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Physical review. B, Condensed matter

Kinetic energy and the Born-Green-Yvon method for fermion quantum fluids

C. E. Campbell, K. E. Kürten, E. Krotscheck

Published Feb 1, 1982• 14 citations
Publisher

Abstract

The Born-Green-Yvon (BGY) equations for two-body distribution functions of fermion-Jastrow many-body trial functions are derived using a diagrammatic method. Also derived are the Jackson-Feenberg and Pandharipande-Bethe expressions for the kinetic energy of this function in terms of partial two- and three-body distribution functions. Simple approximations for these three-body functions are then used in the BGY equations and the kinetic energies and are solved for the ground state of liquid $^{3}\mathrm{He}$.

DOI
10.1103/physrevb.25.1633

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The Born-Green-Yvon (BGY) equations for two-body distribution functions of fermion-Jastrow many-body trial functions are derived using a diagrammatic method. Also derived are the Jackson-Feenberg and Pandharipande-Bethe expressions for the kinetic energy of this function in terms of partial two- and three-body distribution functions. Simple approximations for these three-body functions are then used in the BGY equations and the kinetic energies and are solved for the ground state of liquid 3He^{3}\mathrm{He}3He.
The rigorous evolution equations for the one-body Green's functions (g>,g<{g}^{>},{g}^{<}g>,g<) describing a system of interacting fermions are not closed (an analog of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for many-body distribution functions). The analysis of these equations is made in terms of proper connected diagrams. It is established that in the bulk limit, NNN (particle number) $\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\Omega}$ (volume) $\ensuremath{\rightarrow}\ensuremath{\infty}$ while $\frac{N}{\ensuremath{\Omega}}$ remains finite, these equations can rigorously be transformed into closed equations with respect to (g>,g<{g}^{>},{g}^{<}g>,g<).
From the quantum kinetic equation of Bogoliubov, Gurov, Born, and Green, the equations of motion for macroscopic variables are examined in many different ways. First, in the configuration space a hierarchy of generalized fluid-dynamical equations can be obtained by taking the appropriate limits of the quantum kinetic equation. The resultant equations of continuity, of momentum, and of energy are similar in form to those one encounters in classical fluid dynamics with the exception of additional terms proportional to ${\ensuremath{\hbar}}^{2}$ Secondly, the quantum kinetic equation is examined in phase space. The same set of equations of continuity, of momentum, and of energy can be derived by taking the first three moments of the quantum kinetic equation. The exact results we obtained are utilized to form the starting point for many simplifying approximations for the investigation of the dynamics of quantum many-body systems such as the elastic response limit, the hydrodynamical limit, Landau's Fermi-liquid theory, and finally the time-dependent Hartree-Fock and the multideterminant time-dependent Hartree-Fock approximations. The fact that all these different dynamical descriptions can be traced to a common origin provides a unifying viewpoint to the present approach with the quantum kinetic equation.NUCLEAR STRUCTURE Dynamics of nuclear fluid, quantum kinetic equation, and generalized fluid-dynamical equation. Elastic response, hydrodynamical limit, Landau's Fermi-liquid theory, time-dependent Hartree-Fock approximation, and multideterminant time-dependent Hartree-Fock approximation.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Phi-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Phi-derivable approximation, i.e., an issue of how to handle the "Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy".
The microscopic study of the two-body density matrix ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$,${\mathbf{r}}_{2}^{\ensuremath{'}}$) initiated for uniform Bose fluids in an earlier paper is continued for the Fermi case. We present formal results on the structure of the generalized momentum distribution n(p,q)=${\mathrm{\ensuremath{\Sigma}}}_{\mathbf{k}\mathbf{\^{}}}$〈\ensuremath{\Psi}|${\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}+\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$${\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}\mathrm{\ensuremath{-}}\mathbf{q}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$apˆ{\mathrm{a}}_{\mathbf{p}\mathbf{\^{}}}apˆ​akˆ{\mathrm{a}}_{\mathbf{k}\mathbf{\^{}}}akˆ​|\ensuremath{\Psi}〉, and its Fourier inverse ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$,r2{\mathbf{r}}_{2}r2​)\ensuremath{\equiv}${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), based on a variational ground-state wave function of Jastrow-Slater form. The structural relations are inferred from the cluster expansions of these objects, from the asymptotic condition relating ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) to the particle density and the one-body density matrix ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{1}r1​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), and from formal diagrammatic connections with the Bose problem. The two-body density-matrix elements ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) are thereby expressed in closed form in terms of certain sums of irreducible cluster diagrams. Some of these diagram sums are familiar from the analogous theory of the one-body density matrix; all can be evaluated quantitatively by solving a set of Fermi-hypernetted-chain (FHNC) equations. Upon invoking the sequential relation between ${\mathrm{\ensuremath{\rho}}}_{2}$(r1{\mathbf{r}}_{1}r1​,r2{\mathbf{r}}_{2}r2​,${\mathbf{r}}_{1}^{\ensuremath{'}}$) and ${\mathrm{\ensuremath{\rho}}}_{1}$(r1{\mathbf{r}}_{1}r1​,${\mathbf{r}}_{1}^{\ensuremath{'}}$), the corresponding result for the generalized momentum distribution n(p,q) effects a resolution into contributions from various scattering processes occurring in the many-body medium, specified by form factors that are susceptible to FHNC evaluation. This decomposition is comparable to that derived earlier for the Bose-fluid ground state but is complicated by contributions from exchange scattering and by a dynamically dressed Pauli kinematic correction. Silver has proposed a simple expression for the generalized momentum distribution n(p,q), a function which plays an essential role in his theory of final-state effects in deep-inelastic neutron scattering from the helium liquids. Based on the present microscopic treatment, the quality of Silver's estimate is assessed for the case of normal liquid He3{\mathrm{He}}^{3}He3, by evaluating the necessary distribution functions and form factors within the FHNC/0 approximation.
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