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Journal of Physics A Mathematical and General

High-temperature expansions for the strongly correlated Hubbard model in the limit of infinite dimension

Colin J. Thompson, Sam Y. S. Yang, Anthony John Guttmann, M F Sykes

Published Mar 21, 1991• 16 citations
Publisher

Abstract

High-temperature expansions for the strongly correlated Hubbard model in the limit of infinite spatial dimension d, with hopping scaled by d -1 2/, are developed for the Gibbs free energy and susceptibility to order T -10 for arbitrary values of the density. From Pade and other analysis the authors find no evidence of a phase transition at a finite critical temperature for any value of the density.

DOI
10.1088/0305-4470/24/6/019

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High-temperature expansions for the strongly correlated Hubbard model in the limit of infinite spatial dimension d, with hopping scaled by d -1 2/, are developed for the Gibbs free energy and susceptibility to order T -10 for arbitrary values of the density. From Pade and other analysis the authors find no evidence of a phase transition at a finite critical temperature for any value of the density.
High-temperature expansions for the specific heat and susceptibility of the strongly correlated Hubbard model are compared with corresponding quantities for spinless free fermions and free spins. It is found numerically that, when the temperature is high, the ratio of the specific heat of the strongly correlated Hubbard model to that of spinless free fermions approaches a particle-density-dependent constant. For fixed temperature, the susceptibility of the Hubbard model is less (greater) than that of free spins when the particle density is below (above) a certain threshold density of approximately 0.7. The ratio of the susceptibilities for the two systems, however, appears to be finite for any value of the particle density. Given that the free spin system does not have a ferromagnetic state at any finite temperature, it is concluded that the strongly correlated Hubbard model does not have a finite phase transition temperature. This conclusion is consistent with recent high-temperature expansion studies.
I observe that the high-temperature expansions of the U=\ensuremath{\infty} Hubbard model on two- and three-dimensional bipartite lattices are consistent with a strong separation of energy scales for spin and translational degrees of freedom at an electron density n near 8/11. The high-temperature expansion of the specific heat is nearly identical with the high-temperature expansion of spinless fermionic holes of density 1-n, while the expansion of the uniform magnetic susceptibility is nearly identical with the susceptibility of free spins. Previous finite-size calculations are also consistent with this result.
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.
We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.
The Falicov-Kimball model with correlated hopping is studied in the limit of infinite spatial dimensions.Dynamical susceptibilities are calculated using the generalization of the Dynamical Mean-Field Theory (DMFT), which is based on expansion over electron hopping around the single-site limit.A special case of semi-elliptic density of states and diagonal correlated hopping is considered numerically and the absence of phase transition for all temperatures except T = 0 is demonstrated, which corresponds to the known results.