No abstract

The sequences of ground states in frustrated antiferromagnetic rings with odd number of local spins characterized by a single bond defect or by arbitrary uniform couplings to an additional spin located at the center are determined. The sequences provide firm constraints on the total ground-state quantum numbers, which are more stringent than those arising from the Lieb-Mattis theorem for bipartite quantum spin systems. Apart from their theoretical importance, they suggest the possibility of tailoring a given class of the molecular nanomagnets with desired ground-state properties by tuning the relevant couplings. In particular, they predict the spin S = 1/2 ground state for the centered rings composed of the half-integer spins with approximately uniform interactions. They confirm the applicability of the recent classification of spin frustration in both types of molecular nanomagnets. The classification is also discussed in the classical limit for the first class of the rings, providing a direct picture of frustration types. The Lieb-Mattis energy-level ordering and an analog of the Lande band, i.e., the energy spectra properties simplifying the characterization of the rings using the bulk magnetic or NMR measurements, are briefly discussed

Tle magnetic translation group was introduced as a set of operatorsHowever, these operators commute with the H a m i l t o n i a n f o r a n e l e c t r o n i n a p e r i o d i c p o t e n t i a l a n d a u n i f o r m m a g n e t i c field if the vector potential Α (the gauge) is chosen in a symmetric way. It is showed that a local gauge field AR (r) on a crystal lattice leads to operators, which commute with the Hamiltonian for any (global) gauge field _ Α (r). Such choice of the local gauge determines a factor system ω(11, R') = T(R)T(R')T(R + R') -1 , which depends on a global gauge only. Moreover, for any potential Α a commutator Τ(R)T(R')T(R) -1 T(R') -1 depends onlyon the magnetic field and not on the gauge.

Extensions of a direct product T of two cyclic groups Z Z N1 and Z Z N2 by an Abelian (gauge) group G with the trivial action of T on G are considered. All possible (nonequivalent) factor systems are determined using the Mac Lane method. Some of resulting groups describe magnetic translation groups. As examples extensions with G = U (1) and G = Z Z N are considered and discussed.