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Physical review. B, Condensed matter

High-temperature expansion for the single-band Hubbard model

Jon Henderson, J. Oitmaa, M. C. B. Ashley

Published Sep 1, 1992• 38 citations
Publisher

Abstract

We study the single-band Hubbard model via high-temperature expansions to order (\ensuremath{\beta}t${)}^{9}$ for the standard two- and three-dimensional lattices. Series are derived for the grand potential and for the ferromagnetic susceptibility, for general values of the Coulomb repulsion parameter U and electron density n. The series are difficult to analyze with any degree of precision, but estimates are obtained for the ferromagnetic critical temperature.

DOI
10.1103/physrevb.46.6328

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We study the single-band Hubbard model via high-temperature expansions to order (\ensuremath{\beta}t)9{)}^{9})9 for the standard two- and three-dimensional lattices. Series are derived for the grand potential and for the ferromagnetic susceptibility, for general values of the Coulomb repulsion parameter U and electron density n. The series are difficult to analyze with any degree of precision, but estimates are obtained for the ferromagnetic critical temperature.
A method is described to obtain high-temperature series expansions for the grand potential and the nearest-neighbor correlation of the z-component of the spin in the Hubbard model. The method is applied to the square and simple cubic lattices. The series have been calculated up to eighth order in \ensuremath{\beta}t. The convergence of the series is discussed and a comparison with previous results in the literature is made. The present calculations contain an indication for the occurrence of a Nagaoka-like phase.
Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.
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 high-temperature expansion coefficients of the ordinary and the higher susceptibilities of the spin-1/2 nearest-neighbor Ising model are calculated exactly up to the 20th order for the general d-dimensional (hyper)simple-cubical lattices. These series are analyzed to study the dependence of critical parameters on the lattice dimensionality. Using the general d expression of the ordinary susceptibility, we have more than doubled the length of the existing series expansion of the critical temperature in powers of 1/d.