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Journal of the Physical Society of Japan

Superfluid Flow and Analogy to Dirac's Monopole

Masakazu Ichiyanagi

Published Sep 15, 1979• 4 citations
Publisher

Abstract

Taking the depletion effect into consideration in terms of the depletion velocity, superfluid flow in an annulus is discussed. It is shown that in annular regions superfluid flow is the gauge-invariant manifestation of a nonintegrable phase factor. The circulation is defined in terms of the depletion velocity which is essentially rotational. Arguments akin to Dirac's theory of magnetic monopole are used to obtain quantum mechanical vortex lines which have their ends in fluid.

DOI
10.1143/jpsj.47.755

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Taking the depletion effect into consideration in terms of the depletion velocity, superfluid flow in an annulus is discussed. It is shown that in annular regions superfluid flow is the gauge-invariant manifestation of a nonintegrable phase factor. The circulation is defined in terms of the depletion velocity which is essentially rotational. Arguments akin to Dirac's theory of magnetic monopole are used to obtain quantum mechanical vortex lines which have their ends in fluid.
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By identifying the Schrödinger equation with the hydrodynamic equations in superfluid 3{^3}3He, the effective potential is introduced in the Schrödinger equation to solve the quantum pressure in steady state. The pure gauge velocity solutions of hydrodynamic equations provide an analogous Wu-Yang monopole potential. The two cases of velocity in superfluid are equivalent to the two regions of Wu-Yang monopole potential. Due to the compressibility of superfluid, the physical models are limited, such as hard core and harmonic oscillator. It is important that the constraint condition of ∇ρ⋅v=0\nablaρ\cdot \textbf{v}=0∇ρ⋅v=0 plays a key role, which determines that the integral quantum numbers are selected in monopole harmonics and shows the special analogous Wu-Yang monopole. The results provide a new possibility for the simulation of Wu-Yang monopole.
This chapter focuses on "steady" superfluid flow, in which the superfluid velocity, vs, satisfies the equation div vs=0. Such flow can, in principle, occur only in multiply-connected systems. However, it is possible to achieve an equivalent situation in the bulk of the liquid by setting up vortex lines. The major new feature brought in by superfluidity, and a most important one indeed, is the quantization of vortex lines, arising as a consequence of the quantization of circulation. The square terms in correspond to the energy of the single vortex lines, while the cross term describes an interaction energy between two vortex lines. In the absence of vortex lines, the superfluid and normal components of a Bose liquid are completely decoupled. The velocity pattern of the vortex gives rise to a position-dependent energy, which may be used to calculate the scattering. The dynamics of superfluid flow is thus appreciably modified by the presence of vortex lines.
Magnetic monopoles lie at the crossroads of gauge fields, topology, geometric phases, and charge quantization, yet they remain elusive as fundamental particles. Here we show that an emergent Dirac-monopole framework arises naturally from the dynamics of massive quantum vortices in a spherical superfluid shell. Their dynamics is formally equivalent to that of interacting charged particles constrained to a sphere in the field of a magnetic monopole. The monopole charge is fixed by the superfluid density and automatically satisfies Dirac's quantization condition. The emergent monopole description predicts cyclotron-like vortex motion, in quantitative agreement with Gross--Pitaevskii simulations. We further show that topological frustration induced by two like-charged polar vortices gives rise to the formation of an equatorial vortex necklace, a configuration reminiscent of the polygonal cyclone clusters observed around Jupiter's poles, before its subsequent breakup through a Kelvin--Helmholtz-like instability. Within this framework, the equatorial vortex necklace emerges as the quantized realization of the effective electrostatic line charge that restores global charge neutrality on the sphere. Our results establish spherical superfluids as a versatile platform for realizing and exploring fundamental aspects of Dirac-monopole physics.
An alternative approach to the derivation of the force on a vortex based in an adiabatic approximation in the action of the superfluid system is developed. Assuming that the vortex motion is relatively slow compared with the characteristic times involved in the microscopic degrees of freedom, the effect of the superfluid and its excitations is reduced to a gauge potential, and the associated transverse force. The vortex velocity part of the transverse force is found in terms of the thermal expectation of the angular momentum of the fluid around the vortex. The excitations countercirculate the vortex reducing the effective density to that of the superfluid part only. Non-adiabatic contributions appear in this formalism as a non-Abelian gauge potential connecting different microscopic states, in particular the renormalization of the vortex mass is given by the lowest order diagonal correction to the adiabatic theory, and the long wavelength contributions found to depend only on the static structure factor for the fluid.