**Between
Localization and Ergodicity in Quantum Systems**

Boris
Altshuler

Columbia University

Strictly speaking the laws of conventional Statistical Physics, in particular the Equipartition Postulate, apply only in the presence of a thermostat. For a long time this restriction did not look crucial for realistic systems. The recent development of two new classes of quantum many-body system in which the coupling to the outside world is (or is hoped to be) negligible: (1) cold quantum gases and (2) systems of qubits, is forcing our community to return to the very foundations of Statistical Mechanics. A first step in this direction is the concept of Many-Body Localization (MBL) [1], according to which the states of a many-body system can be localized in Hilbert space via a process that resembles the celebrated Anderson Localization of single particle states in a random potential. Moreover, the one-particle localization in an Anderson tight-binding model (on-site disorder) on regular random graphs (RRG) strongly resembles a generic MBL.

MBL implies that the state of the system, decoupled from the thermostat depends on initial conditions: time averaging does not result in an equipartition distribution, the entropy never reaches its thermodynamic value i.e. the ergodicity is violated. Variations of e.g. temperature can delocalize many body states. However, the recovery of the equipartition is not likely to follow the delocalization immediately: numerical analysis of the RRG problem suggests that the extended states are multi-fractal at any finite disorder [2]. Moreover, regular (no disorder!) Josephson junction arrays (JJA) under the conditions that are feasible to implement and control experimentally demonstrate both MBL and non-ergodic behavior [3].

1. D. Basko, I. Aleiner, and B. Altshuler, Ann. Phys. 321, 1126 (2006).

2. A. De Luca, B.L. Altshuler, V.E. Kravtsov, & A. Scardicchio, PRL 113, 046806, (2014)

3. M. Pino, B.L. Altshuler and L.B. Ioffe, arXiv:1501.03853, PNAS to be published.

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