Uniqueness of Equilibrium State for A Quantum Spin System On a Graph

In this paper we study a quantum spin glass [3,17] as a quantum spin system on an infinite graph. We construct the global dynamics for the quantum spin system on an infinite graph with deterministic nearest neighbour type of interactions. As expected, at a fixed temperature T , the equilibrium state [6,7,12] satisfies the kms [14,18,12] condition and is invariant under the action of the time evolution group. Furthermore, it is shown that the equilibrium state is unique if additional assumptions are made regarding the structure of the graph. Such results have been proved in the past mainly for translation invariant interactions on lattices, see [1,2,8,13]. More recently Matsui has established the uniqueness of the the Gibbs States for quantum spin models on lattices based on the ergodicity of Markov semigroups through the purification of quantum Gibbs states [10]. In our case the nearest neighbour type of interaction we have considered is more general than similar type of interactions considered in some earlier works referred to above.