This chapter derives the time-dependent Ginzburg–Landau equation from first principles, by relying on the same formal approach that has led to the time-dependent Gross–Pitaevskii equation of Chapter 28. Specifically, the time-dependent Ginzburg–Landau equation holds close to the critical temperature of the initial equilibrium preparation and in the (extreme) BCS limit of the BCS–BEC crossover, when the Cooper pairs are largely overlapping with each other. Care has to be exerted when dealing with the analytic properties in the wave-vector and frequency space of the normal and anomalous particle–particle bubbles.

This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two (t and t′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.

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.

The nonequilibrium diagrammatics and the Dyson equations contain integrations over the time variables that run over a generic contour. In both cases, the time variables run first forward and then backward along the ordinary time axis. For computational purposes, it is then required to convert these time integrals into ordinary time integrals. To this end, it is first necessary to single out all possible combinations of the pair of time variables in the contour single-particle Green’s function. This is what is done in the present chapter.

This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.

Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.

This chapter provides a concise account of the salient features of the BCS–BEC crossover. After a brief historical review of the topic, the key features of the BCS–BEC crossover are recalled. In particular, the BCS wave function for the ground state is shown to contain the BEC state of composite bosons as a limiting situation, and the special role played in this context by the chemical potential is pointed out. The need for pairing fluctuations beyond mean field is also emphasized, together with the occurrence of two (coupling and temperature dependent) lengths, which measure the correlation either within a pair of fermions with opposite spins or among different pairs. The limiting physical situations corresponding to the Ginzburg–Landau and Gross–Pitaevskii equations are finally considered.

Abstract This chapter extends the treatment of the previous chapters to the case when the system is initially prepared in an ensemble average. This requires adding a “vertical track” to the oriented contour. Alternative formalisms are considered in this context, depending on the way the vertical track is dealt with. The relationship between transient phenomena and the adiabatic assumption is also considered.

This chapter considers the expectation value of an operator (or of products of two operators) over the ground state of the interacting system, when the time-dependent part of the Hamiltonian is switched off. These limitations apply to systems in equilibrium at zero temperature, which include important cases like insulators and semiconductors as well as Fermi liquids, for which the energy gap and the Fermi energy are, respectively, much larger than the available thermal energy. The ensuing formalism for ground-state averages at zero temperature relies on an “adiabatic assumption,” which cannot be applied as it is when excited states are involved in the ensemble averages.

In the theory of the contour-ordered Green’s functions, one encounters convolutions and products. The task of this chapter is to obtain the corresponding expressions in terms of the real-time functions. This task is accomplished in terms of the so-called Langreth–Wilkins rules, which are here discussed in detail for convolutions as well as for particle–hole-type and particle–particle-type products. A preliminary introduction to what is referred to as the Keldysh space is also provided.

In the Schrödinger and Heisenberg representations, the time-evolution operator depends on the full time-dependent Hamiltonian, which includes an external time-dependent potential. Out of this full-time dependence, it is useful to isolate the time dependences due to either the full system Hamiltonian or its noninteracting part. These two cases, referred to as the Heisenberg and interaction pictures, respectively, are considered separately.

The t-matrix approximation applies to a low-density (or dilute) Fermi gas with a short-range interparticle interaction, either attractive or repulsive. This chapter considers the nonequilibrium (time-dependent) version of the t-matrix approximation for fermions in the normal phase, in the perspective of applying it to the BCS–BEC crossover.

This chapter first recalls the time-independent Bogoliubov–deGennes equations for the equilibrium case and shows their equivalence to the Gor’kov approach for inhomogeneous fermionic superfluidity at equilibrium. It then considers the extension of the Bogoliubov–deGennes approach to the nonequilibrium case in the framework of the Kadanoff–Baym equations, once implemented at the mean-field level. Properties of the solutions are considered in detail.

This chapter introduces the contour Schwinger–Keldysh method for time-dependent averages, in light of its relevance to nonequilibrium processes. A key feature of this approach is that it leaves open the possibility that no state of a system in the future can be identified with any of its states in the past. This method is here illustrated in detail with reference to time-dependent quantum averages, whereby for definiteness the system is initially prepared at the reference time t₀ in a definite quantum state.