Statics and dynamics of quantum XY and Heisenberg systems on graphs

We consider the statics and dynamics of distinguishable spin-1/2 systems on an arbitrary graph G with N vertices. In particular, we consider systems of quantum spins evolving according to one of two Hamiltonians: (i) the XY Hamiltonian H-XY, which contains an XY interaction for every pair of spins connected by an edge in G; and (ii) the Heisenberg Hamiltonian H-Heis, which contains a Heisenberg interaction term for every pair of spins connected by an edge in G. We find that the action of the XY (respectively, Heisenberg) Hamiltonian on state space is equivalent to the action of the adjacency matrix (respectively, combinatorial Laplacian) of a sequence G(k), k=0,...,N of graphs derived from G (with G(1)=G). This equivalence of actions demonstrates that the dynamics of these two models is the same as the evolution of a free particle hopping on the graphs G(k). Thus we show how to replace the complicated dynamics of the original spin model with simpler dynamics on a more complicated geometry. A simple corollary of our approach allows us to write an explicit spectral decomposition of the XY model in a magnetic field on the path consisting of N vertices. We also use our approach to utilize results from spectral graph theory to solve new spin models: the XY model and Heisenberg model in a magnetic field on the complete graph.