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Physical Review A

Microscopic Theory of Quantum Fluids. II. Λ Transformation

Franz Mohling, Indira Ramarao, Dion W. J. Shea

Published Jan 1, 1970• 9 citations
Publisher

Abstract

In a previous paper, the master-graph formulation of the quantum-statistical theory of quantum fluids was developed. If this formulation is used to calculate the equilibrium properties of quantum fluids, apparent divergences are encountered in the low-temperature limit. In the present paper, we transform this theory by means of a $\ensuremath{\Lambda}$ transformation to overcome these apparent low-temperature divergences. In this transformation, the terms in the theory which gave rise to the apparent low-temperature divergences and which represent the dominant low-temperature contributions are summed explicitly to obtain well-behaved expressions. In addition, a consistent method is developed to obtain the corrections to the dominant low-temperature contributions. Explicit expressions for the $\ensuremath{\Lambda}$-transformed theory are given for the cases of a Bose fluid above the Bose-Einstein condensation temperature and for a Fermi fluid. Finally, the physical implications of the $\ensuremath{\Lambda}$ transformation are discussed.

DOI
10.1103/physreva.1.192

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Progress of Theoretical Physics Supplement•Jan 1, 1969•2 citations
In a previous paper, the master-graph formulation of the quantum-statistical theory of quantum fluids was developed. If this formulation is used to calculate the equilibrium properties of quantum fluids, apparent divergences are encountered in the low-temperature limit. In the present paper, we transform this theory by means of a \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ transformation to overcome these apparent low-temperature divergences. In this transformation, the terms in the theory which gave rise to the apparent low-temperature divergences and which represent the dominant low-temperature contributions are summed explicitly to obtain well-behaved expressions. In addition, a consistent method is developed to obtain the corrections to the dominant low-temperature contributions. Explicit expressions for the \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-transformed theory are given for the cases of a Bose fluid above the Bose-Einstein condensation temperature and for a Fermi fluid. Finally, the physical implications of the \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ transformation are discussed.
A simplified derivation of the master-graph formulation of the quantum-statistical theory of an interacting quantum fluid is presented. A complete analysis of the self-energy structures arising from both particle statistics and particle interactions simplifies the master-graph expressions for the thermodynamic properties of the fluid. The xxx-ensemble formulation of Lee and Yang is used to treat the degenerate Bose fluid below its Bose-Einstein condensation temperature. The results for a Bose fluid above its Bose-Einstein condensation temperature and for a Fermi fluid are obtained as special cases.
In the first paper of this series, the master-graph formulation of the Lee-Yang quantum statistical theory for a degenerate Bose system was derived. In the present paper, this entire theory is transformed from a free-particle description to a quasiparticle description by means of a $\ensuremath{\Lambda}$ transformation. This transformation leaves the master-graph formulation of the theory essentially unchanged, while at the same time introducing the quasiparticle energy-momentum relation and the quasiparticle interaction function into the theory. The transformation is motivated by a study of the very low-temperature behavior of the fundamental integral equation of the theory. An interesting feature of the transformed theory is that it contains two quasiparticle energy-momentum relations, ${\ensuremath{\epsilon}}_{+}(k)$ and ${\ensuremath{\epsilon}}_{\ensuremath{-}}(k)$. The understanding of this result is not achieved in this paper, although the possibility of a double quasiparticle spectrum has previously been suggested by Lieb.
In a previous paper, a Hartree-Fock theory of the quantum Ursell expansion for Fermi fluids was developed. The same techniques are used in this paper to derive a theory of degenerate Bose fluids from the xxx-ensemble theory of Lee and Yang. It is shown that this theory is equivalent to the $\ensuremath{\Lambda}$-transformed theory recently derived by Mohling, RamaRao, and Shea. The present approach considerably simplifies the formalism and eliminates the need for the $\ensuremath{\Lambda}$ transformation. By working entirely in a quasiparticle framework, we also avoid the problems with unphysical line factors which have appeared in earlier theories. Finally, a greatly simplified $\ensuremath{\Lambda}$ transformation is obtained for use in problems in which the potential contains an infinitely repulsive core.
Bose–Einstein condensation is an intensely studied quantum phenomenon that emerges at low temperatures. While preceding Bose–Einstein condensation models do not consider what statistics apply above the condensation temperature, we show that neglecting this question leads to inconsistencies. A mathematically rigorous calculation of Bose–Einstein condensation temperature requires evaluating the thermodynamic balance between coherent and incoherent particle populations. The first part of this work develops such an improved Bose–Einstein condensation temperature calculation, for both three-dimensional and two-dimensional scenarios. The progress over preceding Bose–Einstein condensation models is particularly apparent in the two-dimensional case, where preceding models run into mathematical divergence. In the Discussion section, we compare our mathematical model against experimental superconductivity data. A remarkable match is found between experimental data and the calculated Bose–Einstein condensation temperature formulas. Our mathematical model therefore appears applicable to superconductivity, and may facilitate a rational search for higher-temperature superconductors.
Recently the present author developed a diagram formalism applicable to the degenerate Bose fluids.The present author and Hara applied the formalism to show that the long distance 1/r and 1/r 2 behaviors of the pair distribution function are exactly cancelled for interacting Bose fluids.The basic ideas of the formalism have been given in separate articles.It is the purpose of the present article to present the formalism in a unified form.The formalism is developed in the canonical ensemble.However the generating function is used.Hence a criterion of the equivalence of the canonical and grand canonical ensembles is given and used in the discussions.Replacement of operators ao and at by a c•number na' 2 is performed with the same degree of care as Kromminga and Bolsteri took in treating the zero-temperature problem.Hugenholtz and Pines's relation is generalized to finite temperatures.The cancellation of the long-distance 1/r and 1/r 2 behaviors at and below the 'A temperature is briefly discussed.For some of details, readers are requested to read the preceding articles.§ 1. § 2. § 3. § 4. § 5. § 6. § 7. § 8. § 9. §10.§11.§12.§13.§14.§15.§16.Al.Contents Introduction, 70 Operators for the one-and two-particle distribution functions, 73 Second quantization, 73 Canonical and grand canonical ensemble averages, 76 Equivalence of the canonical and grand canonical ensembles, 80 Ideal Bose gas, 83 Diagram expansion for non-degenerate interacting Bose fluids, 87 Lee and Yang's expansion and binary collision expansion, 90 Equivalent Hamiltonian and the averages equivalent to the canonical averages of degenerate Bose fluids, 91 Diagram expansion for degenerate interacting Bose fluids, 95 Distribution functions of the ideal Bose gas, 99 Chain approximation for the degenerate case, 100 Bogolyubov Hamiltonian and Bogolyubov transformation, 102 Cancellation of 1/r and 1/r 2 terms in pC 2 )(r) for the degenerate case 104 Cancellation of 1/r 2 terms in pC 2 )(r) just above TJ.., 107 Discussions and comparison with related works, 108 Calculation for the second quantization, 111 T. Morita A2.Alternative formulation of the diagram expansiOn for nondegenerate Bose fluids, 112 A3.Alternative formulation of the diagram expansion for degenerate Bose fluids, 115 References, 116 )Comparing this fact with the formula (1•2), Hiroike 4 ) noticed that the divergence of the isothermal compressibility is due to this long-range behavior of the pair distribution function.*)We now compare the above situation with the phase transition of liquid helium.The phase diagram is shown in Fig. 2. The p-v isotherms for *' In a recent paper, Luban and Rezven 5 > expressed a similar point of view.