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Journal of Physics C Solid State Physics

Single-particle spectrum of the half-filled-band Hubbard model

Kurt Schönhammer

Published Oct 7, 1974• 8 citations
Publisher

Abstract

A solution for the Hubbard Hamiltonian in the half-filled-band case is presented, which takes into account the degeneracy of the ground state in the atomic limit and also correctly interpolates to the limit Coulomb energy U to 0. It is shown explicitly which approximations lead to Hubbard's improved decoupling scheme and other existing approximate solutions. Taking into account the antiferromagnetic character of the ground state the splitting of the two sub-bands in the case of large U is enhanced. The band gap is shown to decrease with increasing temperatures.

DOI
10.1088/0022-3719/7/19/012

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A solution for the Hubbard Hamiltonian in the half-filled-band case is presented, which takes into account the degeneracy of the ground state in the atomic limit and also correctly interpolates to the limit Coulomb energy U to 0. It is shown explicitly which approximations lead to Hubbard's improved decoupling scheme and other existing approximate solutions. Taking into account the antiferromagnetic character of the ground state the splitting of the two sub-bands in the case of large U is enhanced. The band gap is shown to decrease with increasing temperatures.
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The finite-cell-scaling method is applied to the one-dimensional Hubbard model. In the half-filled-band case and at zero temperature, the numerical results obtained for the ground-state energy differ from the exact solution by approximately 1%. For the range 1<Ut<51<\frac{U}{t}<51<tU​<5, where UUU is the Coulomb interaction and ttt is the transfer integral, the value of the normalized gap $\frac{\ensuremath{\Delta}}{t}$ agrees well with the exact result while it differs significantly for 0.2<Ut<0.90.2<\frac{U}{t}<0.90.2<tU​<0.9. For the latter range, we show that the gap varies as ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}]$, where aaa and bbb are negative constants which depend on the size of the system, instead of ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})$ as in the exact solution.
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We supplement (and critically overview) the existing extensive analysis of antiferromagnetic solution for the Hubbard model with a detailed discussion of two specific features, namely (i) the evolution of the magnetic (Slater) gap (here renormalized by the electronic correlations) into the Mott-Hubbard or atomic gap, and (ii) a rather weak renormalization of the effective mass by the correlations in the half-filled-band case, which contrasts with that for the paramagnetic case. The mass remains strongly enhanced in the non-half-filled-band case. We also stress the difference between magnetic and non-magnetic contributions to the gap. These results are discussed within the slave boson approach in the saddle-point approximation, in which there appears a non-linear staggered molecular field due to the electronic correlations that leads to the appearance of the magnetic gap. They reproduce correctly the ground-state energy in the limit of strong correlations. A brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly-degenerate-band case with one electron per atom is also made. 1 1