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physica status solidi (b)

Functional Approaches to Superconductivity within an Extended Attractive Hubbard Model

A. Teubel, E. Kolley, Winfried Kolley

Published Jan 1, 1990• 6 citations
Publisher

Abstract

Abstract On the basis of an extended attractive Hubbard model with nearest‐neighbour interaction of opposite spins two functional methods are adopted for deriving a self‐consistent equation for the superconducting order parameter. The functional‐integral formalism is exploited in saddle‐point approximation. The functional‐derivative technique using Schwinger source fields leads via the self‐energy in Hartree‐Fock‐Bogolubov approximation to a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s‐wave pairing.

DOI
10.1002/pssb.2221570139

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Abstract On the basis of an extended attractive Hubbard model with nearest‐neighbour interaction of opposite spins two functional methods are adopted for deriving a self‐consistent equation for the superconducting order parameter. The functional‐integral formalism is exploited in saddle‐point approximation. The functional‐derivative technique using Schwinger source fields leads via the self‐energy in Hartree‐Fock‐Bogolubov approximation to a nonlinear gap equation. The superconducting transition temperature is investigated for the case of s‐wave pairing.
On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.
Abstract The extended Hubbard model can host s-wave, d-wave and p-wave superconducting phases depending on the values of the on-site and nearest-neighbour interactions. Upon detailed examination of the free energy functional of the gap in this model, we show that these symmetries are often dependent on temperature. The critical points of this functional are constrained by symmetry and allow us to formulate stringent conditions on the temperature profile of the gap function, applicable to other models as well. We discuss the finite temperature phase diagram of the extended Hubbard model, and point out the existence of symmetry transitions below T c . Understanding the nature of these transitions is crucial to assessing the symmetry of unconventional superconductors.
To describe comprehensively the superconducting phase together with the other phases realized in the copper oxcides, the normal phase, the antiferro-magnetic and the spin-glass phases, an application of the constrained thermal Hartree-Fock-Bogoliubov (CTHFB) approximation is proposed. A set of the electron number- and the magnetization-constrained formalism is exemplified by employing the one-band Hubbard model. The CTHFB equation is applicable to the microscopic interaction whether it includes the phonon-mediated pairing interaction, or it provides a self-sustaining mechanism of the Cooper pair formation. A new type of the order parameter is introduced by extending the generalized Bogoliubov transformation to mix electron spin projections onto the quantization axis (i.e. S^z\hat{S}_{z}S^z​). This yields the lowered minimum of the grand potential, which corresponds to the physical solution violating time-reversal invariance. The gradient method is proposed to solve the CTHFB equation, the non-linear eigenvalue broblem with constraints.
The phase diagrams and superconducting properties of the extended Hubbard model with pair hopping interaction, i.e., the Penson-Kolb-Hubbard model are studied. The analysis of the model is performed for d-dimensional hypercubic lattices, including d=1d=1d=1 and $d=\ensuremath{\infty},$ by means of the (broken symmetry) Hartree-Fock approximations and, for $d=\ensuremath{\infty},$ by the slave-boson mean-field method. For d=1,d=1,d=1, at half filling the phase diagram is shown to consist of nine different phases including two superconducting states with center-of-mass momentum q=0q=0q=0 and q=Qq=Qq=Q $(\ensuremath{\eta}$ pairing), site and bond-located antiferromagnetic and charge-density wave states as well as three mixed phases with coexisting site and bond orderings. The stability range of the bond-type orderings shrinks with increasing lattice dimensionality d and for $d=\ensuremath{\infty}$ the corresponding diagram consists of four phases only, involving exclusively site-located orderings. Comparing the pair hopping model with the attractive Hubbard model we found in both cases gradual evolution from the BCS-like limit to the tightly bound pairs regime and a monotonic increase of the gap in the excitation spectrum with increasing coupling. However, the dynamics of electron pairs in both models is qualitatively different, which results in different dependences of condensation energies and critical temperatures on interaction parameters as well as in different electrodynamic properties, especially in a strong coupling regime.