Operator thermalisation in $d>2$: Huygens or resurgence

TL;DR

This paper explores how operator thermalisation behaves in higher-dimensional free field theories, highlighting the roles of Huygens' principle and resurgence theory in understanding exponential decay of correlations at non-zero temperature.

Contribution

It extends the operator thermalisation hypothesis to large N free fields in dimensions greater than two, incorporating generalised free fields and analyzing the impact of Huygens' principle and resurgence theory.

Findings

Huygens' principle affects thermalisation in even dimensions.

In odd dimensions, resurgence theory provides insights into exponential relaxation.

Support for thermalisation in odd dimensions is found, though not conclusively.

Abstract

Correlation functions of most composite operators decay exponentially with time at non-zero temperature, even in free field theories. This insight was recently codified in an OTH (operator thermalisation hypothesis). We reconsider an early example, with large $N$ free fields subjected to a singlet constraint. This study in dimensions $d>2$ motivates technical modifications of the original OTH to allow for generalised free fields. Furthermore, Huygens' principle, valid for wave equations only in even dimensions, leads to differences in thermalisation. It works straightforwardly when Huygens' principle applies, but thermalisation is more elusive if it does not apply. Instead, in odd dimensions we find a link to resurgence theory by noting that exponential relaxation is analogous to non-perturbative corrections to an asymptotic perturbation expansion. Without applying the power of resurgence technology we still find support for thermalisation in odd dimensions, although these arguments are incomplete.