Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices

TL;DR

This paper explores how non-Hermiticity in power-law random banded matrices influences localization, revealing a competition between localization and delocalization mechanisms that determines the Anderson transition in open, gain-loss systems.

Contribution

It generalizes the Anderson-Levitov resonance counting technique to non-Hermitian matrices, identifying how non-Hermiticity induces localization and modifies the Anderson transition.

Findings

Non-Hermiticity induces localization in power-law random matrices.

The Anderson transition occurs at specific decay exponents depending on non-Hermiticity.

Wave functions exhibit algebraic localization even below the critical decay exponent.

Abstract

The power-law random banded matrix (PLRBM) is a paradigmatic ensemble to study the Anderson localization transition (AT). In $d$-dimension the PLRBM are random matrices with algebraic decaying off-diagonal elements $H_{\vec{n}\vec{m}}\sim 1/|\vec{n}-\vec{m}|^\alpha$, having AT at $\alpha=d$. In this work, we investigate the fate of the PLRBM to non-Hermiticity. We consider the case where the random on-site diagonal potential takes complex values, mimicking an open system, subject to random gain-loss terms. We provide an analytical understanding of the model by generalizing the Anderson-Levitov resonance counting technique to the non-Hermitian case. This generalization identifies two competing mechanisms due to non-Hermiticity: one favoring localization and the other delocalization. The competition between the two gives rise to AT at $d/2\le \alpha\le d$. The value of the critical $\alpha$ depends on the strength of the on-site potential, reminiscent of Hermitian disordered short-range models in $d>2$. Within the localized phase, the wave functions are algebraically localized with an exponent $\alpha$ even for $\alpha<d$. This result provides an example of non-Hermiticity-induced localization.