APPLICATION OF ALGEBRAIC COMBINATORICS TO FINITE SPIN SYSTEMS

W. Florek
Institute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland

Received: June 11, 2001; in final form July 13, 2001

A finite spin system invariant under a symmetry group G is a very illustrative example of a finite group action on mappings f:X-->Y (X is a set of spin carriers, Y contains spin projections for a given spin number s). Orbits and stabilizers are used as additional indices of the symmetry adapted basis. Their mathematical nature does not decrease a dimension of a given eigenproblem, but they label states in a systematic way. It allows construction of general formulas for vectors of symmetry adapted basis and matrix elements of operators commuting with the action of G in the space of states. The special role is played by double cosets, since they label nonequivalent (from the symmetry point of view) matrix elements < x|H|y> for an operator H between Ising configurations |x>,|y>. Considerations presented in this paper should be followed by a detailed discussion of different symmetry groups (e.g.) cyclic or dihedral ones) and optimal implementation of algorithms. The paradigmatic example, {i.e.} a finite spin system, can be useful in investigations of magnetic macromolecules like Fe₆ or Mn₁₂acetate.

DOI: 10.12693/APhysPolA.100.3
PACS numbers: 02.20.Bb, 75.10.Jm, 75.75.+a