Long-time memory effects in a localizable central spin problem

TL;DR

This paper investigates the long-time memory effects in a central spin model coupled to a many-body localized bath, revealing power-law decay of the memory kernel and potential exponential growth linked to delocalization phenomena.

Contribution

It introduces a method to extract long-time populations from finite-time data using the Nakajima-Zwanzig memory kernel in localized and ergodic regimes.

Findings

Memory kernel decays as a power law in both regimes

Finite-time data can predict long-time populations

Unbounded exponential growth in the memory kernel observed

Abstract

We study the properties of the Nakajima-Zwanzig memory kernel for a qubit immersed in a many-body localized (i.e., disordered and interacting) bath. We argue that the memory kernel decays as a power law in both the localized and ergodic regimes, and show how this can be leveraged to extract $t\to\infty$ populations for the qubit from finite time ($J t \leq 10^2$) data in the thermalizing phase. This allows us to quantify how the long-time values of the populations approach the expected thermalized state as the bath approaches the thermodynamic limit. This approach should provide a good complement to state-of-the-art numerical methods, for which the long-time dynamics with large baths are impossible to simulate in this phase. Additionally, our numerics on finite baths reveal the possibility for unbounded exponential growth in the memory kernel, a phenomenon rooted in the appearance of exceptional points in the projected Liouvillian governing the reduced dynamics. In small systems amenable to exact numerics, we find that these pathologies may have some correlation with delocalization.