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Progress of Theoretical Physics

The High Temparature Series for the Single-Band Hubbard Model in the Strong Correlation Limit. I

Katsunori Kubo, Masahito Tada

Published May 1, 1983• 23 citations
Publisher

Abstract

The high temperature series of the thermodynamic potential and the uniform susceptibility have been investigated for the single-band Hubbard model in the strong correlation limit. It is rigorously shown that the series absolutely converge both in the low density and the half-filled limit. Correct first terms have been obtained up to the ninth order in terms of the inverse temperature. Coefficients of the series are oscillating rapidly as functions of the occupation probability of electrons.

DOI
10.1143/ptp.69.1345

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The high temperature series of the thermodynamic potential and the uniform susceptibility have been investigated for the single-band Hubbard model in the strong correlation limit. It is rigorously shown that the series absolutely converge both in the low density and the half-filled limit. Correct first terms have been obtained up to the ninth order in terms of the inverse temperature. Coefficients of the series are oscillating rapidly as functions of the occupation probability of electrons.
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.
We employ the numerical linked-cluster expansion to study finite-temperature properties of the uniform cubic lattice Hubbard model in the thermodynamic limit for a wide range of interaction strengths and densities. We carry out the expansion to the 9th order and find that the convergence of the series extends to lower temperatures as the strength of the interaction increases, giving us access to regions of the parameter space that are difficult to reach by most other numerical methods. We study the precise trends in the specific heat, the double occupancy, and magnetic correlations at temperatures as low as 0.2 of the hopping amplitude in the strong-coupling regime. We show that in this regime, accurate estimates for transition temperatures to the N\'eel ordered phase, in agreement with the predicted asymptotic behavior, can be deduced from the low-temperature magnetic structure factor. We also find evidence for possible instability to the magnetically ordered phase away from, but close to, half filling. Our results have important implications for parametrizing fermionic systems in optical lattice experiments and for benchmarking other numerical methods.
In order to investigate the thermal properties (specific heat, magnetic susceptibility, entropy, internal energy, and some correlation functions) of the one-dimensional half-filled-band Hubbard model, we have studied linear chains and rings containing two to six atoms, by performing machine calculations. Supplementing the low-temperature behavior obtained from the exact solution for the infinite chain by Lieb and Wu, our results should be suggestive of the properties of the infinite system throughout the entire temperature domain. It is shown that when the ratio of the correlation energy UUU to the total width $\ensuremath{\Delta}$ of the band of single-particle excitations is larger than 1, the specific heat has two peaks. The high-temperature peak arises from the gradual metal-insulator transition (or the gradual formation of local moments), while the low-temperature peak is associated with the antiferromagnetic short-range ordering. When $\frac{U}{\ensuremath{\Delta}}$ becomes small, the two peaks merge into one. This picture is consistent with all other thermal properties, including correlation functions. The high-temperature properties are compared with the results predicted by Hubbard's approximate theory based on the truncation of the equations of motion of the Green's functions.
A new first-principles method for numerically calculating finite-temperature properties of quantum many-body systems is described. It is used to compute the thermodynamic properties and magnetic correlations of the Hubbard model, with negligible error at temperatures near the bandwidth or above. Results for the half-filled band and a simple cubic or one-dimensional lattice confirm the existence of a high-temperature peak in the specific heat which has been associated with a smooth but rapid change in conductivity. Correlations calculated for the non-half-filled band simple cubic Hubbard model suggest condensation, an entirely unexpected phenomenon whose existence is shown (by unrelated methods) in Paper II.