Thouless energy and multifractality across the many-body localization transition

Affiliation

IST Austria

Abstract:

Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across this transition in two different models of disordered spin chains. We show that the behavior of matrix elements can be used to characterize the breakdown of thermalization and to extract the many-body Thouless energy. We find that upon increasing the disorder strength the system enters a critical region around the many-body localization transition. The properties of the system in this region are: (i) the Thouless energy becomes smaller than the level spacing, (ii) the matrix elements show critical dependence on the energy difference, and (iii) the matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. This critical region decreases with the system size, which we interpret as evidence for a diverging correlation length at the many-body localization transition. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values and shed light on the critical behavior near the many-body localization transition.

We acknowledge useful discussions with V. Kravtsov, T.
Grover, and R. Vasseur. M.S. was supported by Gordon and
Betty Moore Foundation’s EPiQS Initiative through Grant No.
GBMF4307. Z.P. acknowledges support by EPSRC Grant No.
EP/P009409/1. Statement of compliance with EPSRC policy
framework on research data: This publication is theoretical
work that does not require supporting research data. M.S.
and D.A. acknowledge hospitality of KITP, where parts of
this work were completed (supported in part by the National
Science Foundation under Grant No. NSF PHY-1125915).
D.A. acknowledges support by the Swiss National Science
Foundation.