Ergodicity breaking in an incommensurate system observed by OTOCs and Loschmidt Echoes: From quantum diffusion to sub-diffusion

TL;DR

This paper introduces a new observable called ZOGE, derived from OTOCs and Loschmidt Echoes, to study ergodicity breaking and the transition from quantum diffusion to sub-diffusion in disordered spin systems.

Contribution

It proposes ZOGE as a novel measure for analyzing many-body localization and ergodicity breaking using out-of-time-ordered correlators in a disordered spin chain.

Findings

ZOGE coincides with the inverse participation ratio in non-interacting cases.

The system exhibits diffusive behavior near the critical region.

Weak disorder leads to quantum diffusion, while strong disorder causes sub-diffusion.

Abstract

The metal-insulator transition (MIT), which includes Anderson localization and Mott insulators as extreme regimes, has received renewed interest as the many-body effects often constitute a limitation for the handling of quantum interference. This resulted in the field dubbed many-body localization (MBL), intensively studied theoretically and experimentally as understanding the appearance of equilibration and thermalization becomes relevant in dealing with finite systems. Here, we propose a new observable to study this transition in a spin chain under the ``disorder'' of a Harper-Hofstadter-Aubry-Andr\'e on-site potential. This quantity, which we call zeroth-order gradient entanglement (ZOGE) is extracted from the fundamental Fourier mode of a family of out-of-time-ordered correlators (OTOCs). These are just Loschmidt Echoes, where a field gradient is applied before the time reversal. In the absence of many-body interactions, the ZOGE coincides with the inverse participation ratio of a Fermionic wave function. By adding an Ising interaction to an XY Hamiltonian, one can explore the MBL phase diagram of the system. Close to the critical region, the excitation dynamics is consistent with a diffusion law. However, for weak disorder, quantum diffusion prevails while for strong disorder the excitation dynamics is sub-diffusive.