Constraints on Superfluid Hydrodynamics from Equilibrium Partition Functions

TL;DR

This work derives and cross-validates equality-type constraints on first-order transport in relativistic s-wave superfluids by requiring consistency between an equilibrium partition function generated by a Goldstone-mode action and the local second law of thermodynamics. By decomposing first-order corrections into parity-even and parity-odd sectors and employing both partition-function and entropy-current formalisms, the authors show that the nondissipative transport coefficients are fixed in terms of a small set of Goldstone-action functions, with CPT invariance further restricting allowable terms. The results demonstrate exact agreement between the two independent constraint methods, reinforcing the conjecture that equilibrium partition functions capture all equality-type hydrodynamic constraints to first order (and likely beyond). The paper also clarifies how field redefinitions of the Goldstone field and the choice of frame influence the mapping between thermodynamic data and transport coefficients, and it discusses implications for broader frameworks such as holographic fluid-gravity dualities and near-critical hydrodynamics.

Abstract

Following up on recent work in the context of ordinary fluids, we study the equilibrium partition function of a 3+1 dimensional superfluid on an arbitrary stationary background spacetime, and with arbitrary stationary background gauge fields, in the long wavelength expansion. We argue that this partition function is generated by a 3 dimensional Euclidean effective action for the massless Goldstone field. We parameterize the general form of this action at first order in the derivative expansion. We demonstrate that the constitutive relations of relativistic superfluid hydrodynamics are significantly constrained by the requirement of consistency with such an effective action. At first order in the derivative expansion we demonstrate that the resultant constraints on constitutive relations coincide precisely with the equalities between hydrodynamical transport coefficients recently derived from the second law of thermodynamics.