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International Journal of Quantum Chemistry

Relation between BCS Hamiltonian and Ginzburg-Landau equation

Shigeyuki Aono

Published Jan 1, 1998• 0 citations
Publisher

Abstract

It is intended to derive the Ginzburg–Landau (GL) equation directly from the Bardeen–Cooper–Schrieffer (BCS) Hamiltonian. By the use of the Hubbard–Stratonovitch transformation, the electron–electron interaction composed of four fermion operators is eliminated to yield an auxiliary boson field. This is an effective field in which electrons behave as if they were free. In applying the path integral method, the electron field is integrated out to remain the Lagrangian for this boson field. The symmetry breaking and the phase transition of the system described by this field are discussed, and it is shown that this boson field turns out to be the GL order parameter. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 693–703, 1998

DOI
10.1002/(sici)1097-461x(1998)69:6<693::aid-qua1>3.0.co;2-q

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It is intended to derive the Ginzburg–Landau (GL) equation directly from the Bardeen–Cooper–Schrieffer (BCS) Hamiltonian. By the use of the Hubbard–Stratonovitch transformation, the electron–electron interaction composed of four fermion operators is eliminated to yield an auxiliary boson field. This is an effective field in which electrons behave as if they were free. In applying the path integral method, the electron field is integrated out to remain the Lagrangian for this boson field. The symmetry breaking and the phase transition of the system described by this field are discussed, and it is shown that this boson field turns out to be the GL order parameter. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 693–703, 1998
In this chapter we consider the quantum statistical mechanics of identical nonrelativistic fermions by introducing the formalism of functional integration for the fermionic field. We then derive the equation of state of non-interacting and weakly-interacting repulsive fermions. We also discuss the pairing of electrons within the mean-field Bardeen-Cooper-Schrieffer (BCS) theory of low-temperature superconductivity and the Ginzburg-Landau theory, which is quite successful near the critical temperature. Finally, we analyze the BCS-BEC crossover that was experimentally achieved with ultracold fermionic atoms twenty years ago and also Josephson junctions, which are the building blocks of superconducting quantum computers.
The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature Tc{T}_{c}Tc​ on both sides of the crossover. In the weakly interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.
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.
A nonperturbative argument is used to derive a one-electron Hamiltonian whose eigenfunctions are in one-to-one correspondence with electron or hole quasiparticles of an N-electron system. Nonlocal exchange is included exactly, and self-energy terms due to electronic correlation are included at the level of approximation of a given N-electron wave function. When parametrized by quasiparticle occupation numbers, this one-electron Hamiltonian is the functional derivative of an exact N-electron energy functional, as in the Landau theory of interacting fermion systems. The present nonperturbative derivation extends the Landau theory to ordered or localized systems for which many-body perturbation theory is not directly applicable. Analysis of the Bardeen-Cooper-Schrieffer (BCS) reduced Hamiltonian for superconductors leads to a temperature-dependent energy gap and a transition temperature in good agreement with the BCS theory, in a formalism that conserves the number of electrons.
The application of functional integral methods and the Hubbard--Stratonovich transformation to the Hubbard model is discussed. For the attractive case, using a simple gauge transformation of the superconducting order parameter field, the effective action for the low-energy phase excitations is derived. Inclusion of the electromagnetic field is straightforward. For the repulsive case at half--filling use of a fluctuating spin reference frame allows a derivation of the effective nonlinear sigma model for arbitrary coupling strength U. Results for the spin-wave velocity and the spin stiffness are obtained for arbitrary U. Away from half--filling, at large U an effective action for doped fermions and the spins is found which is related to the slave fermion formulation of the correlated fermion problem. The formalism is generalized to include the full SU(2)xSU(2) symmetry of the Hubbard model. This allows the derivation of a Ginzburg-Landau like theory that includes both particle-hole and particle-particle excitations. (paper at the NATO workshop on the Hubbard model, San Sebastian, Spain, oct. 1993)
Abstract We have seen in the previous chapters that the concept of the macroscopic wave functionΨ(r), is central to understanding atomic Bose–Einstein condensates (BEC), superfluid He, and even superconductivity within the Ginzburg–Landau (GL) theory. But the connection between these ideas is not at all clear, since the atom condensates and He are bosonic systems, while super-conductivity is associated with the conduction electrons in metals which are fermions. The physical meaning of the GL order parameter was not explained until after 1957 when Bardeen Cooper and Schrieffer (BCS) published the first truly microscopic theory of superconductivity. Soon afterward the connection was finally established by Gor’ kov. He was able to show that, at least in the range of temperatures near Tc, the GL theory can indeed be derived from the BCS theory. Furthermore this provides a physical interpretation of the nature of the order parameter. Essentially it is describing a macroscopic wave function, or condensate, of Cooper pairs.