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arXiv:2510.12824 · Proceedings of the National Academy of Sciences

Vacuum tunneling of vortices in two-dimensional 4 He superfluid films

Michael J. Desrochers, D. J. J. Marchand, Philip C. E. Stamp

Published Sep 2, 2025• 1 citation
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Abstract

At low temperature T we expect vacuum tunneling processes to occur in superfluid 4 He films. We distinguish between extrinsic processes, in which single vortices nucleate by tunneling off boundaries in the system, and intrinsic processes, in which vortex/anti-vortex pairs nucleate far from boundaries. It is crucial to incorporate the varying effective mass of the vortex in tunneling calculations. The intrinsic processes are the superfluid analogue of the Schwinger mechanism in quantum field theory; here they appear as a quantum phase transition at T = 0, driven by an external supercurrent. We calculate the tunneling rate for these processes, and describe a means of testing the predictions using a specific “vortex counting” experiment.

DOI
10.1073/pnas.2421273122

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At low temperature T we expect vacuum tunneling processes to occur in superfluid 4 He films. We distinguish between extrinsic processes, in which single vortices nucleate by tunneling off boundaries in the system, and intrinsic processes, in which vortex/anti-vortex pairs nucleate far from boundaries. It is crucial to incorporate the varying effective mass of the vortex in tunneling calculations. The intrinsic processes are the superfluid analogue of the Schwinger mechanism in quantum field theory; here they appear as a quantum phase transition at T = 0, driven by an external supercurrent. We calculate the tunneling rate for these processes, and describe a means of testing the predictions using a specific “vortex counting” experiment.
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.
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.
The tunneling rate ${t}_{v}∕\ensuremath{\hbar}$ of a vortex between two pinning sites (of strength V‾\overline{V}V separated by ddd) is computed using the Bogoliubov expansion of vortex wave-functions overlap. For BCS vortices, tunneling is suppressed beyond a few Fermi wavelengths. For Bose condensates, ${t}_{v}=\overline{V}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}{n}_{s}{d}^{2}∕2)$, where ns{n}_{s}ns​ is the boson density. The analogy between vortex hopping in a superconducting film and two-dimensional electrons in a perpendicular magnetic field is exploited. We derive the variable range hopping temperature, below which vortex tunneling contributes to magnetoresistance. Using the ``quantum Hall insulator'' analogy we argue that the Hall conductivity (rather than the inverse Hall resistivity) measures the effective carrier density in domains of mobile vortices. Details of vortex wave functions and overlap calculations, and a general derivation of the Magnus coefficient for any wave function on the sphere, are provided in appendixes.
We address the collision of two superfluid 4He droplets at non-zero initial relative velocities and impact parameters within the framework of liquid 4He time-dependent density functional theory at zero temperature. Despite the small size of these droplets (1000 He atoms in the merged droplet) imposed by computational limitations, we have found that quantized vortices may be readily nucleated for reasonable collision parameters. At variance with head-on collisions, where only vortex rings are produced, collisions with a non-zero impact parameter produce linear vortices that are nucleated at indentations appearing on the surface of the deformed merged droplet. Whereas for equal-size droplets, vortices are produced in pairs, an odd number of vortices can appear when the colliding droplet sizes are different. In all cases, vortices coexist with surface capillary waves. The possibility for collisions to be at the origin of vortex nucleation in experiments involving very large droplets is discussed. An additional surprising result is the observation of the drops coalescence even for grazing and distal collisions at relative velocities as high as 80 and 40 m/s, respectively, induced by the long-range van der Waals attraction between the droplets.
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.
A theory of superfluidity in two-dimensional Bose systems is presented. In contrast to the classical theory of Nelson and Kosterlitz, the excitation (rotons, ripplons, etc.) are treated quantum mechanically from the outset, resulting in an explicitly temperature-dependent vortex pair coupling proportional to the total mass density minus the normal mass density of the excitations. An application to thick 4He^{4}\mathrm{He}4He films is presented, and the weakly interacting Bose system is discussed explicitly.