Pre-relaxation in weakly interacting models

TL;DR

This paper investigates pre-relaxation phenomena in weakly interacting models near integrable points, revealing persistent oscillations and quasi-stationary states through mean-field analysis and explicit examples like the XYZ spin chain.

Contribution

It introduces a mean-field approach to analyze pre-relaxation in weakly interacting, near-integrable models, highlighting persistent oscillations and the role of hidden symmetries.

Findings

Identification of a time window where perturbations influence dynamics

Observation of persistent oscillatory behaviour in weakly interacting models

Connection between nonlocal toy models and mean-field dynamics

Abstract

We consider time evolution in models close to integrable points with hidden symmetries that generate infinitely many local conservation laws that do not commute with one another. The system is expected to (locally) relax to a thermal ensemble if integrability is broken, or to a so-called generalised Gibbs ensemble if unbroken. In some circumstances expectation values exhibit quasi-stationary behaviour long before their typical relaxation time. For integrability-breaking perturbations, these are also called pre-thermalisation plateaux, and emerge e.g. in the strong coupling limit of the Bose-Hubbard model. As a result of the hidden symmetries, quasi-stationarity appears also in integrable models, for example in the Ising limit of the XXZ model. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-$\frac{1}{2}$ chain with additional perturbations that break integrability. One of the most intriguing results of the analysis is the appearance of persistent oscillatory behaviour. To unravel its origin, we study in detail a toy model: the transverse-field Ising chain with an additional nonlocal interaction proportional to the square of the transverse spin per unit length [Phys. Rev. Lett. 111, 197203 (2013)]. Despite being nonlocal, this belongs to a class of models that emerge as intermediate steps of the mean-field mapping and shares many dynamical properties with the weakly interacting models under consideration.