Combinatorial aspects of nanoscale magnetism

TL;DR

This paper explores the mathematical and group-theoretical structures underlying finite nanoscale spin systems, providing formulas for symmetry-adapted bases and operator matrix elements, with applications to magnetic molecules.

Contribution

It introduces general formulas for symmetry-adapted bases and matrix elements in finite spin systems, highlighting combinatorial and group-theoretical structures involved.

Findings

Mathematical structures like orbits, stabilizers, double cosets are used.

Formulas for symmetry-adapted basis vectors are derived.

Applications to magnetic macromolecules are discussed.

Abstract

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.