Clustering property for quantum Markov chains on the comb graph

Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an $XX$-Ising model over the comb graph $\mathbb{N}{⊳}_0\mathbb{Z}$. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.