Localization enhancement in gain-loss non-Hermitian disordered models.*

Recently the interest to non-Hermitian (nH) disordered models has been revived, due to the claims of instability of a many-body localization to a coupling to a bath.

To describe such open quantum systems, one often focus on an energy leakage to a bath, using effective nH Hamiltonians. A well-known nH Hatano-Nelson model [1], being a 1d Anderson localization (AL) model, with different hopping amplitudes to the right/left, shows AL breakdown, as nH suppresses the interference.

Unlike this, we consider models with the complex gain-loss disorder and show that in general these systems tend to localization due to nH.

First, we focus on a nH version [2] of a Rosenzweig-Porter model [3], known to carry a fractal phase [4] along with the AL and metallic (Me) ones.

We show that Me and AL phases are stable against the nH matrix entries, while the fractal phase, intact to nH off-diagonal terms, gives a way to AL in a gain-loss disorder.

The understanding of this counterintuitive phenomenon is given in terms of the cavity method and in addition in simple hand-waving terms from the Fermi's golden rule, applicable, strictly speaking, to a Hermitian RP model. The main effect in this model is given by the fact that the generally complex diagonal potential forms an effectively 2d (complex) distribution, which parametrically increases the bare level spacing and suppresses the resonances.

Next, we consider a power-law random banded matrix ensemble (PLRBM) [5], known to show AL transition (AT) at the power of the power-law hopping decay a=d equal to the dimension d. In [6], we show that a nH gain-loss disorder in PLRBM shifts AT to smaller values $d/2<a_{AT}(W)<d$, dependent on the disorder on-site W.

A similar effect of the reduced critical disorder due to the gain-loss complex-valued disorder has been recently observed by us numerically [7].

In order to analytically explain the above numerical results, we derive an effective non-Hermitian resonance counting and show that the delocalization transition is driven by so-called "bad resonances",  which cannot be removed by the wave-function hybridization (e.g., in the renormalization group approach), while the usual "Hermitian" resonances are suppressed in the same way as in the non-Hermitian RP model.