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.
Highlights
The composite nature of an operator is a necessary condition for its effective thermalisation, since only does it couple to a thermal bath, indicated by a temperature dependence of its response functions
In odd dimensions we find a link to resurgence theory by noting that exponential relaxation is analogous to nonperturbative corrections to an asymptotic perturbation expansion
We demonstrate an operator non-thermalisation condition which excludes this kind of partial thermalisation, and state a converse partial operator thermalisation hypothesis
Summary
Sabella-Garnier et al formulated the operator thermalisation hypothesis in [13] and considered the thermalisation properties of operator correlation functions in a fixed background rather than operator expectation values in the presence of assumptions on the energy spectrum, as done by the eigenstate thermalisation hypothesis, ETH [1, 2] They discuss thermalisation in terms of an exponentially fast return to equilibrium of operator expectation values in response to a perturbation by the operator in question. While pure exponential decay occurs in free systems in contrast to naive expectations, it is generally masked by leading power law decay for d > 2, as well as multiplied by inverse powers of time The new definitions are important for consistency with the examples we discuss, but the idea is approximately the same
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