Extremal statistics of entanglement eigenvalues can track the many-body localized to ergodic transition

Abstract

Some interacting disordered many-body systems are unable to thermalize when the quenched disorder becomes larger than a threshold value. Although several properties of nonzero energy density eigenstates (in the middle of the many-body spectrum) exhibit a qualitative change across this many-body localization (MBL) transition, many of the commonly-used diagnostics only do so over a broad transition regime. Here, we provide evidence that the transition can be located precisely even at modest system sizes by sharply-defined changes in the distribution of extremal eigenvalues of the reduced density matrix of subsystems. In particular, our results suggest that $p\ast ={\lim }_{{\lambda }_2\to \ln (2{)}^{+}}{P}_2({\lambda }_2)$, where $P_2({\lambda }_2)$ is the probability distribution of the second lowest entanglement eigenvalue ${\lambda }_2$, behaves as an ''order-parameter'' for the MBL phase: $p\ast >0$ in the MBL phase, while $p\ast =0$ in the ergodic phase with thermalization. Thus, in the MBL phase, there is a nonzero probability that a subsystem is entangled with the rest of the system only via the entanglement of one subsystem qubit with degrees of freedom outside the region. In contrast, this probability vanishes in the thermal phase.

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• Renyi Entropy of Interacting Thermal Bosons in Large $N$ Approximation
• Breaking the chains: extreme value statistics and localization in random spin chains
• Memories of initial states and density imbalance in dynamics of interacting disordered systems
• Characterizing the many-body localization transition through correlations
• Many-body localized to ergodic transitions in a system with correlated disorder
• Local density of states and scattering rates across the many-body localization transition
• Emergence of ghost in once-subtracted on-shell unitarization in glueball-glueball scattering