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arXiv:cond-mat/0509350

Atomic Theory of the Two-fluid Model: Broken Gauge Symmetry in Bose-Einstein condensation

S. J. Han

Published Sep 14, 2005• 0 citations
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Abstract

We discuss the collective excitations in a spatially inhomogeneous (cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature ($T \ll T_λ$). 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 $ω=c k$, where $k=k_θ=m/r$, the wave number and $c=[4 πa ρ\hbar^{2}]^{1/2}/M$, 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.

DOI
10.48550/arxiv.cond-mat/0509350

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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.
A theory of collective excitations in Bose-Einstein condensation in a trap is developed based on the quantum Hamilton-Jacobi equation of Bohm and the phase coherence along with the idea of off-diagonal long range order of Penrose and Onsager. First, we show that a free surface behaves like a normal fluid - a breakdown of superfluidity. Second, inside the free surface it is shown that the spectrum of phonons is of the form ω=ckω=ckω=ck scaled with the external potential, where the speed of (first) sound, c=[4πaρℏ2]1/2/Mc=[4πaρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M and kkk is the wave number. Third, in the limit a→0a\to 0a→0, the hard spheres in the Bose-Einstein condensation collapse to a close-packed classical lattice with the zero-point vibrational motion about fixed points.
This paper presents an elementary theory of the phase-coherent collective excitations in both He II in the gravitational field and atomic Bose-Einstein condensation in a trap. The theory is based on the concept of off-diagonal long-range order by Penrose and Onsager and the quantum theory of Bohm, with emphasis on the broken symmetry in a Bose-Einstein gas with repulsive interactions. It is shown that a spontaneously broken symmetry that accompanies a phonon (Nambu-Goldstone mode) takes place at the surface layer of an inhomogeneous Bose system in the presence of an external field. The spontaneously broken symmetry in a Bose system is described and is shown to manifest itself in both He II and the Bose-Einstein condensation - a shell-like structure of Bose-Einstein condensation in a trap. The broken symmetry gives a coherent explanation for a number of long standing puzzles in He II.
This paper presents an elementary theory of the phase-coherent collective excitations in both He II in the gravitational field and atomic Bose-Einstein condensation in a trap. The theory is based on the concept of off-diagonal long-range order by Penrose and Onsager and the quantum theory of Bohm, with emphasis on the broken symmetry in a Bose-Einstein gas with repulsive interactions. It is shown that a spontaneously broken symmetry that accompanies a phonon (Nambu-Goldstone mode) takes place at the surface layer of an inhomogeneous Bose system in the presence of an external field. The spontaneously broken symmetry in a Bose system is described and is shown to manifest itself in both He II and the Bose-Einstein condensation - a shell-like structure of Bose-Einstein condensation in a trap. The broken symmetry gives a coherent explanation for a number of long standing puzzles in He II.
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.
The fundamental problem is analyzed, the relation between Bose-Einstein condensation and spontaneous gauge symmetry breaking. This relation is largerly misunderstood in physics community. Numerous articles and books contain the statement that, though gauge symmetry breaking helps for describing Bose-Einstein condensation, but the latter, in principle, does not require any symmetry breaking. This, however, is not correct. The analysis is based on the known mathematical theorems. But in order not to overcomplicate the presentation and to make it accessible to all readers, technical details are often omitted here. The emphasis is made on the following basic general facts: Spontaneous breaking of gauge symmetry is the necessary and sufficient condition for Bose-Einstein condensation. Condensate fluctuations, in thermodynamic limit, are negligible. Their catastrophic behavior can arise only as a result of incorrect calculations, when a Bose-condensed system is described without gauge symmetry breaking. It is crucially important to employ the representative statistical ensembles equipped with all conditions that are necessary for a unique and mathematically correct description of the given statistical system. Only then one is able to develop a self-consistent theory, free of paradoxes.