classes D and DIII due to sigma-model manifold topology.

Abstract

We study generalized multifractality characterizing fluctuations and correlations of eigenstates in disordered systems of symmetry classes AII, D, and DIII. Both metallic phases and Andersonlocalization transitions are considered. By using the non-linear sigma-model approach, we construct pure-scaling eigenfunction observables. The construction is verified by numerical simulations of appropriate microscopic models, which also yield numerical values of the corresponding exponents. In the metallic phases, the numerically obtained exponents satisfy Weyl symmetry relations as well as generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator). At the same time, the generalized parabolicity is strongly violated at critical points of metal-insulator transitions, signalling violation of local conformal invariance. Moreover, in classes D and DIII, even the Weyl symmetry breaks down at critical points of metal-insulator transitions. This last feature is related with a peculiarity of the sigma-model manifolds in these symmetry classes: they consist of two disjoint components. Domain walls associated with these additional degrees of freedom are crucial for ensuring Anderson localization and, at the same time, lead to the violation of the Weyl symmetry.