Critical Fidelity

TL;DR

This paper investigates how the fidelity of systems at the Anderson transition decays under small perturbations, revealing three distinct regimes including a novel algebraic decay linked to the eigenstates' correlation dimension.

Contribution

It introduces a detailed analysis of fidelity decay regimes at the Anderson transition using a Wigner Lorentzian Random Matrix ensemble, highlighting the algebraic decay related to the correlation dimension.

Findings

Fidelity exhibits three decay regimes: Gaussian, exponential, and algebraic.

Algebraic decay follows $F(t) o t^{-D_2}$, with $D_2$ as the correlation dimension.

Linear Response Theory describes the initial two decay regimes.

Abstract

Using a Wigner Lorentzian Random Matrix ensemble, we study the fidelity, $F(t)$, of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a gaussian and an exponential decay respectively and can be described using Linear Response Theory. For stronger perturbations $F(t)$ decays algebraically as $F(t)\sim t^{-D_2}$, where $D_2$ is the correlation dimension of the critical eigenstates.