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Physical review. B, Condensed matter

Symmetry breaking, Ward identities, and the two-fluid model

E. Talbot, Adam Griffin

Published Apr 1, 1984• 17 citations
Publisher

Abstract

We derive several important features of a Bose-condensed system, based on a diagrammatic analysis at finite temperatures. Firstly, the symmetry-breaking term necessary to describe Bose condensation is shown to give a rigorous basis for the fictitious gap argument of Gavoret and Nozi\`eres (1964). This puts their well-known results on a firm footing. Secondly, in the presence of a moving condensate with velocity ${v}_{S}$, the current carried by the noncondensate atoms is shown to be $({\ensuremath{\rho}}_{S}\ensuremath{-}{\mathrm{mn}}_{o}){v}_{S}$. The results of this paper are direct consequences of various Ward identities which have been recently discussed for Bose-condensed systems.

DOI
10.1103/physrevb.29.3952

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We derive several important features of a Bose-condensed system, based on a diagrammatic analysis at finite temperatures. Firstly, the symmetry-breaking term necessary to describe Bose condensation is shown to give a rigorous basis for the fictitious gap argument of Gavoret and Nozi`eres (1964). This puts their well-known results on a firm footing. Secondly, in the presence of a moving condensate with velocity vS{v}_{S}vS​, the current carried by the noncondensate atoms is shown to be (\ensuremathρS\ensuremath−mno)vS({\ensuremath{\rho}}_{S}\ensuremath{-}{\mathrm{mn}}_{o}){v}_{S}(\ensuremathρS​\ensuremath−mno​)vS​. The results of this paper are direct consequences of various Ward identities which have been recently discussed for Bose-condensed systems.
A gauge-symmetry-breaking parameter associated with the fluctuation of the zero-mode particles renders a source term to the continuity equation of interacting Bose systems. As a consequence, additional terms appear in the generalized Ward identities, which modify in an essential way the relation between the superfluid and condensate densities. An energy gap results as a function of the symmetry-breaking parameter and interatomic potential. The general results are worked out in the shielded potential approximation, and an equation for the condensate fraction n0{n}_{0}n0​(T) is obtained in closed form. At absolute zero the condensate fractions in dilute Bose systems and superfluid 4He^{4}\mathrm{He}4He are found to be n0{n}_{0}n0​=0.438 and n0{n}_{0}n0​=0.126, respectively.
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.
The usual symmetry-breaking procedures for Bose condensed systems, namely, the Bogoliubov prescription, the symmetry-breaking term added to the Hamiltonian, and the canonical shift transformation are unified into a single formalism. By taking into account the condensate reservoir as a source and sink of excited particles, exact Ward identities are solved in the shielded-potential approximation. A relationship between the condensate density n0{\mathit{n}}_{0}n0​ and the superfluid density nS{\mathit{n}}_{\mathit{S}}nS​ is obtained in closed form. It is shown that the Bogoliubov prescription yields n0{\mathit{n}}_{0}n0​\ensuremath{\simeq}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density, U0{\mathit{U}}_{0}U0​ the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has n0{\mathit{n}}_{0}n0​\ensuremath{\ll}nS{\mathit{n}}_{\mathit{S}}nS​ and nU0{\mathit{nU}}_{0}nU0​\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid 4He^{4}\mathrm{He}4He at saturated vapor pressure, gives excellent agreement between theory and experiment, without any adjustable parameter. The condensate density turns out to be strongly dependent on pressure as observed experimentally. The formalism provides in a natural way a consistent description of Bose systems in arbitrary D-dimensional space.
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.
Major issues arising in the theory of Bose–Einstein condensation are reviewed. These issues, although being principally important, are very often misunderstood, which results in wrong conclusions. The basic point is global gauge symmetry breaking that is a necessary and sufficient condition for Bose–Einstein condensation. Paying no attention to this basic point is a common fallacy leading to a number of confusions. For instance, the attempt of describing Bose condensation without gauge symmetry breaking produces the so-called “grand canonical catastrophe” that actually does not exist in the correct description of Bose condensation accompanied by gauge symmetry breaking. The other common flaw is forgetting to consider the stability of the studied systems. One sometimes accomplishes lengthy calculations and discusses the properties of a system that in reality cannot exist being unstable. In some cases, the seeming instability is caused by the negligence of the simple mathematical reason teaching us that one should not go beyond the approximation applicability. An example of such an artificial instability is related to the appearance of the so-called “thermodynamically anomalous fluctuations” whose arising is due to the use of a second-order approximation for calculating fourth-order terms, in this way distorting the O(2)-class model of a Bose-condensed system to the Gaussian-class model. These and other principal points, important for the correct treatment of Bose-condensed systems, are reviewed, including the resolution of the Hohenberg–Martin dilemma of gapless versus conserving theories for Bose-condensed systems and the problem of statistical ensemble equivalence.
This lecture describes certain properties of interacting Bose gases and superconductors which have recently been considered at Harvard. It is a brief resumé of work by the lecturer, P.C. Hohenberg, A. Fetter, R. Lange, and C. De Dominicis, which will be reported in mathematical detail elsewhere. It concerns attempts to derive from fundamental principles, several aspects of the macroscopic two-fluid model of London, Landau, and Tisza, the Landau phonon spectrum for bosons, and the structured condensate envisaged by Onsager and Feynman. The lecture describes new results which have been obtained concerning: (1) methods for calculating properties of large, but not necessarily infinite, condensed systems at finite temperatures; (2) methods for performing calculations in boson systems consistent with current conservation and sum rules; (3) results obtained by these methods for the phonon spectrum; (4) more accurate results for the less physical single-particle spectrum at finite temperatures; (5) results obtained for the macroscopically structured condensate in rotating systems which agree with the macroscopically inferred results of Feynman and Onsager; (6) results which treat depletion of the condensate as a result of interaction consistently and verify that nonetheless, the superfluid density, as operationally determined, is not correspondingly depleted; (7) methods in which the entropy plays a central role, for casting the macroscopic theory in a form which lays stress on the renormalized excitations of renormalized interactions; (8) the almost identical features of the mathematics for condensed Fermi and Bose systems; (9) a restatement, in this microscopic transcription, of the Feynman-Onsager argument for flux and vorticity quantization.