On the theory of superfluidity in two dimensions

Abstract

The superfluid phase transition of the general vortex gas, in which the circulations may be any non-zero integer, is studied. When the net circulation of the system is not zero the absence of a superfluid phase is shown. When the net circulation of the vortices vanishes, the presence of off-diagonal long range order is demonstrated and the existence of an order parameter is proposed. The transition temperature for the general vortex gas is shown to be the Kosterlitz-Thouless temperature. An upper bound for the average vortex number density is established for the general vortex gas and an exact expression is derived for the Kosterlitz-Thouless ensemble.

The Landau­Ginzberg Theory for the Two-Dimensional Bose Gas

“…The description of the two-dimensional quantum vortices used in [9,15,16,13] and [17] follows in direct analogy with the description of vortex lines used in three-dimensional superfluids. In this description vortices are characterized by {κ i , z i }, where κ i is the circulation associated with the i th vortex at the position z i (in complex coordinates) on a Riemann surface M. Vortices are formed due to a current flow j z in the fluid.…”

“…In the phenomenological theory given in [17] the superfluid state is marked by the presence of vortices while the normal fluid state does not have vortices. Consequently, ξ = 1 when T > T c and when T < T c , ξ = 0.…”

“…There are effectively no free vortices in the superfluid state in this theory. In theory given in [17], on the other hand, vortices are only present in the superfluid state and the phase transition is due to the annihilation of oppositely charged vortices at the transition temperature. In this case, there are no quantum vortices in the normal fluid state.…”

“…We do so by using the results from analysing the classical equations of motion for the vortices given in [9,15,16,13] and [17] as a motivation for studying sheaf theory. Results from sheaf theory are then combined with an effective-field description of the system to construct the order parameter.…”

Eigenstates of the Vorticity Operator: The Creation and Annihilation of Vortex States in Two Dimensions

“…The purpose of this paper, therefore, is to develope a quantum mechanical description of vortices in superfluid 4 He films. There are two approaches that one can take.…”