By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.
By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.
Making use of the inversion method, which is invented to discuss the characteristics of phase transition in a perturbative manner, a fundamental equation for nonequilibrium superconductivity is derived. In the course of the calculation, the generalized contour-ordered Green's-function formalism is used to treat the initial time correlation function and real-time Green's function in a unified manner. The result is a set of self-consistent integral equations and the Gorkov's equation is given a posteriori as one of these components describing the initial time correlation. By virtue of the contour-ordered Green's-function formalism, it is not necessary to employ any assumption to consider the effect of the initial time correlation on the system at subsequent time. Owing to this nature, our result helps in discussing transient behavior of the system induced by a supercritical current that appears in some kinds of superconducting device.
We present some basic Green’s functions of superconductivity, making emphasis on their geneology and analytic properties. From calculations, we note that the temperature dependence of the Green’s functions for fermionic (and bosonic) systems limits and defines the extent of their applications and results. Furthermore, the Gorkov interaction term of the four field fermion operators, is examined and interpreted in terms of the Landau condensate. Finally we show that the Gorkov interaction under a certain condition sustains superconductivity and spin density wave in the system.KEYWORDS: Green’s functions, Bethe-Salpeter equation, Landau condensate, superconductivity, spin density wave.
Functions which can be expressed as an antisymmetrized power of a single two-particle function (a geminal) occur in the BCS ansatz. They constitute a comparatively tractable generalization of a single Slater determinant. The second-order reduced density matrix and bounds on its eigenvalues are obtained for the most general such antisymmetrized geminal power (AGP). These eigenvalues are physically of great importance, being analogs, in an analysis of the wavefunction into pair states, of the one-particle occupation numbers of the independent-particle model. A necessary and sufficient condition is given that an alleged first-order reduced density matrix be derivable from an N-particle AGP function. It is shown that only for exceptional functions (called extreme) is the generating geminal of the AGP function an eigenfunction of the 2-matrix. The extreme 2-matrix is diagonalized explicitly and provides a vivid example of the fact that one-particle occupation numbers alone are unable to convey certain information of decisive significance about the wavefunction. The extreme 2-matrix satisfies the assumption basic to current theories of superconductivity.
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.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
The exact reduced density-matrix functional is derived from the Luttinger-Ward functional of the single-particle Green's function. Thereby, a formal link is provided between diagrammatic many-body approaches using Green's functions on the one hand and theories based on many-body wave functions on the other. This link can be used to explicitly construct approximations for the density-matrix functional that are equivalent to standard diagrammatic resummation techniques and to nonperturbative dynamical mean field theory in particular. Contrary to functionals of the Green's function, the exact density-matrix functional is convex and thus provides a true minimum principle which facilitates the calculation of the grand potential and derived equilibrium properties. The benefits of the proposed Green's-function-based density-matrix functional theory for geometrical structure optimization of strongly correlated materials are discussed.