CONSTRUCTION OF ANALYTICALLY TRACTABLE MEAN-FIELD THEORIES FOR QUANTUM MODELS .2. SUSCEPTIBILITIES AND ENERGY BOUNDS FOR THE HUBBARD-MODEL

The mean-field theory for the Hubbard model constructed in the preceding paper is worked out in detail. The ferro- and antiferromagnetic susceptibilities are calculated explicitly. It is shown that the singularities suggested by the Stoner theory are spurious since two-particle correlation effects are neglected. At T = 0, n = 1 and large U the correct Curie-Weiss behavior is recovered. For U = infinity the antiferromagnetic susceptibility is found to vanish logarithmically at a generic density n = 2/3, corresponding to a half-filled lower Hubbard band. The magnetic U - n phase diagram is determined for T = 0; below a critical density n(c) = 0.77 the antiferromagnetic phase is found to be unstable for all U. Rigorous lower and upper bounds on the ground state energy of the Hubbard model in d = infinity are calculated; they almost coincide for n = 1 and thus determine the exact result very accurately. An investigation of the ground state spectral properties in the ordered phase at fixed density n(c) < n < 1 reveals an interesting transition scenario for increasing U, namely transitions from a paramagnetic metal to an antiferromagnetic metal and then to an antiferromagnetic insulator; further increase of U leads to a reversal of this sequence back to the paramagnetic metal. The theory described here generalizes the well-known, but ad hoc, alloy-analogy approximation and puts it on a firm, thermodynamically consistent basis.