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Physical review. B, Solid state

Low-density thermodynamics of the Hubbard chain

John C. Kimball, Mukesh Eswaran

Published Feb 15, 1974• 0 citations
Publisher

Abstract

All the three-electron wave functions and energy levels are obtained for the one-dimensional Hubbard model. The second and third virial coefficients for the system are calculated exactly. The virial expansion is converted into an electron density expansion which appears to converge much better than the virial series. The two-particle and three-particle contributions to the specific heat, entropy, intra-atomic correlation function, and magnetic susceptibility of the system are obtained and presented graphically.

DOI
10.1103/physrevb.9.1402

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All the three-electron wave functions and energy levels are obtained for the one-dimensional Hubbard model. The second and third virial coefficients for the system are calculated exactly. The virial expansion is converted into an electron density expansion which appears to converge much better than the virial series. The two-particle and three-particle contributions to the specific heat, entropy, intra-atomic correlation function, and magnetic susceptibility of the system are obtained and presented graphically.
A perturbation expansion for the thermodynamic potential is performed to the third order by HTSE method. Entropy, internal energy, specific heat, and magnetic susceptibility of the one-dimensional extended Hubbard model are computed as functions of temperature, electron density, and t, U, W. Thermodynamics can be discussed in two temperature regions. The high T-region corresponds to increase in number of double occupied states. The low T-region corresponds to the distribution of electron spin. The effect of t on thermodynamic properties is obvious.
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.
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.
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.
We present a calculation of the thermodynamic properties of a two-dimensional negative-U Hubbard model in the low-density limit. We show that the standard expansion of the thermodynamic potential in terms of particle-particle ladder diagrams evaluated in the undressed fermion propagator, leads to unphysical results like negative entropy and/or specific heat. We propose a phenomenological pair susceptibility that avoids these problems and calculate the entropy and specific heat for different values of the parameters.