Ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2(σN+1 ≡ σ1, σN+2 ≡ σ2) are studied for both signs of J and −1 ≤ α ≤ 1 to gain insight into the stability of the ground state with nearest-neighbor interactions only (α = 0) in the presence of the next-nearest-neighbor interaction. Short chains of up to 8 particles have been exactly studied. For J > 0, the ground state for even N belongs always to spin zero, but its symmetry changes for certain values of α. For J < 0, the ground state belongs either to the highest spin (ferromagnetic state) or to the lowest spin and so to zero for even N. The trend of the results suggests that these facts are true for arbitrary N and that the critical value of α is probably zero. Upper and lower bounds to the ground-state energy per spin of the above Hamiltonian are obtained. Such bounds can also be obtained for the square lattice with the nearest- as well as the next-nearest-neighbor interaction.

Continuing our work on the ground-state properties of the Hamiltonian H=12J ∑ i=1N σi·σi+1 + 12Jα ∑ i=1N σi·σi+2, −1≤α≤1,we have completed the study of 10 spins. The results of short-chain calculations provide better upper and lower bounds of the ground-state energy per particle as N → ∞, but no simple formula can be fitted to the data to get this limit for all α. For J > 0 and α = ½, however, this is exactly found to be −¾J. Some upper and lower bounds for the free energy are also derived.

No abstract

No abstract

A soluble model of a gas of independent fermions in the presence of an attractive localized potential \v(t) is considered. It is shown that the properties of the system as a whole are smooth (analytic) functions of X, even at those values of X where new single-particle bound states appear. Thus for the system as a whole, the transition from "screening" to "binding" is smooth and the concept of a bound state cannot be given a sharp meaning. Implications for certain metals and alloys are discussed.