KINEMATICS OF THE ONE-DIMENSIONAL FINITE HEISENBERG MAGNET WITH IMPURITIES

The kinematics of the one-dimensional finite Heisenberg magnetic ring with impurities is discussed in the light of the general recipe of Weyl. The cyclic group generating the ring, and the group of all its automorphisms, play the roles of the obvious and hidden symmetry, respectively. The hidden symmetry group imposes a property of the distribution of quantum states of a translationally invariant ensemble of magnets over the finite Brillouin zone. Namely, this distribution is constant on each orbit of the action of the hidden symmetry group. It is shown that the above property cannot be broken, neither by any chemical composition of impurities nor by change of their localization. Inhomogeneities of the distribution arise from irregular orbits of the action of the cyclic group on the set of all magnetic configurations of the ensemble. Irregular orbits are indicators of the reference system of 'absolute rest' in the crystal, corresponding to the centre of the Brillouin zone, whereas the case of exclusively regular orbits does not provide such a distinction, so that all quasimomenta enter the theory on the same footing. Size-dependent effects, like rarefied bands, are classified using the arithmetic structure of integers (prime numbers, socles and arithmetic exponents), which is more appropriate than the linear order in the ring of integers.