In a recent paper, Lai introduced a lattice-gas model. In this paper we generalize Lai s model, making application to various systems such as dilute Heisenberg magnets, higher-spin systems, and a lattice of SU(3) triplets. By a careful consideration of general thermodynamic stability, and by variational arguments, we demonstrate Lai s solution to be incorrect, and in turn produce the correct solution in this case and in other cases including higher-dimensional models. The remaining cases we treat in one dimension by Bethe s ansatz, reducing the problem to coupled integral equations. We locate the singularities of the ground-state energy in the phase plane; and we explicitly calculate the absolute-ground-state energy, excitations above the absolute ground state, and the first correction to the absolute ground state for small concentrations of impurities.

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We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e. , the Mott-Hubbard transition or density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.

The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show "critical" (or "exotic") behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ae which takes a value in the range [cte'", az'"]. The fractal dimensions f(ae) of these singularities in the Cantor set are calculated. This function f(ae) represents the global scaling properties of the Cantor-set spectrum.