Non-Hermiticity induces localization: Good and bad resonances in power-law random banded matrices
Giuseppe De Tomasi, Ivan M. Khaymovich
Abstract
The power-law random banded matrix (PLRBM) is a paradigmatic ensemble to study the Anderson localization transition (AT). In $d$ dimensions, the PLRBMs are random matrices with algebraic decaying off-diagonal elements ${H}_{\mathbf{n}\mathbf{m}}\ensuremath{\sim}1/{|\mathbf{n}\ensuremath{-}\mathbf{m}|}^{\ensuremath{\alpha}}$, having AT at $\ensuremath{\alpha}=d$. In this work, we investigate the fate of the PLRBM to non-Hermiticity (nH). 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 understand the model analytically by generalizing the Anderson-Levitov resonance counting technique to the nH case. We identify two competing mechanisms due to nH: favoring localization and delocalization. The competition between the two gives rise to AT at $d/2\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The value of the critical $\ensuremath{\alpha}$ depends on the strength of the on-site potential, like in Hermitian disordered short-range models in $d>2$. Within the localized phase, the wave functions are algebraically localized with an exponent $\ensuremath{\alpha}$ even for $\ensuremath{\alpha}<d$. This result provides an example of non-Hermiticity-induced localization and finds immediate application in phase transitions driven by weak measurements.