The pairon field operator ψ(r,t) evolves, following Heisenberg’s equation of motion. If the Hamiltonian H contains a condensation energy α0(, , the evolution equation for ψ is non-linear, from which we derive the Ginzburg-Landau (GL) equation: for the GL wave function where σdenotes the state of the condensed Cooper pairs (pairons), and n the pairon density operator (u and are kind of square root density operators). The GL equation with holds for all temperatures (T) below the critical temperature Tc, where εg(T) is the T-dependent pairon energy gap. Its solution yields the condensed pairon density . The T-dependence of the expansion parameters near Tc obtained by GL: constant is confirmed.
The pairon field operator ψ(r,t) evolves, following Heisenberg’s equation of motion. If the Hamiltonian H contains a condensation energy α0(, , the evolution equation for ψ is non-linear, from which we derive the Ginzburg-Landau (GL) equation: for the GL wave function where σdenotes the state of the condensed Cooper pairs (pairons), and n the pairon density operator (u and are kind of square root density operators). The GL equation with holds for all temperatures (T) below the critical temperature Tc, where εg(T) is the T-dependent pairon energy gap. Its solution yields the condensed pairon density . The T-dependence of the expansion parameters near Tc obtained by GL: constant is confirmed.
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.
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.
We show that the eigenvalues of the number operator [Formula: see text] in the pair state (k1, k2), where c's and c†'s are fermion operators, are limited to 0 or 1, while the eigenvalues of the pair-number operator with the net momentum q≡ k1+k2 fixed, [Formula: see text], are 0, 1, 2,…. Both fermionic and bosonic natures of the Cooper pairs (pairons) must be used in the superconductivity theory. The Bose–Einstein condensation (BEC) of the pairons moving with a linear energy–momentum relation occurs at [Formula: see text] (3D), [Formula: see text] (2D), where n0 is the pairon density and v F the Fermi velocity. The connection between this [Formula: see text] and the BCS superconducting temperature is discussed.
The fourth order linear differential equation is obtained for the probability density considering the non-Hermitian Hamiltonian (the case of quasistationary states—complexity of energy). Third order nonlinear differential equation for the square of the modulus of the order parameter and for the phase is obtained by making use of Ginzburg–Landau equations. Three integrals of “motion” are found in the absence of the external magnetic field and two integrals are found in the presence of the external magnetic field. The analysis of these integrals is conducted. New analytical solutions are obtained.
We derive a Ginzburg–Landau (GL) expansion in the disordered attractive Hubbard model within the combined Nozieres–Schmitt-Rink and DMFT+Σ approximation. Restricting ourselves to the homogeneous expansion, we analyze the disorder dependence of GL expansion coefficients for a wide range of attractive potentials U, from the weak BCS coupling region to the strong-coupling limit, where superconductivity is described by Bose–Einstein condensation (BEC) of preformed Cooper pairs. We show that for the a semielliptic “bare” density of states of the conduction band, the disorder influence on the GL coefficients A and B before quadratic and quartic terms of the order parameter, as well as on the specific heat discontinuity at the superconducting transition, is of a universal nature at any strength of the attractive interaction and is related only to the general widening of the conduction band by disorder. In general, disorder growth increases the values of the coefficients A and B, leading either to a suppression of the specific heat discontinuity (in the weak-coupling limit), or to its significant growth (in the strong-coupling region). However, this behavior actually confirms the validity of the generalized Anderson theorem, because the disorder dependence of the superconducting transition temperature T c, is also controlled only by disorder widening of the conduction band (density of states).