Breaking the chains: extreme value statistics and localization in random spin chains

Abstract

Despite a very good understanding of single-particle Anderson localization in one-dimensional (1D) disordered systems, many-body effects are still full of surprises, a famous example being the interaction-driven many-body localization (MBL) problem, about which much has been written, and perhaps the best is yet to come. Interestingly enough the non-interacting limit provides a natural playground to study non-trivial multiparticle physics, offering the possibility to test some general mechanisms with very large-scale exact diagonalization simulations. In this work, we first revisit the 1D many-body Anderson insulator through the lens of extreme value theory, focusing on the extreme polarizations of the equivalent spin chain model in a random magnetic field. A many-body-induced chain breaking mechanism is explored numerically, and compared to an analytically solvable toy model. A unified description, from weak to large disorder strengths $W$ emerges, where the disorder-dependent average localization length $\xi (W)$ governs the extreme events leading to chain breaks. In particular, tails of the local magnetization distributions are controlled by $\xi (W)$. Remarkably, we also obtain a quantitative understanding of the full distribution of the extreme polarizations, which is given by a Fréchet-type law. In a second part, we explore finite interaction physics and the MBL question. For the available system sizes, we numerically quantify the difference in the extreme value distributions between the interacting problem and the non-interacting Anderson case. Strikingly, we observe a sharp "extreme-statistics transition" as $W$ changes, which may coincide with the MBL transition.

arXiv comment18 + 6 pages; 13 + 4 figures; Accepted version

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