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Physical Review Letters

Pair Theory of the Hubbard Hamiltonian

H. Barentzen

Published Oct 18, 1982• 7 citations
Publisher

Abstract

A pair theory of the Hubbard Hamiltonian is presented for the case of a half-filled band. The original Hamiltonian is reformulated exactly in terms of pair operators which satisfy boson commutation relations. The pair energies and wave functions are obtained explicitly. In the one-dimensional case, these reduce to the results derived by Lieb and Wu for $M=1$. The pair Hamiltonian offers a better starting point for perturbation treatments or variational calculations.

DOI
10.1103/physrevlett.49.1206

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Pair theory of the Hubbard Hamiltonian

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•Jan 1, 1998•8 citations
A pair theory of the Hubbard Hamiltonian is presented for the case of a half-filled band. The original Hamiltonian is reformulated exactly in terms of pair operators which satisfy boson commutation relations. The pair energies and wave functions are obtained explicitly. In the one-dimensional case, these reduce to the results derived by Lieb and Wu for M=1M=1M=1. The pair Hamiltonian offers a better starting point for perturbation treatments or variational calculations.
A detailed account is given of a recently developed pair theory of the Hubbard Hamiltonian for the case of a half-filled band. The resulting pair Hamiltonian is an exact reformulation of the original Hubbard Hamiltonian in terms of pair operators which satisfy elementary boson commutation relations. The Hamiltonian takes the familiar form of a sum of a quadratic part representing independent-pair energies and a quartic part representing pair interactions. The matrix elements entering the pair Hamiltonian are the pair energies and products of pair wave functions, which are formally given as the solutions of a certain eigenvalue equation. By the solving of the eigenvalue problem, explicit analytical expressions are obtained for both pair energies and wave functions for an arbitrary number of dimensions. For the case M=1M=1M=1, where MMM is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.
A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
The thermodynamic and correlation functions for one- and three- dimensional systems described by the half-filled band Hubbard Hamiltonians are evaluated in the pair-approximation of cluster variation method. The energy, specific heat, magnetization, susceptibility and various equilibrium correlation functions are computed as functions of temperature as well as the external magnetic field for different values of coulomb energies. Our solutions correspond to Bethe approximation in which the relevant pair-correlation functions are included in our formulation. It is observed that the chemical potential for the half-filled band is exactly half of the coulomb energy at all temperatures, irrespective of the value of the applied magnetic field.
PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model