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arXiv:cond-mat/9504100 · International Journal of Modern Physics B

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

Achilles D. Speliotopoulos

Published Apr 20, 1997• 2 citations
arXivHTMLPDF

Abstract

A quantum mechanical description of vortices in two-dimensional superfluid 4 He films is presented. Single vortex creation and annihilation operators are defined and wavefunctions for these states are explicitly constructed.
DOI
10.1142/s0217979297000654

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We present a general method for calculating vortex creation and annihilation rates at arbitrary boundaries in two-dimensional superfluids and give explicit results for the cases of parallel film edges and cylinders. We explain why cylindrical boundaries may provide a reasonable description of vortex shedding and pinning in regions of a superfluid film where the superfluid density is reduced locally to zero, for example, by substrate imperfections. The relation of our work to other recent contributions is discussed, and the current status of the comparison between theoretical predictions and experimental results for the decay of persistent currents in superfluid films is discussed and clarified.
In addition to the acoustic (phonon) excitations described above, specific excitations called ‘quantum vortices’ may exist in superfluid Bose systems. Attempts to incorporate the vortex excitations have already been made in the original version of Landau's theory (Landau, 1941). The existence of a periodical lattice of quantum vortices in a superconductor located in a magnetic field was conjectured by Abrikosov (1952, 1957). The idea of quantum vortices occurring in rotating helium below the λ-point was introduced independently by Onsager (1949) and Feynman (1955). In this section we present the description of quantum vortices in the functional integral formalism. We will also discuss the role of quantum vortices in the phase transition for the superfluid to the normal state. Our starting point is the functional integration scheme over fast and slow variables. We will apply this formalism mainly to two-dimensional Bose systems and will try to take into account vortex-like configurations of the Bose field. We thus come to the conclusion that the system of vortices is equivalent to the system of charged particles interacting via a new field which play the role of electromagnetic field. At small temperatures there are only pairs with opposite signs, bounded by the long-range logarithm potential. It turns out that the phase transition in the two-dimensional Bose system is nothing but the dissociation of bounded pairs. A similar approach to the three-dimensional systems leads to the conclusion that phase transition is accompanied here by creation of long vortex filaments.
Time-dependent processes accompanied by vortex creations and annihilations are investigated in terms of the eigenstates in conjugate spaces of Gel'fand triplets in 2-dimensions. Creations and annihilations of vortices are described by the insertions of unstable eigenstates with complex-energy eigenvalues into stable states written by the superposition of eigenstates with zero-energy eigenvalues. Some concrete examples are presented in terms of the eigenfunctions of the 2-dimensional parabolic potential barrier, i.e., −mγ2(x2+y2)/2-m γ^2 (x^2+y^2)/2−mγ2(x2+y2)/2. We show that the processes accompanied by vortex creations and annihilations can be analyzed in terms of the eigenfunctions in the conjugate spaces of Gel'fand triplets. Throughout these examinations we point out three interesting properties of the zero-energy flows. (i) Mechanisms using the zero-energy flows are absolutely economical from the viewpoint of energy consumption. (ii) An enormous amount of informations can be discriminated in terms of the infinite variety of the zero-energy flows. (iii) The zero-energy flow patterns are absolutely stable in any disturbance by inserting arbitrary decaying flows with complex-energy eigenvalues.
A family of exact analytical solutions of vortices in quantum fluid governed by a two-dimensional time-dependent Schrödinger equation is presented, which describes different kinds of vortex structures. The dynamics of different vortex clusters, such as the single vortex, vortex pair, vortex dipole and vortex trimer in a two-dimensional quantum fluid are analytically studied based on these exact solutions. The time evolutions of the wave of such vortices are demonstrated, and the orbits of motion of singular points in the vortices are also explored. The interactions of vortices in many-vortex clusters are discussed. A repulsive interaction between vortices with the same topological charge, and inter-annihilation and inter-creation of vortices with opposite topological charge, are shown.
Quantum vortices in superfluids have been an important research area for many decades. Naturally, research on this topic has focused on two and three-dimensional superfluids, in which vortex cores form points and lines, respectively. Very recently, however, there has been growing interest in the quantum simulation of systems with four spatial dimensions; this raises the question of how vortices would behave in a higher-dimensional superfluid. In this paper, we begin to establish the phenomenology of vortices in 4D superfluids under rotation, where the vortex core can form a plane. In 4D, the most generic type of rotation is a "double rotation" with two angles (or frequencies). We show, by solving the Gross-Pitaesvkii equation, that the simplest case of equal-frequency double rotation can stabilise a pair of vortex planes intersecting at a point. This opens up a wide number of future research topics, including unequal-frequency double rotations; the stability and reconnection dynamics of intersecting vortex surfaces; and the possibility of closed vortex surfaces.
The phase change of the wave function of a superfluid film as a vortex moves around a closed path counts the number of superfluid particles enclosed by that path. This result is used to investigate whether such vortices obey ``fractional statistics.'' We conclude that this is not the case in compressible superfluids such as 4He^{4}\mathrm{He}4He films.