Progress of Theoretical Physics

Symmetry Properties of the Ring Heisenberg Hamiltonian and New Sum Rule for Its Energy Eigenvalues

K. Fujii

Published Sep 1, 1980• 2 citations

Abstract

Employing the method of a symmetry lowering (form the permutation group Pn to the dihedral group Cnv), the eigenstates of the ring Heisenberg Hamiltonian H=-2J ∑ni=1Si·Si+1 are classified not only by total spin S and its z-component M, but also by wave number k=2 πL/n, where L= ±0, ±1, ±2, ∼, ±p for n=2p+1 or 2p. By the characters of irreducible representation of Pn and Cnv, we express the sum of the energy eigenvalues at fixed S, M and L, together with the number of the eigenstates contained in this sum. With respect to the spin configurations in the ring, it is revealed that there exist invariant subspaces under all symmetry operations of Cnv in the space of the spin-functions specified by M. By connecting this fact with the Wigner coefficients, the eigenfunctions are easily obtained.

DOI: 10.1143/ptp.64.770

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