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arXiv:cond-mat/0212557

The Functional Schrödinger Picture Approach to Many-Particle Systems

Chul Koo Kim, Sang Koo You

Published Dec 23, 2002• 2 citations
arXivHTMLPDF

Abstract

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 $λϕ^4$ model to yield the effective potential to the third order. The optimized perturbation theory is also applied to the $λϕ^4$ model to yield a general expression up to the second order.

DOI
10.48550/arxiv.cond-mat/0212557

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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 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.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
Following previous work in which a variational method for the many-body problem was proposed wherein an effective potential $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ was introduced as a variational parameter to construct a trial wave function, we apply the method to the two-dimensional electron gas. In the method, every physical quantity is expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ instead of the bare potential as in the usual perturbation-theoretic approach. Because of this choice of expansion parameter, the expansion series converges rapidly even in the strong-coupling region. The result for the correlation energy in the two-dimensional electron gas agrees very well with that given by the variational Monte Carlo method even in the lowest-order calculation (i.e., to second order in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$). The difference is within several percent for 1<rs<1001<{r}_{s}<1001<rs​<100.
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.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
In this primer to the many-body theory of condensed-matter systems, the authors introduce the subject to the non-specialist in a broad, concise, and up-to-date manner. A wide range of topics are covered including the second quantization of operators, coherent states, quantum-mechanical Green's functions, linear response theory, and Feynman diagrammatic perturbation theory. Material is also incorporated from quantum optics, low-dimensional systems such as graphene, and localized excitations in systems with boundaries as in nanoscale materials. Over 100 problems are included at the end of chapters, which are used both to consolidate concepts and to introduce new material. This book is suitable as a teaching tool for graduate courses and is ideal for non-specialist students and researchers working in physics, materials science, chemistry, or applied mathematics who want to use the tools of many-body theory.