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Physical Review A

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

E. R. Tuttle

Published Apr 1, 1970• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.1.1243

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

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An elaborate cluster analysis of the single-particle occupation probability ${n}_{\stackrel{^}{q}}$ and associated one-body density matrix $n(r)$ is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that ${n}_{\stackrel{^}{q}}$ may be written as $n[N(q)+{N}_{1}(q)]$, where $n$ is a strength factor independent of wave number $q$ and the quantities $N(q)$ and ${N}_{1}(q)$ may be expressed as series of irreducible cluster contributions. The strength factor $n$ has the form $n={e}^{Q}$, where $Q$ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for $Q$ in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of ${n}_{\stackrel{^}{q}}$, it is further demonstrated that $n(r)$ may be cast in the form $\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)$, where $\ensuremath{\rho}$ is the particle density and the functions ${N}_{1}(r)$, ${N}_{2}(r)$, and $\mathcal{Q}(r)$ are all given by irreducible cluster series. Massive partial summations are executed in the $\mathcal{Q}(r)$ series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has $\mathcal{Q}(0)=Q$. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid $^{3}\mathrm{He}$ and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid $^{4}\mathrm{He}$ to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}$ being just the strength factor $n$. It may also be shown (under certain reasonable assumptions) that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlations implies a singular behavior ${n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}$ of the Bose momentum distribution for small $q$.

Physical review. B, Solid state•Oct 1, 1976•70 citations
Hartree-Fock theory is used in conjunction with the quantum Ursell expansion to obtain the grand potential f(\ensuremathβ,g,\ensuremathΩ)f(\ensuremath{\beta}, g, \ensuremath{\Omega})f(\ensuremathβ,g,\ensuremathΩ) 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(\ensuremathβ,g,\ensuremathΩ)f(\ensuremath{\beta}, g, \ensuremath{\Omega})f(\ensuremathβ,g,\ensuremathΩ) and 〈n(k)〉〈n(k)〉〈n(k)〉 take on forms identical to those obtained in the \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ-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 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.
The widely used thermal Hartree-Fock (HF) theory is generalized to include the effect of electron correlation while maintaining its quasi-independent-particle framework. An electron-correlated internal energy (or grand potential) is postulated in consultation with the second-order finite-temperature many-body perturbation theory (MBPT), which then dictates the corresponding thermal orbital (quasiparticle) energies in such a way that all fundamental thermodynamic relations are obeyed. The associated density matrix is of a one-electron type, whose diagonal elements take the form of the Fermi-Dirac distribution functions, when the grand potential is minimized. The formulas for the entropy and chemical potential are unchanged from those of Fermi-Dirac or thermal HF theory. The theory thus stipulates a finite-temperature extension of the second-order Dyson self-energy of one-particle many-body Green's function theory and can be viewed as a second-order, diagonal, frequency-independent, thermal inverse Dyson equation. At low temperatures, the theory approaches finite-temperature MBPT of the same order, but it may outperform the latter at intermediate temperatures by including additional electron-correlation effects through orbital energies. A physical meaning of these thermal orbital energies is proposed (encompassing that of thermal HF orbital energies, which has been elusive) as a finite-temperature version of Janak's theorem.
Employing an infinite hierarchy of equations for the Green functions of a many particle system in its ground state mod psi N ), the well known non-relativistic, time-dependent, ground-state perturbation theory of a normal system of interacting fermions is derived, without introducing the hypothesis that mod psi N ) is the adiabatic transform of a non-interacting state. This new formulation of the problem allows us to renormalise conventional perturbation theory by introducing an effective interaction Gamma and the final result is a highly summed and manifestly self-consistent perturbation expansion in powers of Gamma . The authors calculate Gamma to second order and demonstrate that many of the well known results of time-dependent perturbation theory may be obtained by choosing the simplest approximations for Gamma . The results are readily generalised to finite temperatures and the formalism provides a generalised form of Hartree-Fock theory which may have important applications in many areas of physics.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
A theory of a Fermi superfluid is formulated based on the condensate wave function of the configuration space. First, two basic ingredients are extracted from the wave function, i.e. the effective condensate wave function v ( x 1 , x 2 ) and the quasiparticle field γ( x ) with x denoting the space and spin coordinates r α. After transforming a Hamiltonian into a normal-ordered product of γ, a perturbation expansion with respect to the quasiparticle interaction is formulated for both the thermodynamic potential Ω and the one-particle density matrix ρ. The thermodynamic potential thus obtained includes the normal-state expression of Luttinger and Ward as the limit of v → 0. The density matrix is written as a sum of the coherent part ρ ( c) and the quasiparticle part ρ ( q) , where ρ ( c) is expressed only with respect to v ( x 1 , x 2 ). It is shown that v and γ are responsible for the coherent and thermodynamic properties respectively. Thus the present formalism gives a microscopic foundation for the two-fluid model. Connections with the Gor'kov equations and the Bogoliubov-de Gennes equations are also clarified. It is found that the one-particle density matrices of those formalisms are not equivalent to ρ of the present formalism. The origin of the angular momentum paradox of 3 He - A is traced to this difference.
An elaborate cluster analysis of the single-particle occupation probability ${n}_{\stackrel{^}{q}}$ and associated one-body density matrix n(r)n(r)n(r) is performed for a Fermi system described by a Jastrow wave function. A diagrammatic formalism rooted in Ursell-Mayer theory facilitates the analysis. It is conjectured, and demonstrated to convincingly high cluster order, that ${n}_{\stackrel{^}{q}}$ may be written as n[N(q)+N1(q)]n[N(q)+{N}_{1}(q)]n[N(q)+N1​(q)], where nnn is a strength factor independent of wave number qqq and the quantities N(q)N(q)N(q) and N1(q){N}_{1}(q)N1​(q) may be expressed as series of irreducible cluster contributions. The strength factor nnn has the form n=eQn={e}^{Q}n=eQ, where QQQ may also be expressed as a series of irreducible cluster contributions. Massive partial summations on the latter series yield a compact expression for QQQ in terms of the spatial distribution functions corresponding to the Jastrow wave function. Working with the Fourier inverse of ${n}_{\stackrel{^}{q}}$, it is further demonstrated that n(r)n(r)n(r) may be cast in the form $\ensuremath{\rho}n[{N}_{1}(r)+{N}_{2}(r)]\mathrm{exp}\mathcal{Q}(r)$, where $\ensuremath{\rho}$ is the particle density and the functions N1(r){N}_{1}(r)N1​(r), N2(r){N}_{2}(r)N2​(r), and Q(r)\mathcal{Q}(r)Q(r) are all given by irreducible cluster series. Massive partial summations are executed in the Q(r)\mathcal{Q}(r)Q(r) series to achieve a compact expression of this quantity in terms of the aforementioned spatial distribution functions. One has Q(0)=Q\mathcal{Q}(0)=QQ(0)=Q. The leading diagrams necessary for a quantitative evaluation of the momentum distribution of liquid 3He^{3}\mathrm{He}3He and nuclear matter are displayed. Specialization to infinite degeneracy of the single-particle levels, while shrinking the Fermi wave number to zero (Bose limit), allows liquid 4He^{4}\mathrm{He}4He to be treated as well. In this limit off-diagonal long-range order appears, the condensate fraction ${\ensuremath{\rho}}^{\ensuremath{-}1}n(\ensuremath{\infty})={n}_{c}$ being just the strength factor nnn. It may also be shown (under certain reasonable assumptions) that the customary ${r}^{\ensuremath{-}2}$ long-range behavior of the two-body correlations implies a singular behavior ${n}_{q}={n}_{c}(\frac{\mathrm{mc}}{2\ensuremath{\hbar}}){q}^{\ensuremath{-}1}$ of the Bose momentum distribution for small qqq.