Sound-induced vortex interactions in a zero temperature two-dimensional superfluid

Recent experimental work\footnote{T. W. Neely \textit{et al}., Physical Review Letters 111 235301 (2013).}\footnote{W. J. Kwon \textit{et al}., arXiv:1403.4658.} on two-dimensional turbulent superfluids have increased the importance of resolving fundamental theoretical questions about the nature of superfluid turbulence in two dimensions. Crucial to this task is a proper understanding of the effective dynamics of vortices in the superfluid. We present a systematic calculation of the effective action of $N > 1$ superfluid vortices, assuming that the underlying continuum action is the Gross-Pitaevskii action. Our calculation is valid at next-to-leading order in the ratio of the vortex core size to intervortex spacing, and so takes into account the leading-order dressing of superfluid vortices by sound. We are able to exactly determine the action for a pair of vortices and we find that it demonstrates no instability to annihilation. This is suggestive that the inverse cascade picture of classical turbulence is qualitatively correct for a turbulent zero temperature superfluid.