Non-ergodic delocalized phase with Poisson level statistics

Weichen Tang1 and Ivan M. Khaymovich2,3,4

1Department of Physics, University of California, Berkeley, California 94720, USA
2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany
3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia
4Nordita, Stockholm University and KTH Royal Institute of Technology Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

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Motivated by the many-body localization (MBL) phase in generic interacting disordered quantum systems, we develop a model simulating the same eigenstate structure like in MBL, but in the random-matrix setting. Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space. On the above example, we formulate general conditions to a single-particle and random-matrix models in order to carry such states, based on the transparent generalization of the Anderson localization of single-particle states and multiple resonances.

Many-body localization is a phenomenon, providing a unique possibility for an isolated interacting quantum system to avoid thermalization and to keep the information about its initial state.
Being the localization in the real space, many-body localization not only suppresses the transport, but also avoids repulsion of energy levels and makes eigenstates of an interacting system to be non-ergodic, but extended in the space of configurations (Hilbert space).
Facing the difficulties of describing many-body quantum systems, many researchers focus on the universal statistical description of the above phenomenon in a random-matrix setting.

Motivated by the many-body localization phase, we develop a random-matrix model simulating the same eigenstate structure.
Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space.
Using the above example and a transparent generalization of the Anderson localization to multiple resonances, we formulate general conditions to realize non-ergodic states with Poisson level statistics.

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[3] Xiaolong Deng, Alexander Burin, and Ivan Khaymovich, "Anisotropy-mediated reentrant localization", arXiv:2002.00013, SciPost Physics 13 5, 116 (2022).

[4] Xiaolong Deng, Ivan Khaymovich, and Alexander L. Burin, "Superdiffusion in random two dimensional system with ubiquitous long-range hopping", arXiv:2205.14715.

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[6] M. Tarzia, "Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime", Physical Review B 105 17, 174201 (2022).

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