The Antiferromagnetic Heisenberg Model on Clusters with Icosahedral Symmetry

TL;DR

This study investigates the antiferromagnetic Heisenberg model on icosahedral clusters, revealing frustration effects, singlet excitations, and magnetization discontinuities, with implications for larger symmetric clusters.

Contribution

The paper provides a full diagonalization analysis of the Heisenberg model on dodecahedron and icosahedron clusters, highlighting frustration effects and magnetic properties with symmetry considerations.

Findings

Presence of singlet excited states indicating frustration

Doubly-peaked specific heat as a function of temperature

Discontinuities and plateaux in magnetization curves

Abstract

The antiferromagnetic Heisenberg model is considered for spins $s_{i}={1/2}$ located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group $I_{h}$. Taking into account the permutational and spin inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low energy spectrum, where the first excited states are singlets. Frustration also results in a doubly-peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. The calculation is also extended for $s_{i}=1$ where both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.