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arXiv:2001.08696 · Physics Letters B

On effective theory of superfluid phonons

Lasha Berezhiani

Published Apr 27, 2020• 11 citations
arXivHTMLPDF

Abstract

It is shown that the standard effective field theory of non-relativistic superfluids, which includes only the leading order action in derivative expansion, is in general inadequate for studying supersonic processes. Namely, for a proper treatment of the latter the phonons with momenta greater than mcs are relevant at times (with m being the mass of the superfluid constituent and cs denoting the sound-speed), which lie beyond the regime of validity of the naive effective theory mentioned above. This is demonstrated on a concrete example by recovering the known expression for the friction force as a process of radiating phonons by a supersonic perturber. In particular, when the impurity is heavier than superfluid constituents, it is indeed necessary to extend the non-relativistic effective theory of superfluids to include high-derivative corrections. For the light perturber, however, there is a window of velocities for which high-derivative corrections are irrelevant.

DOI
10.1016/j.physletb.2020.135451

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It is shown that the standard effective field theory of non-relativistic superfluids, which includes only the leading order action in derivative expansion, is in general inadequate for studying supersonic processes. Namely, for a proper treatment of the latter the phonons with momenta greater than mcs are relevant at times (with m being the mass of the superfluid constituent and cs denoting the sound-speed), which lie beyond the regime of validity of the naive effective theory mentioned above. This is demonstrated on a concrete example by recovering the known expression for the friction force as a process of radiating phonons by a supersonic perturber. In particular, when the impurity is heavier than superfluid constituents, it is indeed necessary to extend the non-relativistic effective theory of superfluids to include high-derivative corrections. For the light perturber, however, there is a window of velocities for which high-derivative corrections are irrelevant.
We study the recently proposed effective-field theory for the phonon of an arbitrary nonrelativistic superfluid. After computing the one-loop phonon self-energy, we obtain the low-temperature TTT contributions to the phonon dispersion law at low momentum and see that the real part of those can be parametrized as a thermal correction to the phonon velocity. Because the phonons are the quanta of the sound waves, at low momentum their velocity should agree with the speed of sound. We find that our results match at order T4lnT{T}^{4}\mathrm{ln}TT4lnT with those predicted by Andreev and Khalatnikov for the speed of sound, derived from the superfluid hydrodynamical equations and the phonon kinetic theory. We get also higher-order corrections of order T4{T}^{4}T4, which are not reproduced pushing naively the kinetic theory computation. Finally, as an application, we consider the cold Fermi gas in the unitarity limit and find a universal expression for the low-TTT relative correction to the speed of sound for these systems.
We develop a unified effective field theory framework for phonon dynamics in partially coherent quantum fluids, explicitly incorporating the role of a coherence parameter that encodes the fraction of the superfluid component. Starting from a microscopic description and systematically deriving the low-energy effective action, we show that coherence modifies the kinetic structure of the theory, leading to a renormalized sound speed and an emergent effective geometry governing phonon propagation. A key result of this work is the identification of a coherence dependent effective wave operator, from which the physically relevant metric is reconstructed. This resolves ambiguities associated with conformal rescalings and establishes a direct link between microscopic coherence and emergent spacetime structure. At the nonlinear level, we derive a temperature and coherence dependent effective mass for phonon excitations. Importantly, this mass does not correspond to a gap in the dispersion relation, but rather to an inertial response arising from thermodynamic gradients. We show that this effect is strongly enhanced near the superfluid transition and depends sensitively on the thermodynamic path. Order of magnitude estimates for ultracold atomic gases indicate that the effective mass can become comparable to the atomic mass in experimentally accessible regimes. This suggests that coherence-induced inertial effects may be observable through phonon dynamics in inhomogeneous or near critical systems. Our results establish a quantitative bridge between microscopic coherence, effective geometry, and emergent inertial phenomena, opening new avenues for probing collective excitations in quantum fluids.
An alternative approach to the derivation of the force on a vortex based in an adiabatic approximation in the action of the superfluid system is developed. Assuming that the vortex motion is relatively slow compared with the characteristic times involved in the microscopic degrees of freedom, the effect of the superfluid and its excitations is reduced to a gauge potential, and the associated transverse force. The vortex velocity part of the transverse force is found in terms of the thermal expectation of the angular momentum of the fluid around the vortex. The excitations countercirculate the vortex reducing the effective density to that of the superfluid part only. Non-adiabatic contributions appear in this formalism as a non-Abelian gauge potential connecting different microscopic states, in particular the renormalization of the vortex mass is given by the lowest order diagonal correction to the adiabatic theory, and the long wavelength contributions found to depend only on the static structure factor for the fluid.
A strongly interacting Fermi gas, such as that of cold atoms operative near a Feshbach resonance, is difficult to study by perturbative many-body theory to go beyond mean-field approximation. Here I develop an effective field theory for the resonant superfluid based on broken symmetry. The theory retains both fermionic quasiparticles and superfluid phonons, the interaction between them being derived nonperturbatively. The theory converges and can be improved order by order, in a manner governed by a low energy expansion rather than by a coupling constant. I apply the effective theory to calculate the specific heat and discuss the theory with a recent heat capacity experiment.
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 H3H_3H3​ 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 H3H_3H3​, 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.
We construct an effective field theory description of relativistic superfluids with nonzero angular momentum density. At first order in the derivative expansion, the effective action contains a term with a quantized coefficient, which encodes the Berry phase for the angular momentum of the superfluid condensate. From this single term we derive a number of physical effects, including the relativistic Mermin-Ho relation, an anomalous Ettingshausen effect (exchangeable for an anomalous Hall effect), and an anomalous Hall viscosity.