Generalized multifractality in 2D disordered systems of chiral symmetry classes

TL;DR

This paper investigates the complex behavior of eigenstate fluctuations in 2D disordered systems with chiral symmetry, combining analytical sigma-model theory with numerical simulations to reveal unique symmetry properties and deviations at phase transitions.

Contribution

It introduces a novel framework for generalized multifractality in chiral classes, including new composite operators and symmetry analyses, supported by numerical validation.

Findings

Eigenfunction exponents satisfy Weyl symmetry relations.

Generalized parabolicity holds in the metallic phase.

Parabolicity is violated at the Anderson transition.

Abstract

We study generalized multifractality that characterizes eigenstate fluctuations and correlations in disordered systems of chiral symmetry classes AIII, BDI, and CII. By using the non-linear sigmamodel field theory, we construct pure-scaling composite operators and eigenfunction observables that satisfy Abelian fusion rules. The observables are labelled by two multi-indices $\lambda$, $\lambda'$ referring to two sublattices, at variance with other symmetry classes, where a single multi-index $\lambda$ (that can be viewed as a generalized version of a Young diagram) is needed. Further, we analyze Weyl symmetries of multifractal exponents, which are also peculiar in chiral classes, in view of a distinct root system associated with the sigma-model symmetric space. The analytical results are supported and complemented by numerical simulations that are performed for a 2D lattice Hamiltonian of class AIII, both in the metallic phase and at the Anderson transition. Both in the metallic phase and at the transition, the numerically obtained exponents satisfy Weyl symmetry relations, confirming that the sigma-model is the right theory of the problem. Furthermore, in the metallic phase, we observe the generalized parabolicity (proportionality to eigenvalues of the quadratic Casimir operator), as expected in the one-loop approximation. On the other hand, the generalized parabolicity is strongly violated at the metal-insulator transition, implying violation of local conformal invariance.