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Journal of Physics C Solid State Physics

Thermodynamics of the Hubbard chain: temperature extension of ground-state renormalisation for fermions

Carlo Vanderzande, Attilio L. Stella

Published Apr 30, 1984• 6 citations
Publisher

Abstract

A recently proposed ground-state renormalisation-group approach to the Hubbard model is generalised to finite temperatures. This new method provides a global and consistent description of the thermodynamics of the model at all temperatures. Application to the Hubbard chain with a half-filled band gives values for various thermodynamic functions which compare remarkably well with existing exact or numerical data, over all temperature ranges. The difficulties posed by Fermi-Dirac statistics in the set-up of the transformation are discussed to some extent and it is shown that these do not prevent one, at least in principle, from extending the approach to higher dimensions.

DOI
10.1088/0022-3719/17/12/007

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A recently proposed ground-state renormalisation-group approach to the Hubbard model is generalised to finite temperatures. This new method provides a global and consistent description of the thermodynamics of the model at all temperatures. Application to the Hubbard chain with a half-filled band gives values for various thermodynamic functions which compare remarkably well with existing exact or numerical data, over all temperature ranges. The difficulties posed by Fermi-Dirac statistics in the set-up of the transformation are discussed to some extent and it is shown that these do not prevent one, at least in principle, from extending the approach to higher dimensions.
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.
The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.
A new real-space renormalisation group method is developed and tested for the case of the Anderson-Hubbard model. The development is concentrated on obtaining and subsequent analysis of one-dimensional recurrence relations which take into account the effects of interaction and temperature. The results are presented in a form that naturally permits the extension of the given treatment to the case of higher dimensions and off-diagonal bond percolation type of disorder. Although the method itself is not limited to low temperatures, actual computations were made for the case of low temperatures and weak-electron-electron coupling. For this case, the scaling expressions obtained for the ground-state energy and the energy gap are in satisfactory agreement with available exact results.
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.