Antiferromagnetic Heisenberg Model on the Icosahedron: Influence of Connectivity and the Transition from the Classical to the Quantum Limit
The antiferromagnetic Heisenberg model on the icosahedron, which consists of 20 edge-sharing triangles and belongs to the icosahedral $I_h$ symmetry group, presents unconventional properties at the classical and quantum level. These originate in the frustrated nature of the interactions between the spins. For classical spins the magnetization is discontinuous in a magnetic field. Here we examine the importance of the connectivity of the icosahedron for the appearance of the magnetization discontinuity, and also investigate the transition from the classical to the quantum limit. The influence of connectivity on the magnetic properties is revealed by considering the cluster as being made up of a closed strip of a triangular lattice with two additional spins attached. The classical magnetization discontinuity is shown to evolve continuously from the discontinuity effected by these two spins when they are uncoupled to the cluster. In the second part the transition from the classical to the quantum limit is examined by focusing on the low energy spectrum taking fully into account the spatial and the spin symmetry of the model in the characterization of the states. A symmetry analysis of the highly degenerate due to the connectivity lowest energy classical manifold identifies as its direct fingerprint the low energy quantum states for spin magnitude as low as $s=1$, with the latter following a tower of states behavior which relates to the icosahedron having a structure reminiscent of a depleted triangular lattice. The classical character of the AHM for small $s$ is also detected on the ground state energy and correlation functions. On the other hand the classical magnetization discontinuity in a field eventually disappears for small $s$, after a weak reentrant behavior.