Theory of the Many-Body Localization Transition in One-Dimensional Systems

TLDR

The authors develop a theory of the many‑body localization transition using a novel real‑space renormalization‑group approach. It analyzes the transition via a real‑space renormalization‑group scheme that captures the critical behavior. The theory predicts that the delocalized state near the transition is a Griffiths phase with sub‑diffusive transport and sub‑ballistic entanglement spreading, that the transition is governed by an infinite‑randomness RG fixed point with logarithmic entanglement growth, and that entanglement‑to‑thermal‑entropy ratios and their moments are universal scaling functions of system size over correlation length, with the correlation length diverging at criticality.

Abstract

We formulate a theory of the many-body localization transition based on a novel real space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the "badly conducting" state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits sub-diffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent $\alpha < 1/2$ vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as $\sim\log t$ at the critical point, as it also does in the localized phase. (iii) In the vicinity of the critical point the ratio of the entanglement entropy to the thermal entropy, and its variance (and in fact all moments) are scaling functions of $L/\xi$, where $L$ is the length of the system and $\xi$ is the correlation length, which has a power-law divergence at the critical point.