An Exactly Soluble 2-Dimensional Quantum-Mechanical Heisenberg-Model - Quantum Fluctuations Versus Magnetic Order

Using some simple geometries composed solely of interconnecting 'diamonds', we study the competition between long-range magnetic order and quantum fluctuations. Since a four-atom 'diamond' is formed from two-edge sharing triangles, a 'diamond' is topologically frustrated, and hence magnetic order is energetically less favourable than usual. The classical limit yields a ferrimagnetic state with equally large ferromagnetic and antiferromagnetic moments. Symptomatic of the topological problems, there are some zero energy 'spin wave' modes in the classical limit. For low-spin systems these low energy modes become excited in the ground state, which has none of the properties predicted by the classical solution. For spin 1/2, the ground state has only short-range correlations, a broken translational symmetry, and a gap to localised spin-1/2 excitations, which also have a topological quantum number.