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Progress of Theoretical Physics

Thermodynamic Properties of the One-Dimensional Half-Filled-Band Hubbard Model. II

Hiroyuki Shiba

Published Dec 1, 1972• 280 citations
Publisher

Abstract

In order to determine finite-temperature properties of the infinite one-dimensional Hubbard model for the half-filled band, finite systems such as chains with 2 to 5 atoms and rings with 4 and 6 atoms are studied with the application of the grand canonical ensemble. The obtained temperature dependence of the specific heat, the entropy, the magnetic susceptibility and some correlation functions clearly shows a rearrangement of the electronic state of the system with the increase of the strength of the Coulomb interaction relative to the transfer integral. In the light of our new results the functional integral approach recently proposed by Kimball and Schrieffer as well as Hubbard's approximate theory is examined.

DOI
10.1143/ptp.48.2171

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In order to determine finite-temperature properties of the infinite one-dimensional Hubbard model for the half-filled band, finite systems such as chains with 2 to 5 atoms and rings with 4 and 6 atoms are studied with the application of the grand canonical ensemble. The obtained temperature dependence of the specific heat, the entropy, the magnetic susceptibility and some correlation functions clearly shows a rearrangement of the electronic state of the system with the increase of the strength of the Coulomb interaction relative to the transfer integral. In the light of our new results the functional integral approach recently proposed by Kimball and Schrieffer as well as Hubbard's approximate theory is examined.
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With the aim of contributing to a better understanding of the finite-temperature properties of low-dimensional strongly correlated systems we study a one-dimensional half-filled-band extended Hubbard model with a term of exchange interaction between neighboring sites. We analyze the temperature dependence of the specific heat, internal energy, entropy, number of doubly occupied sites, magnetic susceptibility, spin-spin correlation functions, and also investigate a detailed diagram of magnetic correlations. We use the method of small-cluster exact diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined model exhibits a rich structure of magnetic properties due to the competition between the onsite Coulombian repulsion and the exchange interaction.
In order to contribute to understand well the finite-temperature properties of the Falicov-Kimball model, we study the temperature dependence of the specific heat, internal energy, entropy, and mean square fluctuation of the difference between the ion and electron numbers, and of the correlation functions of the one-dimensional half-filled-band Falicov-Kimball model. For this purpose we used the method of small-cluster exact-diagonalization calculations with the application of the grand canonical ensemble and of extrapolation techniques to the infinite chain. Our results show that the examined Falicov-Kimball model exhibits behavior of alternate electron-ion short-range ordering at low temperature, and gradual metal-insulator transition at high temperature, but, does not exhibit the crossing point of the specific heat curves, characteristic of the Hubbard model.
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.
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.