Critical properties of a dilute gas of vortex rings in three dimensions and the<i>λ</i>transition in liquid helium

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&lt;\ensuremath{\nu}&lt;0.53, 0.94&lt;\ensuremath{\alpha}&lt;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.