Self-adjoint symmetry operators connected with the magnetic Heisenberg ring

TL;DR

This paper explores the construction and decomposition of self-adjoint symmetry operators in the Heisenberg magnetic ring, providing algorithms for their calculation using group theory and computer algebra.

Contribution

It introduces methods to construct and decompose self-adjoint idempotent symmetry operators in the Heisenberg model, including algorithms utilizing discrete Fourier transforms.

Findings

Constructed self-adjoint idempotents from subgroup characters.

Identified unique self-adjoint idempotents in minimal right ideals.

Developed algorithms for decomposing self-adjoint idempotents.

Abstract

We consider symmetry operators a from the group ring C[S_N] which act on the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We investigate such symmetry operators a which are self-adjoint (in a sence defined in the paper) and which yield consequently observables of the Heisenberg model. We prove the following results: (i) One can construct a self-adjoint idempotent symmetry operator from every irreducible character of every subgroup of S_N. This leads to a big manifold of observables. In particular every commutation symmetry yields such an idempotent. (ii) The set of all generating idempotents of a minimal right ideal R of C[S_N] contains one and only one idempotent which ist self-adjoint. (iii) Every self-adjoint idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k which are also self-adjoint and pairwise orthogonal. We give a computer algorithm for the calculation of such decompositions. Furthermore we present 3 additional algorithms which are helpful for the calculation of self-adjoint operators by means of discrete Fourier transforms of S_N. In our investigations we use computer calculations by means of our Mathematica packages PERMS and HRing.