Eigenstates hybridize on all length scales at the many-body localization transition

TL;DR

This paper investigates how eigenstates in a disordered quantum system hybridize across all length scales at the many-body localization transition, revealing a diverging length scale and the formation of resonating states.

Contribution

It introduces a detailed analysis of eigenstate hybridization probability as a function of range and disorder, uncovering a diverging length scale at the MBL transition and the presence of resonating cat states.

Findings

Hybridization probability decays exponentially with range in the MBL phase.

A length scale η(W) diverges at the critical disorder strength W_c.

Range invariance at the transition indicates resonating cat states at all scales.

Abstract

An interacting quantum system can transition from an ergodic to a many-body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian. Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in the MBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstates of the disordered Heisenberg model as the disorder strength, $W$, is varied adiabatically. We focus on the probability that two `colliding' eigenstates hybridize as a function of both the range $R$ at which they differ as well as the strength of their hybridization. We find, in the MBL phase, that the probability of a colliding eigenstate hybridizing strongly at range $R$ decays as $Pr(R)\propto \exp [-R/\eta]$, with a length scale $\eta(W) = 1 / (B \log(W / W_c) )$ which diverges at the critical disorder strength $W_c$. This leads to range-invariance at the transition, suggesting the formation of resonating cat states at all ranges. This range invariance does not survive to the ergodic phase, where hybridization is exponentially more likely at large range, a fact that can be understood with simple combinatorial arguments. In fact, compensating for these combinatorial effects allows us to define an additional correlation length $\xi$ in the MBL phase which is in excellent agreement with previous works and which takes the critical value $1 / \log(2)$ at the transition, found in previous works to destabilize the MBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-level collisions of eigenstates close in energy.