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Journal of the Physical Society of Japan

Superfluid Flow and Local Gauge Invariance

Masakazu Ichiyanagi

Published Mar 15, 1980• 1 citation
Publisher

Abstract

A Boss system has two broken symmetries, namely the gauge symmetry and the Galilei translation symmetry. We discuss the microscopic background of the ansatz that the superfluid flow is the manifestation of the local gauge invariance. To do this we show that the depletion velocity fields are the Goldstone fields due to the breakdown of the Galilei translation symmetry. A connection between a Bose-Einstein condensation and superfluid flow is also discussed.

DOI
10.1143/jpsj.48.793

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A Boss system has two broken symmetries, namely the gauge symmetry and the Galilei translation symmetry. We discuss the microscopic background of the ansatz that the superfluid flow is the manifestation of the local gauge invariance. To do this we show that the depletion velocity fields are the Goldstone fields due to the breakdown of the Galilei translation symmetry. A connection between a Bose-Einstein condensation and superfluid flow is also discussed.
The Bose-Einstein condensates of alkali atomic gases are spinor fields with local ``spin-gauge'' symmetry. This symmetry is manifested by a superfluid velocity us{\mathbf{u}}_{s}us​ (or gauge field) generated by the Berry phase of the spin field. In ``static'' traps, us{\mathbf{u}}_{s}us​ splits the degeneracy of the harmonic energy levels, breaks the inversion symmetry of the vortex nucleation frequency ${\mathbf{\ensuremath{\Omega}}}_{c1}$, and can lead to vortex ground states.
The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
The interplay between spontaneously broken gauge symmetries and Bose–Einstein condensation has long been controversially discussed in science, since the equations of motion are invariant under phase transformations. Within the present model, it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose–Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
This paper represents the full version of a paper published earlier in Physica A [246 (1997), 275]. The present paper includes argumentation, proofs and details omitted in the shortened version. The papers are a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose-Einstein condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the papers superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The equations obtained are solved in two simple cases. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well.
We discuss the collective excitations in a spatially inhomogeneous (cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature (T≪TλT \ll T_λT≪Tλ​). The main result is the dispersion relation for a (first) sound wave that is obtained by describing the perturbation as a Lagrangian coordinate. The dispersion curve is in good agreement with the Bogoliubov phonon spectrum ω=ckω=c kω=ck, where k=kθ=m/rk=k_θ=m/rk=kθ​=m/r, the wave number and c=[4πaρℏ2]1/2/Mc=[4 πa ρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M, the speed of first sound. Based on Bohm's quantum theory, a spontaneously broken gauge symmetry in a quantum fluid is discussed in terms of the quantum fluctuation-dissipation, from which it is shown that the symmetry breaking takes place at the free surface of BEC in an external field.