Extremal States of Multipartite Quantum Spin Systems

We consider two quantum spin models on $k$-partite graphs. The system is assumed to be invariant under all permutations within any of the $k$ sets. In the limit of infinitely many particles, the Gibbs state is described as a mixture of independent spins due to a version of the Quantum de Finetti theorem. First, we study a spin-1/2 antiferromagnetic Heisenberg model and identify its three phases. At low temperature all $k$ sets are magnetised but cancel each other, at medium temperature there is a net magnetic field and at high temperature there is no magnetisation. The magnetisation temperature is a solution to a $k$-degree polynomial. Second, we allow a general spin $s$ and identify all orthogonally invariant models which have frustration. The criterium for a frustration-free model is that the $k$-sets can be partitioned into two parts, where the only antiferromagnetic interaction is between the two parts.