Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Hartree-Fock Theory and the Λ Transformation in Quantum Statistical Mechanics. II. Degenerate Bose System

E. R. Tuttle

Published Jun 1, 1970• 1 citation
Publisher

Abstract

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 $x$-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.

DOI
10.1103/physreva.1.1758

Related papers

Hartree-Fock Theory and the Δ Transformation in Quantum-Statistical Mechanics. I. Normal Fermi Systems

E. R. Tuttle

Hartree-Fock theory is used in conjunction with the quantum Ursell expansion to obtain the grand potential $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and the momentum distribution $〈n(k)〉$ for a Fermi fluid. It is shown that the perturbation expansion of the quasiparticle Ursell functions thus obtained yields a formal expansion of the grand potential which is easily related to the theory of de Dominicis. After the summation of certain ladder diagrams in the perturbation expansion, $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and $〈n(k)〉$ take on forms identical to those obtained in the $\ensuremath{\Lambda}$-transformed theory derived by Mohling, RamaRao, and Shea. It is therefore possible to identify the Hartree-Fock single-particle energy with the quasiparticle energy of those authors' work.

Physical Review A•Apr 1, 1970•12 citations

Degenerate Bose System. II. Λ Transformation of Quantum Statistical Theory

Franz Mohling

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.

Physical Review•Aug 17, 1964•7 citations

Microscopic Theory of Quantum Fluids. III. Quasiparticle Ensemble for Degenerate Bose Fluid

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

The analysis of the preceding two papers in this series is applied to the specific case of a degenerate Bose fluid, in which there can be macroscopic occupation of the zero-momentum state (in a system at rest). The quasiparticle ensemble of Mohling and Tuttle is incorporated into the theory in order to guarantee that Nernst's theorem for the entropy is satisfied. It is shown that a double-quasiparticle model emerges naturally from the theory, as developed earlier, and this feature is then incorporated into the quasiparticle ensemble. The general theory is prepared, with the aid of the $\ensuremath{\Lambda}$ transformation, for calculational applications. A phenomenological theory of the degenerate Bose fluid, based on the double-quasiparticle model, is presented, but no significant physical implications are deduced from it.

Physical Review A•Sep 1, 1970•4 citations

Degenerate Bose System. I. Quantum Statistical Theory

Franz Mohling

The theory of a degenerate system of interacting bosons is developed using the $x$-ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.

Physical Review•Aug 17, 1964•17 citations

Theory of Degenerate and Non-Degenerate Bose Fluids and Calculation of the Pair Distribution Function

Tohru Morita

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.

Progress of Theoretical Physics Supplement•Jan 1, 1969•2 citations

Microscopic Theory of Quantum Fluids. II. Λ Transformation

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

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.

Physical Review A•Jan 1, 1970•9 citations
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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-transformed theory recently derived by Mohling, RamaRao, and Shea. The present approach considerably simplifies the formalism and eliminates the need for the \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ 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 \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ transformation is obtained for use in problems in which the potential contains an infinitely repulsive core.
Hartree-Fock theory is used in conjunction with the quantum Ursell expansion to obtain the grand potential $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and the momentum distribution 〈n(k)〉〈n(k)〉〈n(k)〉 for a Fermi fluid. It is shown that the perturbation expansion of the quasiparticle Ursell functions thus obtained yields a formal expansion of the grand potential which is easily related to the theory of de Dominicis. After the summation of certain ladder diagrams in the perturbation expansion, $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and 〈n(k)〉〈n(k)〉〈n(k)〉 take on forms identical to those obtained in the $\ensuremath{\Lambda}$-transformed theory derived by Mohling, RamaRao, and Shea. It is therefore possible to identify the Hartree-Fock single-particle energy with the quasiparticle energy of those authors' work.
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.
The analysis of the preceding two papers in this series is applied to the specific case of a degenerate Bose fluid, in which there can be macroscopic occupation of the zero-momentum state (in a system at rest). The quasiparticle ensemble of Mohling and Tuttle is incorporated into the theory in order to guarantee that Nernst's theorem for the entropy is satisfied. It is shown that a double-quasiparticle model emerges naturally from the theory, as developed earlier, and this feature is then incorporated into the quasiparticle ensemble. The general theory is prepared, with the aid of the $\ensuremath{\Lambda}$ transformation, for calculational applications. A phenomenological theory of the degenerate Bose fluid, based on the double-quasiparticle model, is presented, but no significant physical implications are deduced from it.
The theory of a degenerate system of interacting bosons is developed using the xxx-ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.
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.
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.