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arXiv:1306.2103

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

Yu. M. Poluéktov

Published Jun 10, 2013• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1306.2103

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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.
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 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.
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.
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.
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.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by the latest high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green's function approach allows for calculating the excitation spectrum at finite temperature and for determining the effective masses of particles and holes.