Symmetry Violation of Quantum Multifractality: Gaussian fluctuations versus Algebraic Localization

TL;DR

This paper investigates the causes of symmetry violations in quantum multifractality, identifying Gaussian fluctuations and algebraic localization as two distinct mechanisms affecting the anomalous multifractal dimension.

Contribution

It reveals a previously unexplored mechanism involving algebraic localization that causes symmetry violations in quantum multifractality.

Findings

Gaussian fluctuations can be removed by coarse graining.

Algebraic localization causes robust symmetry violations.

The violation mechanism is demonstrated in two different quantum systems.

Abstract

Quantum multifractality is a fundamental property of systems such as non-interacting disordered systems at an Anderson transition and many-body systems in Hilbert space. Here we discuss the origin of the presence or absence of a fundamental symmetry related to this property. The anomalous multifractal dimension $\Delta_q$ is used to characterize the structure of quantum states in such systems. Although the multifractal symmetry relation \mbox{$\Delta_q=\Delta_{1-q}$} is universally fulfilled in many known systems, recently some important examples have emerged where it does not hold. We show that this is the result of two different mechanisms. The first one was already known and is related to Gaussian fluctuations well described by random matrix theory. The second one, not previously explored, is related to the presence of an algebraically localized envelope. While the effect of Gaussian fluctuations can be removed by coarse graining, the second mechanism is robust to such a procedure. We illustrate the violation of the symmetry due to algebraic localization on two systems of very different nature, a 1D Floquet critical system and a model corresponding to Anderson localization on random graphs.