Abstract
We compute the critical properties of a dilute gas of circular and elliptical vortex rings in a three-dimensional superfluid using a scale-dependent mean-field approximation of the Kosterlitz-Thouless type. A set of scaling differential equations is derived and solved numerically. The system has a second-order phase transition with superfluid density critical exponent \ensuremath{\nu}=0.57 and specific-heat critical exponent \ensuremath{\alpha}=0.64 for circular rings and 0.50<\ensuremath{\nu}<0.53, 0.94<\ensuremath{\alpha}<0.99 for elliptical rings of small eccentricity. The relevance of these results for the understanding of the \ensuremath{\lambda} transition in liquid helium is discussed.
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