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Journal of Physics Condensed Matter

From a Hubbard dimer to a Hubbard model in d dimensions: a path integral solution

Arkady L. Kholodenko

Published Jan 9, 1989• 2 citations
Publisher

Abstract

A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.

DOI
10.1088/0953-8984/1/1/008

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A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.
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 non-zero 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.
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We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.
The one-dimensional Hubbard model is studied for general electron densities and large U. On the basis of the coupled integral equations of Lieb and Wu exact asymptotic expansions are derived to fifth order in ${U}^{\mathrm{\ensuremath{-}}1}$ for the ground-state energy, the kinetic energy, and the number of doubly occupied sites. A closed-form expression for the distribution functions of charge and spin degrees of freedom is proposed which yields the same large-U expansion as that derived directly from the Lieb-Wu equations. Our analytical results are in excellent agreement with previous numerical studies.
The gap equation at zero temperature for the half-filled Hubbard chain obtained by Lieb and Wu is rederived with use of the formalism of Yang and Yang and then the results are extended to the case of finite temperatures. For the values of the coupling constant c in the range 0<c\ensuremath{\le}1.5 the gap \ensuremath{\Delta}(T) exhibits BCS-like behavior \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​)1/2{)}^{1/2})1/2 while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​) with ${\ensuremath{\Delta}}_{0}$ being the zero-temperature result of Lieb and Wu for the gap and Tc{T}_{c}Tc​ some critical transition temperature.
The ``string'' structure of one-dimensional Hubbard model eigenstates is studied numerically and is used to suggest the validity of thermodynamic Bethe-ansatz equations (TBAE). We furthermore solve TBAE in a strong-coupling expansion series form and obtain the thermodynamic potential which agrees with the known high-temperature expansion series. We also calculate various thermodynamic quantities using our solution and provide some additional features of the strongly correlated one-dimensional Hubbard model.