Abstract
The low-energy dynamics of a zero temperature superfluid or of the compressional modes of an ordinary fluid can be described by a simple effective theory for a scalar field — the superfluid ‘phase’. However, when vortex lines are present, to describe all interactions in a local fashion one has to switch to a magnetic-type dual two-form description, which comes with six degrees of freedom (in place of one) and an associated gauge redundancy, and is thus considerably more complicated. Here we show that, in the case of vortex rings and for bulk modes that are much longer than the typical ring size, one can perform a systematic multipole expansion of the effective action and recast it into the simpler scalar field language. In a sense, in the presence of vortex rings the non-single valuedness of the scalar can be hidden inside the rings, and thus out of the reach of the multipole expansion. As an application of our techniques, we compute by standard effective field theory methods the sound emitted by an oscillating vortex ring.
Highlights
Scales in the system — say the radius of curvature of the vortex line, or the typical wavelength of sound waves in the surrounding fluid — one can take in first approximation the zero-thickness limit, and parametrize finite-thickness corrections according to the standard philosophy of effective field theories
Considerable progress has been made in studying their dynamics via effective field theory techniques [4,5,6,7,8]: one can couple systematically the bulk degrees of freedom that parametrize generic fluid flows and long-wavelength perturbations such as sound waves to the embedding coordinates X(t, σ) of a zero-thickness string — the vortex line — with σ being an arbitrary coordinate along the string
We have developed an effective field theory for small vortex rings interacting with long wavelength fluid flows and sound waves, which we organized as a multipole expansion
Summary
From a QFT viewpoint, a superfluid at equilibrium can be defined as a system with a conserved U(1) charge Q in a homogeneous state |ψ such that (i) Q has a nonzero density, and (ii) Q is spontaneously broken:. This is the QFT analog of the statement that the ground state is a Bose-Einstein condensate Which is a solution of the equations of motion following from (2.4), with μ(x) = μ It corresponds to the fluid being at rest, U μ = δ0μ, and with a non-zero charge density, n ≡ p (μ) = 0.
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