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 H 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 SN such as decompositions of ideals of the group ring C[SN ], idempotents of C[SN ], discrete Fourier transforms of SN , 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[SN ]. Let U (r1,r2) μ be a subspace of eigenvectors of a 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. 1. The model of the magnetic Heisenberg ring We summarize essential concepts of the one-dimensional (1D) spin-1/2 Heisenberg model of a magnetic ring (see e.g. [9, 10, 11]). We denote by N̂ the set N̂ := {1, . . . , N} of the integers 1, 2, . . . , N and by K̂ b N the set of all functions σ : N̂ → K̂. Definition 1.1 We assign to every function σ ∈ 2̂ b N the sequence |σ〉 := |σ(1), σ(2), . . . , σ(N)〉 of its values over the set N̂ . Then the Hilbert space of the 1D spin-1/2 Heisenberg ring with N sites is the set of all formal complex linear combinations H := LC { |σ〉 | σ ∈ 2̂ b N } of the |σ〉 in which the set B := { |σ〉 | σ ∈ 2̂ b N } of all |σ〉 is considered a set of linearly independent elements. We equip H with the scalar product 〈σ|σ〉 := δσ,σ′ and 〈u|v〉 := ∑ σ∈b2 b N uσvσ for u = ∑ σ∈b2 b N uσ|σ〉 , v = ∑ σ∈b2 b N vσ|σ〉 ∈ H . (1) Obviously, H is an Hilbert space of dimension dimH = 2 , in which B is an orthonormal basis. The states |σ〉 ∈ H describemagnetic configurations of the Heissenberg ring. σ(k) = 2 represents an up spin σ(k) =↑ and σ(k) = 1 a down spin σ(k) =↓ at site k. Definition 1.2 According to [9, 10], the Hamiltonians HF (HA) of a 1D spin1 2 Heisenberg ferromagnet (antiferromagnet) of N sites with periodic boundary conditions S N+1 := S α 1 , α ∈ {+,−, z}, are defined by