It is proposed that some of the formulae derived from the Gibbs-Duhem relation may serve as a simple test of consistency for approximation employed in solving many-body problems. When two of the intensive parameters are partly developed in terms of normalized extensive parameters, such as density, simple connections among the expansion coefficients follow directly from the Gibbs-Duhem relation. As an example chemical potential and external magnetic field are chosen as the two intensive parameters, to apply the relation to the Hubbard model. It is shown that the Hartree-Fock approximation surely satisfies the relation, while Hubbard's first approximation does not. Thus the latter approximation is thermodynamically inconsistent, and its validity should be seriously questioned.
The students interested in the study of strongly correlated electron systems normally feel, in the beginning of their studies, a great difficulty in understanding the complex correlations of these systems. We consider that for these students it is very important to have texts that analyze pedagogically the theoretical research methods of the subject. We show how to determine the thermodynamic properties of the Hubbard model for the two-dimensional square lattice, utilizing the Hartree-Fock approximation. All the procedures and peculiarities which lead to the determination of the thermodynamic properties and to the comprehension of the meaning of the Hartree-Fock approximation applied to models of strongly correlated electron systems are pedagogically presented in detail. We obtain the magnetization, the magnetic susceptibility, the internal energy, the specific heat, and also a magnetic phase diagram of the studied system. Our approach clearly shows that the use of the Hartree-Fock approximation, in this case, is very simple.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard
I result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.
Models of strongly interacting Fermi systems are sufficiently intractable that simple mean-field approximations (typically Hartree-Fock) are often employed to obtain qualitative insight into their properties, and also to set parameter values by comparing with band-structure calculations. However, if the correlations are accounted for with use of more sophisticated (but still mean-field-like) methods, qualitative changes in charge densities and magnetic properties can result. We compare four ``mean-field'' approximations for an extended Hubbard model and show that they yield quite different results. Implications for the choice of model for high-temperature superconductors are discussed.
The atomic limit of the Hubbard model is a simple single-site problem which can be solved exactly, and all one- and two-particle Green's functions can be obtained analytically. These solutions can thus serve as a means of critiquing the success of various approximate theories which might be applied to the full Hubbard model. In particular, we have examined the
T-matrix approximation for the attractive Hubbard model in the atomic limit, which should give reasonable results at low electronic densities, if one can avoid the spurious phase transition that results when a fully non-self-consistent
T-matrix approximation is employed---previously we have shown that any level of self-consistency guarantees that this phase transition is correctly suppressed to zero temperature in two dimensions or less. Here, a minimally self-consistent
T-matrix approximation is shown to be successful in reproducing the exact results for the atomic limit, while fully self-consistent
T-matrix results do not agree with the known solutions. Of particular note is that the minimally self-consistent
T-matrix approximation reproduces not only one- and two-particle (static) thermodynamic quantities, but it also exactly reproduces the one-particle spectral function at low but nonzero temperatures. We also make a comparison to the two-particle self-consistent approach of Vilk and Tremblay, and find that the minimally self-consistent
T-matrix theory can give better results over a broader temperature range.