Symmetry classes connected with the magnetic Heisenberg ring

TL;DR

This paper explores the symmetry properties of eigenvectors in the quantum Heisenberg ring using advanced group representation theory, revealing how symmetry classes vary within eigenvector subspaces and developing algorithms for their explicit determination.

Contribution

It introduces an algorithm to compute explicit symmetry classes and stability subgroups for eigenvectors of the Heisenberg ring, connecting representation theory with quantum symmetry analysis.

Findings

Smallest symmetry classes can be explicitly calculated.

Eigenvector subspaces exhibit symmetry class jumps.

Computer algebra tools facilitate these symmetry computations.

Abstract

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 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 of the magnet. 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 be a subspace of eigenvectors of a a fixed eigenvalue \mu of the Hamiltonian with weight (r_1,r_2). If one determines the smallest symmetry class for every v in U then one can observe jumps of the symmetry behaviour. For ''generic'' v all smallest symmetry classes have the same maximal dimension d and structure. But U 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.