Combinatorial aspects of nanoscale magnetism

Details Combinatorial aspects of nanoscale magnetism Combinatorial aspects of nanoscale magnetism

2002-12-14

arxiv.org/abs/cond-mat/0212341v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

7 pages, 1 figure, 3 tables, presented as an invited talk at 7th WigSym (College Park, MD, USA, Aug 2001), revTeX 4

Scientific paper

A finite spin system invariant under a symmetry group G is a very illustrative example of the finite group action on a set of mappings f:X->Y. In the case of spin systems X is a set of spin carriers and Y contains 2s+1 z-components -s<=m<=s for a given spin number s. Orbits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not lead to smaller eigenproblems, but they label states in a systematic way. Some combinatorial and group-theoretical structures, like double cosets and transitive representations, appear in a natural way. In such a system one can construct general formulas for vectors of symmetry adapted basis and matrix elements of operators commuting with the action of $G$ in the space of states. Considerations presented in this paper should be followed by detailed discussion of different symmetry groups (e.g. cyclic of dihedral ones) and optimal implementation of algorithms. The paradigmatic example, i.e. a finite spin system, can be useful in investigation of magnetic macromolecules.

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