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Cambridge University Press eBooks

The T-matrix Approximation in the Superfluid Phase

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1017/9781009411509.026

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Cambridge University Press eBooks•Feb 6, 2026•0 citations
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.
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 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.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
The effects of the coupling to the environment can also manifest itself in a superfluid Fermi system. This chapter explicitly considers this case, by addressing the time-dependent behavior of the gap parameter following a sharp quench of the coupling parameter of the contact interaction. In this case, coupling the system to the environment is important for reaching equilibrium eventually. Several simplifying assumptions are adopted along the way for treating the problem in an as simple as possible way.
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.