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

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.

4.5 pages, 4 figures, 57 references + 5 pages, 4 figures in Appendices

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• Higher-order gap ratios of singular values in open quantum systems
• Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry
• Non-equilibrium dynamics of localization phase transition in the non-Hermitian Disorder-Aubry-André model
• Localization of Lindbladian Fermions
• Robust extended states in Anderson model on partially disordered random regular graphs
• Spectral form factor and energy correlations in banded random matrices
• Eigenvalue spectral tails and localization properties of asymmetric networks
• Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians
• Ergodicity-breaking phase diagram and fractal dimensions in long-range models with generically correlated disorder
• Theory of the Anderson transition in three-dimensional chiral symmetry classes: Connection to type-II superconductors