Vortex-ring model of the superfluidλtransition

Abstract

An initial model of the superfluid X transition is constructed with use of vortex-ring excitations, as originally suggested by Onsager and Feynman. A real-space renormalization technique generates a screened vortex energy and core size, and gives rise to a transition where rings of infinite diameter are excited as the superfluid density approaches zero at T c . Although the model satisfies the Josephson hyperscaling relation, it is not yet a complete theory: The superfluid density exponent is v=0.53, and does not match the known value v = 0.67. PACS numbers: 67.40.Kh, 67.40.Db, 67.40.Vs The superfluid transition of liquid He has been studied intensively for nearly 50 years. In spite of this, there is still little known about the underlying physical mechanism. Onsager and Feynman l proposed in their original papers that vortex excitations might be responsible for the transition. Since then many authors have reiterated this proposal, and at least a partial listing of these papers is given in the references of Kotsubo and Williams. 2 The key role of vortex-ring excitations, in particular, was emphasized by Popov, 3 Wiegel, 4 Banks, Myerson, and Kogut, 5 and Nelson and Toner. 6 These papers gave a qualitative and physical picture of the transition, but quantitative predictions remained elusive. On the other hand, perturbation-series techniques such as the high-temperature expansion and the 4 -e expansion proved to be successful in the calculation of the critical exponents of the transition. 7 However, these are formal and mathematical procedures that are hard to apply in many situations, and the momentum-space representation that is used makes it difficult to identify the excitations causing the fluctuations. 8 The role of vortices in the superfluid transition has recently been made much more concrete by the numerical results of Kohring, Shrock, and Wills, 9 who used a Monte Carlo simulation of the three-dimensional (3D) XY model to give strong evidence that vortices play a crucial part in the transition. Vortices had previously only been known to be important in two dimensions (2D), as shown theoretically by Kosterlitz and Thouless 10 and as confirmed by experiments in thin helium films.' x The question of how the transition evolves as one goes from 2D to 3D geometries has recently arisen in experiments on helium films adsorbed in packed powders. 2 The results there led to speculation that the nature of the transition did not seem to change much as one varied the film thickness to the point where the powder pores fill completely with helium. The finite-size 2D transition appears to merge smoothly with the finitesize 3D transition, with little sign that the vortices cease to play a role in the 3D case. This conclusion has now been corroborated by the Monte Carlo results.In this Letter the superfluid density is calculated near the transition by consideration of the screening effects of circular vortex rings on an applied superflow. The starting point is to realize that vortex rings are eigenfunctions of the Landa...

Phase transitions and connectivity in three-dimensional vortex equilibria

“…The energy per link rises more slowly with temperature above the transition than below. This qualitative behavior is consistent with the renormalization arguments of Shenoy [130] and Williams [144]. A modification to the Note that the f3 axis is reversed so that low temperature is on the left and high temperature is on the right.…”

Vortex phase transitions in 21/2 dimensions

“…The general formulation of magnetization variables and their gauge invariance can be found in [28]. Earlier work can be found in [1], [2], [29], [30]. Consider the Euler equation for incompressible flow: (5) where U = (UI' U2, U3) is the velocity, p is the pressure, t is the time, and ~ is the differentiation vector.…”

“…(Note that in [11] the authors also map out the f{ < 0 region which, as far as is known, is not dual to a vortex system with T < 0.) A widely believed vortex picture of the transition from a superfluid state to a normal helium state is as follows [1], [2], [27]: As T increases towards T)., the number of vortex excitations increases, and they allay each other's energy through polarization, i.e. large loops arrange smaller loops so as to reduce the energy.…”

“…In the superfiuid case, a three-dimensional analogue of the Kosterlitz-Thouless renormalization analysis of the A transition has been offered by Williams [1] and Shenoy [2]; it relies on Shenoy's ansatz which assumes a polymer-like structure for vortex filament near the critical temperature, and uses a "magnet" representations, see below. (A "polymer", here and below, means "equal probability self-avoiding walk").…”

Liquid 4He: Contributions to First Principles Theory. I. Quantized Vortices and Thermohydrodynamic Properties

“…(4) Williams [142] used an incorrect sign in the term that takes into account the interaction of a vortex ring with an applied superfluid velocity field. It is clear that this is not just a typographical error because that same sign is used in his expression [142] for the polarizability of a large ring, as one will find by working through the derivation of his formula for polarizability.…”