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

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 Hₙₘ~1/|n-m|^α, having AT at α=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≤α≤d. The value of the critical α 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 α even for α<d. This result provides an example of non-Hermiticity-induced localization and finds immediate application in phase transitions driven by weak measurements.