Symmetry classes connected with the magnetic Heisenberg ring

It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8].

We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[S_N] such as decompositions of ideals of the group ring C[S_N], idempotents of C[S_N], discrete Fourier transforms of S_N, Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U, respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[S_N].

Let U^(r1,r2)_μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight (r₁,r₂). If one determines the smallest symmetry class for every v U^(r1,r2)_μ then one can observe jumps of the symmetry behaviour. For "generic" v U^(r1,r2)_μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U^(r1,r2)_μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely.

In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.