Phonon damping in a 2D superfluid: insufficiency of Fermi's golden rule at low temperature

TL;DR

This paper investigates phonon damping in 2D superfluids and finds that Fermi's golden rule underestimates damping rates at low temperatures, unlike in 3D, due to non-perturbative effects.

Contribution

It demonstrates the insufficiency of Fermi's golden rule for phonon damping in 2D superfluids at low temperatures using Green's function and non-perturbative methods.

Findings

Fermi's golden rule underestimates damping by a factor of about three in 2D.

Classical simulations confirm the non-perturbative predictions.

Damping rates differ significantly from 3D superfluids at low temperatures.

Abstract

It is generally accepted that the phonon gas of a superfluid always enters a weak coupling regime at sufficiently low temperatures, whatever the strength of the interactions between the underlying particles (constitutive of the superfluid). Thus, in this limit, we should always be able to calculate the damping rate of thermal phonons by applying Fermi's golden rule to the Hamiltonian $H_3$ of cubic phonon-phonon coupling taken from quantum hydrodynamics, at least in the case of a convex acoustic branch and in the collisionless regime (where the eigenfrequency of the considered phonons remains much greater than the gas thermalization rate). Using the many-body Green's function method, we predict that, unexpectedly, this is not true in two dimensions, contrary to the three-dimensional case. We confirm this prediction with classical phonon-field simulations and a non-perturbative theory in $H_3$, where the fourth order is regularized by hand, giving a complex energy to the virtual phonons of the four-phonon collisional processes. For a weakly interacting fluid and a phonon mode in the long-wavelength limit, we predict a damping rate about three times lower than that of the golden rule.