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arXiv:1303.4913

On the quantum-field description of many-particle Fermi systems with spontaneously broken symmetry

Yu. M. Poluéktov

Published Mar 20, 2013• 4 citations
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Abstract

A quantum-field approach for describing many-particle Fermi systems at finite temperatures and with spontaneously broken symmetry has been proposed. A generalized model of self-consistent field (SCF), which allows one to describe the states eligible for this system with various symmetries, is used as the initial approximation. A perturbation theory has been developed, and a diagram technique for temperature Green's functions (GFs) has been constructed. The Dyson's equation for the self-energy and vertex parts has been deduced.

DOI
10.48550/arxiv.1303.4913

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A quantum-field approach for describing many-particle Fermi systems at finite temperatures and with spontaneously broken symmetry has been proposed. A generalized model of self-consistent field (SCF), which allows one to describe the states eligible for this system with various symmetries, is used as the initial approximation. A perturbation theory has been developed, and a diagram technique for temperature Green's functions (GFs) has been constructed. The Dyson's equation for the self-energy and vertex parts has been deduced.
A quantum-field approach to studying the Bose systems at finite temperatures and in states with spontaneously broken symmetry, in particular in a superfluid state, is proposed. A generalized model of a self-consistent field (SCF) for spatially inhomogeneous many-particle Bose systems is used as the initial approximation. A perturbation theory has been developed, and a diagram technique for temperature Green's functions (GFs) has been constructed. The Dyson's equations joining the eigenenergy and vertex functions have been deduced.
The theory of fermion phase transitions is reviewed from a unified field theoretic standpoint, based on the diagrammatic perturbation expansion of a generalized matrix propagator. Transitions from a normal to a condensed phase are characterized by the spontaneous appearance of long-range order and (in the presence of a suitable infinitesimal external field) broken symmetry. This is illustrated by the ferromagnetic, solid, superconducting and spindensity wave ground states. The phenomenon is explained qualitatively as caused by the creation of a long-range internal field, F, due to the interactions between particles. This field establishes long-range order in the system, and is in turn itself established by the long-range order, in a self-consistent fashion. The mechanism here is expressed quantitatively in terms of a self-consistent Dyson equation relating a generalized matrix propagator, G, to a proper self-energy matrix, Σ. The off-diagonal elements of G describe ‘anomalous’ propagation processes which are characteristic for the condensed phase, and they yield directly the long-range order parameters. The Σ-matrix is just the potential of the internal field. The method is illustrated by applying it to the ferromagnetic phase of a system with δ-function interaction between particles. Finally, the technique is used to derive the vertex part equation for the transition point.
The development of methods of quantum statistical mechanics is considered in light of their applications to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green’s functions, which is widely used in various physical problems of many-particle systems with interaction. Quantum cooperative effects and quasi-particle dynamics in the basic microscopic models of quantum theory of magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spinfermion model are considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description of complex magnetic materials. The concepts of broken symmetry, quantum protectorate, and quasi-averages are analyzed in the context of quantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presented within the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the results of investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects, are presented.
A self-consistent field (SCF) model is constructed for spatially inhomogeneous Bose systems with broken symmetry. A system of self-consistent equations is obtained for the wave functions of the quasiparticles and the wave function of the condensate particles and another system of equations for the normal and anomalous single-particle density matrices. The many-particle wave function is found. The thermodynamics of multiparticle Bose systems is constructed on the basis of a microscopic treatment in the SCF approximation. It is stressed that Bose condensation in the ideal gas model is substantially different from Bose condensation in a system of interacting Bose particles because a pair condensate is necessarily present as well as the single-particle condensate, even for an arbitrarily weak interaction. The role of single-particle and collective excitations in Bose systems is 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.
A new approach to the one-particle Green's functions G\mathit{G}G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M\mathit{M}M related to G\mathit{G}G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (nthn\mathrm{th}nth-order ADC schemes) that represent infinite partial summations for M\mathit{M}M and (via the Dyson equation) for G\mathit{G}G being complete through nthn\mathrm{th}nth order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.