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Journal of Physics Condensed Matter

Superconducting properties of the attractive Hubbard model in the slave-boson approach

M. Bąk, R. Micnas

Published Oct 12, 1998• 22 citations
Publisher

Abstract

The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.

DOI
10.1088/0953-8984/10/40/009

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The attractive Hubbard model for normal and superconducting ground states is examined for arbitrary electron concentration by the use of the slave-boson mean-field approximation (SBMFA) technique. This approach, at a saddle-point level, is equivalent to the Gutzwiller approximation. Several superfluid characteristics of the model are shown for hypercubic lattices of all dimensions, including , and a comparison with the Bardeen-Cooper-Schrieffer Hartree-Fock approximation (BCS-HFA) calculations and exact results is made. Our results show quantitative and qualitative corrections of the SBMFA to the HFA. The improvement of the SBMFA over the HFA diminishes with increasing lattice dimensionality. We have also evaluated the energy difference between the superconducting and the normal states; this is compared with available results for the superconducting critical temperatures in various dimensions and Uemura-type plots are obtained. The results confirm and substantiate the assertion of a continuous evolution of the superfluid properties of the model from the weak-coupling (BCS-like) to the strong-coupling (composite-boson superconductivity) limit.
The superfluid characteristics of the attractive Hubbard model are analyzed for any coupling |U| and arbitrary electron concentration (0n2) by means of the slave-boson mean-field method and also by the perturbative treatment of the strong-coupling limit. The slave boson method takes into account correlations of electrons and yields a reliable description of the crossover from BCS-type superconductivity to local pair (composite bosons) superconductivity with increasing |U|. The results for the ground state (the free energy, the gap in the excitation spectrum) and the electromagnetic characteristics (the critical magnetic field, the London penetration depth, the coherence length) are compared with those obtained by the Hartree-Fock approximation and by the self-consistent second-order perturbation theory in the weak-coupling limit as well as with those obtained using perturbational approaches in the strong-coupling limit. We show that the slave-boson method, in contrast to the Hartree-Fock approximation, gives credible results for all investigated quantities in the whole interaction range, interpolating smoothly between the BCS and local pair regimes. A comparison of theoretical predictions for our simple model with experimental data for various families of short-coherence-length superconductors suggests that the best agreement can be obtained for intermediate values of the local attraction. \textcopyright{} 1996 The American Physical Society.
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.
In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).
We study the negative-U Hubbard model for arbitrary values of the attractive interaction U and band filling by means of the slave-boson mean-field theory. We analyze both the normal and the superconducting states. We show that for the superconducting state our approach gives results similar to those obtained in the usual BCS mean field; only small renormalization of the bare parameters are obtained for U of the order of the bandwidth W. The energy difference between the normal and the superconducting states is maximum for U\ensuremath{\simeq}W. This energy difference is a measure of the superconducting critical temperature, which increases exponentially for small U and decreases as 1/U for large U. We discuss briefly the pair-breaking excitations.
We show that the extended attractive Hubbard model on a square lattice allows for a variety of superconducting phases, including exotic mixed-symmetry phases with [Formula: see text] and [Formula: see text] symmetries, and a novel [Formula: see text] state. The calculations are performed within the Hartree-Fock Bardeen-Cooper-Schrieffer framework. The ground states of the mean-field Hamiltonian are obtained via a minimization scheme that relaxes the symmetry constraints on the superconducting solutions, hence allowing for a mixing of s-, p- and d-wave order parameters. The results are obtained within the assumption of uniform-density states. Our results show that extended attractive Hubbard model can serve as an effective model for investigating properties of exotic superconductors.
We extend previous studies of the (BCS) canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard model and a simpler reduced Hamiltonian. Using direct diagonalization, exact one- and two-dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one-dimensional findings, the BCS approach is an excellent approximation for the Hubbard ground state energy but not the energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wave functions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. Even for weak coupling pronounced differences are found between the BCS and Hubbard on-site correlators. Significantly, in the extreme strong coupling limit the Hubbard model exhibits no dimensional crossover, or density induced transitions, to long-range order in two dimensions. However, for the reduced Hamiltonian the BCS approach is an excellent approximation for all observables in both one and two dimensions.