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Cambridge University Press eBooks

Time-Dependent BCS Hamiltonian and Gor’kov Equations

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter considers the analog of the time-dependent Hartree–Fock (mean-field) decoupling treated in Part I and extends it to the broken-symmetry phase for superfluid fermions. Two coupled equations for the “normal” and “anomalous” time-dependent single-particle Green’s functions are obtained, which extend to nonequilibrium situations the equations originally obtained at equilibrium by Gor’kov, soon after the BCS original article on the theory of superconductivity. Accordingly, the time-dependent gap (order) parameter is also introduced.

DOI
10.1017/9781009411509.023

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arXiv:nucl-th/9706041•Jun 17, 1997•0 citations
This chapter considers the analog of the time-dependent Hartree–Fock (mean-field) decoupling treated in Part I and extends it to the broken-symmetry phase for superfluid fermions. Two coupled equations for the “normal” and “anomalous” time-dependent single-particle Green’s functions are obtained, which extend to nonequilibrium situations the equations originally obtained at equilibrium by Gor’kov, soon after the BCS original article on the theory of superconductivity. Accordingly, the time-dependent gap (order) parameter is also introduced.
Mean-field decouplings can also be utilized in time-dependent (nonequilibrium) situations. This chapter considers the time-dependent Hartree–Fock approximation for fermions in the normal phase, as obtained in terms of a time-dependent mean-field decoupling (postponing to Part II the Gor’kov generalization to the superfluid phase). A connection is also established with a more standard formulation of the time-dependent Hartree–Fock approximation in terms of a set of time-dependent single-particle wave functions.
This chapter considers the extension of the t -matrix approximation to the superfluid phase, for which it is convenient to restrict from the outset to a contact-type interparticle interaction. This is because, when addressing nonequilibrium (time-dependent) situations, the extension of the fermionic t -matrix approach from the normal to the superfluid phase requires a careful account for the Nambu indices in the two-particle channels, owing to the presence of the “anomalous” single-particle Green’s functions. The ladder approximation for the many-particle T-matrix is specifically considered.
This chapter introduces the Nambu representation for the pseudo-spinor fields and expresses the system Hamiltonian in terms of them. In this way, the anomalous single-particle Green’s function is made to match the form of the single-particle Green’s functions treated in Part I, where an even number of creation and destruction operators appear. On physical grounds, this approach exploits the fact that opposite-spin fermions are coupled in pairs. The special role played by the Hartree–Fock self-energy for a superfluid Fermi system is duly emphasized.
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we rely on a single variational principle which optimizes the characteristic function. The Bloch equation is used as a constraint to define the equilibrium state, and the trial quantities are a density operator and a Lagrangian multiplier matrix which is akin to an observable. The conditions of stationarity are respectively a Schrödinger-like equation and a Heisenberg-- like equation with an imaginary time running backwards. General conditions for the trial spaces are stated that warrant the preservation of thermodynamic relations. When the trial spaces are chosen to be of the independent-quasi-particle type, the ensuing coupled equations provide an extension of the Hartree-Fock-Bogoliubov approximation, which optimizes the characteristic function. Variational expressions for thermodynamic quantities or characteristic functions are also obtained with projected trial states, whether an invariance symmetry is broken or not. In particular, the projection on even or odd particle number is worked out for a pairing Hamiltonian, which leads to new equations replacing the BCS ones. Qualitative differences between even and odd systems, depending on the temperature TTT, the level density and the strength of the pairing force, are investigated analytically and numerically.