Quantum statistical derivation of the Ginzburg-Landau equation. Energy gap, condensed pairon density and penetration depth

Abstract

The Cooper pair (pairon) field operator psi(dagger) (r, t) changes, following Heisenberg's equation of motion. If the Hamiltonian H contains pairon kinetic energies h(0), a condensation energy alpha(< 0) and a repulsive point-like interpairon interaction beta delta(r(1) - r(2)), beta > 0, the evolution equation for psi is nonlinear, from which we obtain the Ginzburg-Landau (GL) equation: h(0)(r -i (h) over bar del)Psi(sigma)(r)+alpha Psi(sigma)(r)+beta \Psi(sigma)(r)\(2) Psi(sigma)(r)=0 for the GL wave function Psi(sigma)(r)drop(r \ n(1/2) \sigma), where a denotes the state of the condensed pairons, and n the density operator. The GL equation with alpha = -epsilon g(T) is shown to hold for all temperatures (T) below T-c, where epsilon(g) is the pairon energy gap. Its equilibrium solution yields that the condensed pairon density n(0)(T) = \Psi(sigma)(r)\(2) is proportional to epsilon(g)(T) The original GL T-dependence of the expansion parameters near T-c : alpha = -b(T-c-T), beta = constant is justified. With the assumption of h(0), a new formula for the penetration depth is obtained.