Energy spectrum and quantum phase transition of the coupled single spin and an infinitely coordinated Ising chain

We consider a spin model, composed of a single spin, connected to an infinitely coordinated Ising chain. Theoretical models of this type arise in various fields of theoretical physics, such as theory of open systems, quantum control and quantum computations. In the thermodynamic limit of infinite chain, we map the chain Hamiltonian to the Hamiltonian of the Lipkin-Meshkov-Glik model and the system as a whole is described by a generalized Rabi Hamiltonian. Next the effective Hamiltonian is obtained using Foulton-Gouterman transformation. In thermodynamic limit we obtain the spectrum of the whole system and study the properties of the ground state quantum phase transition.