Exact ground state properties of antiferromagnetic Heisenberg spin rings with isotropic next neighbour interaction are presented for various numbers of spin sites and spin quantum numbers. Earlier work by Peierls, Marshall, Lieb, Schultz and Mattis focused on bipartite lattices and is not applicable to rings with an odd number of spins. With the help of exact diagonalization methods we find a more general systematic behaviour which for instance relates the number of spin sites and the individual spin quantum numbers to the degeneracy of the ground state. These numerical findings all comply with rigorous proofs in the cases where a general analysis could be carried out. Therefore it can be plausibly conjectured that the ascertained properties hold for ground states of arbitrary antiferromagnetic Heisenberg spin rings. These general rules help to explain the low temperature behaviour of recently synthesized spin rings.
We suggest a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings. This rule generalizes the well-known results of Marshall, Peierls, Lieb, Schultz, and Mattis for even rings. Our rule is confirmed by numerical investigations and rigorous proofs for special cases, including systems with a Haldane gap for N→ϱ. Implications for the total spin quantum number S of relative ground states are discussed as well as generalizations to the XXZ model.
For the diagonalization of the Hamilton matrix in the Heisenberg model relevant dimensions are determined depending on the applicable symmetries. Results are presented, both, by general formulae in closed form and by the respective numbers for a variety of special systems. In the case of cyclic symmetry, diagonalizations for Heisenberg spin rings are performed with the use of so-called magnon states. Analytically solvable cases of small spin rings are singled out and evaluated.
A quasi-particle theory for monatomic gases in equilibrium is formulated and evaluated to yield the exact virial contributions to the thermodynamic state functions in lowest order of the density. Van der Waals blocking has necessarily to be accounted for in occupation number statistics. The quasi-particle distribution function differs from the Wigner function by a bilinear functional thereof. The progress made so far is promising with respect to a corresponding version of kinetic theory.
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