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arXiv:cond-mat/9703101

Finite temperature many-particle theory of condensed matter systems in the functional Schroedinger picture

Hyun Sik Noh, Sang Koo You, Chul Koo Kim

Published Mar 11, 1997• 0 citations
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Abstract

A finite temperature many-particle theory of condensed matter systems is formulated using the functional Schroedinger picture. Using the interacting electron gas as a model system, we solve the equation of motion for the density matrix variationally with a Gaussian type trial density matrix. We show that the present formalism yields the finite temperature Hartree-Fock results both for the para- and ferromagnetic states in a simple and convenient fashion. Implications of the present results and future prospects are also discussed.

DOI
10.48550/arxiv.cond-mat/9703101

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A finite temperature many-particle theory of condensed matter systems is formulated using the functional Schroedinger picture. Using the interacting electron gas as a model system, we solve the equation of motion for the density matrix variationally with a Gaussian type trial density matrix. We show that the present formalism yields the finite temperature Hartree-Fock results both for the para- and ferromagnetic states in a simple and convenient fashion. Implications of the present results and future prospects are also discussed.
A finite temperature many-particle theory of condensed matter systems is formulated using the functional Schrödinger picture. Using the interacting electron gas as a model system, we solve the equation of motion for the density matrix variationally with a Gaussian type trial density matrix. We show that the present formalism yields the finite temperature Hartree–Fock results both for the para- and ferromagnetic states in a simple and convenient fashion. Implications of the present results and future prospects are also discussed.
A self-contained pedagogical introduction to the functional Schrödinger picture method of many-body theory is given at a level suitable for graduate students and also for many-body physicists who have not been exposed to the functional Schrödinger picture method previously. Mathematical tools necessary for the functional Schrödinger picture calculation are introduced. The method is first applied to various condensed matter problems including the electron gas, the Hubbard model, the BCS superconductivity, and dilute bose gas within the variational approximation. It is shown that the variational approximation with the Gaussian trial functional invariably leads into the Hartree-Fock results for both zero and finite temperature cases. In order to go beyond the Gaussian results, concepts of variational and optimized perturbation theories are introduced using simple quantum mechanical languages. Then, the variational perturbation theory in the functional Schrödinger picture is applied to the λϕ4λϕ^4λϕ4 model to yield the effective potential to the third order. The optimized perturbation theory is also applied to the λϕ4λϕ^4λϕ4 model to yield a general expression up to the second order.
A time-dependent many-body theory for many-particle systems is formulated using the functional Schrödinger picture. Taking a Gaussian trial wave functional and using the variational principle, the equations of motion are obtained. The equations of motion are solved in the small oscillation regime to yield the generalized RPA results. The electron–hole excitation, the collective plasma oscillation for the paramagnetic state, the Stoner excitation and the spin wave for the ferromagnetic state are derived and discussed. As a further test on the versatility of the present method, the formalism is applied to the bound states of the exciton problem and shown to yield the governing equation in a straightforward manner.
The variational density-matrix approach introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.
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.