Spin-Boson Model with Diagonal and Off-Diagonal Coupling to Two   Independent Baths: Ground-State Phase Transition in the Deep Sub-Ohmic Regime
  We investigate a spin-boson model with two boson baths that are coupled to two perpendicular components of the spin by employing the density matrix renormalization group method with an optimized boson basis. It is revealed that in the deep sub-Ohmic regime there exists a novel second-order phase transition between two types of doubly degenerate states, which is reduced to one of the usual type for nonzero tunneling. In addition, it is found that expectation values of the spin components display jumps at the phase boundary in the absence of bias and tunneling. 
Yang Zhao, Yao Yao, Vladimir Chernyak, Yang Zhao
quant-ph cond-mat.str-el physics.chem-ph