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arXiv:cond-mat/9609275 · Physical review. B, Condensed matter

Evaluation of the BCS approximation for the attractive Hubbard model in one dimension

F. Marsiglio

Published Jan 1, 1997• 39 citations
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Abstract

The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.

DOI
10.1103/physrevb.55.575

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The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.
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We test the canonical BCS wave functions for fixed number of electrons for the attractive Hubbard model. We present results in one dimension for various chain lengths, electron densities, and coupling strengths. The ground-state energy and energy gap to the first excited state are compared with the exact solutions obtained by the Bethe ansatz as well as with the results from the conventional grand-canonical BCS approximations. While the canonical and grand-canonical BCS results are both in very good agreement with the exact results for the ground state energies, improvements due to conserving the electron number in finite systems are manifest in the energy gap. As the system size is increased, the canonical results converge to the grand-canonical ones. The ``parity'' effect that arises from the number parity (even or odd) of electrons are studied with our canonical scheme.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t.U/t.U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/tU/tU/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/tU/tU/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/tU/tU/t and n.
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.
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).
The canonical BCS wave function is tested for the attractive Hubbard model. Results are presented for one dimension, and are compared with the exact solutions by the Bethe ansatz and the results from the conventional grand canonical BCS approximation, for various chain lengths, electron densities, and coupling strengths. While the exact ground state energies are reproduced very well both by the canonical and grand canonical BCS approximations, the canonical method significantly improves the energy gaps for small systems and weak coupling. The ``parity'' effect emerges naturally in our canonical results, as a positive gap for an even number of electrons and a negative ``gap'' for an odd number of electrons. Furthermore, we find a ``super-even'' effect: the energy gap oscillates as a function of even electron number, depending on whether the number of electrons is 4m4m4m or 4m+2(m4m+2 (m4m+2(m integer). Such oscillations as a function of electron number should be observable with tunneling measurements in ultrasmall metallic grains.