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arXiv:2105.14966 · Physics of Particles and Nuclei

Hartree–Fock–Bogolubov Method in the Theory of Bose-Condensed Systems

Vyacheslav I. Yukalov, Elizaveta P. Yukalova

Published Jul 1, 2020• 2 citations
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Abstract

Abstract The Hohenberg–Martin dilemma of conserving versus gapless theories for systems with Bose–Einstein condensate is considered. This dilemma states that, generally, a theory characterizing a system with broken global gauge symmetry, which is necessary for Bose–Einstein condensation, is either conserving, but has a gap in its spectrum, or is gapless, but does not obey conservation laws. In other words, such a system either displays a gapless spectrum, which is necessary for condensate existence, but is not conserving, which implies that it corresponds to an unstable system, or it respects conservation laws, describing a stable system, but the spectrum acquires a gap, which means that the condensate cannot appear. An approach is described, resolving this dilemma, and it is shown to give good quantitative agreement with experimental data. Calculations are accomplished in the Hartree–Fock–Bogolubov approximation.

DOI
10.1134/s1063779620040772

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Abstract The Hohenberg–Martin dilemma of conserving versus gapless theories for systems with Bose–Einstein condensate is considered. This dilemma states that, generally, a theory characterizing a system with broken global gauge symmetry, which is necessary for Bose–Einstein condensation, is either conserving, but has a gap in its spectrum, or is gapless, but does not obey conservation laws. In other words, such a system either displays a gapless spectrum, which is necessary for condensate existence, but is not conserving, which implies that it corresponds to an unstable system, or it respects conservation laws, describing a stable system, but the spectrum acquires a gap, which means that the condensate cannot appear. An approach is described, resolving this dilemma, and it is shown to give good quantitative agreement with experimental data. Calculations are accomplished in the Hartree–Fock–Bogolubov approximation.
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.
The review is devoted to the elucidation of the basic problems arising in the theoretical investigation of systems with Bose-Einstein condensate. Understanding these challenging problems is necessary for the correct description of Bose-condensed systems. The principal problems considered in the review are as follows: (i) What is the relation between Bose-Einstein condensation and global gauge symmetry breaking? (ii) How to resolve the Hohenberg-Martin dilemma of conserving versus gapless theories? (iii) How to describe Bose-condensed systems in strong spatially random potentials? (iv) Whether thermodynamically anomalous fluctuations in Bose systems are admissible? (v) How to create nonground-state condensates? Detailed answers to these questions are given in the review. As examples of nonequilibrium condensates, three cases are described: coherent modes, turbulent superfluids, and heterophase fluids.
The review is devoted to the elucidation of the basic problems arising in the theoretical investigation of systems with Bose-Einstein condensate. Understanding these challenging problems is necessary for the correct description of Bose-condensed systems. The principal problems considered in the review are as follows: (i) What is the relation between Bose-Einstein condensation and global gauge symmetry breaking? (ii) How to resolve the Hohenberg-Martin dilemma of conserving versus gapless theories? (iii) How to describe Bose-condensed systems in strong spatially random potentials? (iv) Whether thermodynamically anomalous fluctuations in Bose systems are admissible? (v) How to create nonground-statc condensates? Detailed answers to these questions are given in the review. As examples of nonequilibrium condensates, three cases are described: coherent modes, turbulent superfluids, and heterophase fluids.
We derive and discuss the equations of motion for the condensate and its fluctuations for a dilute, weakly interacting Bose gas in an external potential within the self-consistent Hartree-Fock-Bogoliubov (HFB) approximation. Account is taken of the depletion of the condensate and the anomalous Bose correlations, which are important at finite temperatures. We give a critical analysis of the self-consistent HFB approximation in terms of the Hohenberg-Martin classification of approximations (conserving vs gapless) and point out that the Popov approximation to the full HFB gives a gapless single-particle spectrum at all temperatures. The Beliaev second-order approximation is discussed as the spectrum generated by functional differentiation of the HFB single-particle Green's function. We emphasize that the problem of determining the excitation spectrum of a Bose-condensed gas (homogeneous or inhomogeneous) is difficult because of the need to satisfy several different constraints.
The excitation spectrum for the three-dimensional Bose gas in the Bose-Einstein condensation phase is calculated nonperturbatively with the modified Hartree-Fock-Bogoliubov theory, which is both conserving and gapless. From improved $\ensuremath{\Phi}$-derivable theory, the diagrams needed to preserve the Ward-Takahashi identity are re-summed in a systematic and nonperturbative way. It is valid up to the critical temperature where the dispersion relation of the low-energy excitation spectrum changes from linear to quadratic. Because including the higher-order fluctuation, the results show significant improvement on the calculation of the shift of critical temperature with other conserving and gapless theories.