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## Localization enhancement in gain-loss non-Hermitian disordered models

### Ivan Khaymovich (Nordita Stockholm)

Recently the interest in non-Hermitian 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 focuses on an energy leakage to a bath, using effective non-Hermitian Hamiltonians. A well-known Hatano-Nelson model [1], being a 1d Anderson localization (AL) model, with different hopping amplitudes to the right/left, shows AL breakdown, as non-Hermiticity suppresses the interference.

Unlike this, we consider models with the complex gain-loss disorder and show that in general these systems tend to the localization due to non-Hermiticity. First, we focus on a non-Hermitian version [2] of a Rosenzweig-Porter model [3], known to carry a fractal phase [4] along with the AL and ergodic ones. We show that ergodic and localized phases are stable against the non-Hermitian matrix entries, while the fractal phase, intact to non-Hermiticity of 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 (ALT) at the power of the power-law hopping decay a=d equal to the dimension d. In [6], we show that a non-Hermitian gain-loss disorder in PLRBM shifts ALT 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.

In the last part, if time permits, I will consider the effects of non-Hermitian diagonal disorder on many-body localization in interacting systems and the localization in quantum random energy model [8] and show that the above paradigm also works there.

[1] N. Hatano, D. R. Nelson, « Localization Transitions in Non-Hermitian Quantum Mechanics », PRL 77, 570 (1996).

[2] G. De Tomasi, I. M. K. « Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase », Phys. Rev. B, 106, 094204 (2022).

[3] N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra,” Phys. Rev. B 120, 1698 (1960).

[4] V. E. Kravtsov, I. M. K., E. Cuevas, and M. Amini, “A random matrix model with localization and ergodic transitions,” New J. Phys. 17, 122002 (2015).

[5] A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, “Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices,” Phys. Rev. E 54, 3221–3230 (1996).

[6] G. De Tomasi, I. M. K. « Non-Hermiticity induces localization: good and bad resonances in power-law random banded matrices », Phys. Rev. B 108, L180202 (2023).

[7] L. S. Levitov « Absence of localization of vibrational modes due to dipole-dipole interaction », EPL 9, 83 (1989).

[8] G. De Tomasi, I. M. K. « Stable many-body localization under random continuous measurements in the no-click limit », arXiv:2311.00019 .