Quantum derivation of Ginzburg-Landau equation. New formula for penetration depth

Abstract

The Cooper pair (pairon) field operator psi(r, t) changes in time, following Heisenberg's equation of motion. If the system Hamiltonian H contains the pairon kinetic energies ho, the condensation energy per pairon alpha(< 0) and the repulsive point-like potential beta delta(r(1)-r(2)), beta > 0, the evolution equation for psi is non-linear, from which we obtain the Ginzburg-Landau 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 complex order parameter Psi(sigma)(r) := < r / n(1/2) / sigma >, where sigma denotes the state of the condensed pairons, and n the pairon density operator. The total kinetic energy h(0) for "electron" (1) and "hole" (2) pairons is h(0) Psi(sigma)(r) = {1/2v(F)((1)) / -i (h) over bar del + 2eA(r) / + 1/2v(F)((2)) / -i (h) over bar del - 2eA(r) /} Psi(sigma)(r), where v(F)((J)) = (2 epsilon(F/)m(j))(1/2) are Fermi velocities, and A the vector potential. A new expression for the penetration depth lambda is obtained: lambda = c/e [p/4 pi n<INF>0</INF>(v<INF>F</INF><SUP>(2</SUP>) + v<INF>F</INF><SUP>(1</SUP>))]<SUP>1/2 </SUP>where p and n(0) are respectively the momentum and density of condensed pairons.