Exact Spectral Statistics in Strongly Localised Circuits

TL;DR

This paper derives exact spectral statistics for a class of strongly localised quantum many-body systems, revealing a cascade of correlation regimes and providing insights into many-body localisation phenomena.

Contribution

It introduces strongly localised quantum circuits and analytically characterizes their spectral statistics, bridging a gap in understanding MBL systems from first principles.

Findings

Spectral correlations exhibit three regimes depending on energy scale.

At small scales, spectral statistics are Poissonian.

The results are expected to generalize to generic MBL systems.

Abstract

Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they do become Poissonian.