[Paper Review] On the quantum-field description of many-particle Fermi systems with spontaneously broken symmetry

Yu. M. Poluéktov|arXiv (Cornell University)|Mar 20, 2013

Cold Atom Physics and Bose-Einstein Condensates3 references3 citations

TL;DR

This paper proposes a quantum-field framework for non-relativistic Fermi systems with spontaneously broken symmetry at finite temperature, using a generalized self-consistent field (SCF) model as the foundation. It develops a diagrammatic perturbation theory for temperature Green's functions, derives Dyson equations for self-energy and vertex parts, and establishes a consistent, thermodynamically sound approach based solely on quantum mechanics and statistical physics principles.

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.

Motivation & Objective

• To develop a consistent quantum-field description of many-particle Fermi systems exhibiting spontaneously broken symmetry, such as superconductors or superfluids.
• To address the limitations of standard perturbation theory in systems where symmetry is spontaneously broken, which invalidates standard vacuum-based approaches.
• To construct a thermodynamically consistent framework that preserves fundamental symmetries and thermodynamic relationships in the presence of broken symmetry.
• To extend the applicability of quantum field techniques—previously restricted to symmetric systems—to systems with ordered phases like superconductivity or magnetism.
• To provide a microscopic foundation for studying equilibrium properties and collective excitations in strongly correlated Fermi systems with broken symmetry.

Proposed method

• Adopts a generalized self-consistent field (SCF) model as the zeroth-order approximation, preserving a quasi-one-particle description via quasiparticles with individual wave functions.
• Applies the Bogolyubov concept of quasiaverages to consistently account for long-range order and broken symmetry in the initial approximation.
• Constructs a diagrammatic perturbation theory for finite-temperature Green's functions using the Matsubara formalism.
• Derives the Dyson equation for the full Green's function in terms of self-energy and vertex parts, incorporating both mean-field and correlation effects.
• Introduces a compact diagram technique by expressing the correlation Hamiltonian in normal-ordered form, eliminating unphysical diagrams.
• Uses field operators and second quantization formalism to express the Hamiltonian in terms of single- and two-body interactions, with chemical potential and external fields included.

Experimental results

Research questions

• RQ1 How can a consistent quantum-field approach be formulated for non-relativistic Fermi systems with spontaneously broken symmetry at finite temperature?
• RQ2 What modifications to standard perturbation theory are required to describe systems where the ground state breaks a symmetry present in the Hamiltonian?
• RQ3 How can the self-energy and vertex parts be consistently derived in such systems using a diagrammatic approach?
• RQ4 Can the thermodynamic consistency of the SCF approximation be preserved when extended to include correlations and broken symmetry?
• RQ5 What is the role of quasiaverages and normal-ordered correlation Hamiltonians in simplifying the diagram technique and ensuring physical consistency?

Key findings

• The SCF model used as the zeroth-order approximation yields thermodynamically consistent thermodynamic functions, confirming its internal consistency and validity as a foundation.
• The self-energy function is derived via a Dyson-type equation that includes both mean-field and correlation contributions, with a closed-form expression involving the vertex function.
• The vertex part is shown to govern the dispersion of collective excitations, with poles in the vertex function corresponding to collective modes.
• The diagram technique is compactified by expressing the correlation Hamiltonian in normal-ordered form, which eliminates unphysical diagrams and simplifies calculations.
• The approach is fully based on general principles of non-relativistic quantum theory and statistical mechanics, without additional assumptions or phenomenological inputs.
• The formalism is extendable to Bose systems with broken symmetry and potentially to relativistic field theories, suggesting broad applicability beyond Fermi systems.