Can we study the many-body localisation transition?

TL;DR

This paper investigates the length and timescales relevant to the many-body localization transition, highlighting limitations of current numerical and experimental approaches in studying this phenomenon.

Contribution

It provides a detailed analysis of finite-size and timescale effects near the MBL transition, emphasizing the challenges in observing the transition with current methods.

Findings

Finite-size systems appear localized, but larger systems develop resonances restoring ergodicity.

Transport time increases rapidly with disorder, exceeding the Heisenberg time in small systems.

Current numerical and experimental setups are insufficient to fully explore the MBL transition.

Abstract

We present a detailed analysis of the length- and timescales needed to approach the critical region of MBL from the delocalised phase, studying both eigenstates and the time evolution of an initial state. For the eigenstates we show that in the delocalised region there is a single length, which is a function of disorder strength, controlling the finite-size flow. Small systems look localised, and only for larger systems do resonances develop which restore ergodicity in the form of the eigenstate thermalisation hypothesis. For the transport properties, we study the time necessary to transport a single spin across a domain wall, showing how this grows quickly with increasing disorder, and compare it with the Heisenberg time. For a sufficiently large system the Heisenberg time is always larger than the transport time, but for a smaller system this is not necessarily the case. We conclude that the properties of the MBL transition cannot be explored using the system sizes or times available to current numerical and experimental studies.