Quantum phase transitions of the spin-boson model within   multi-coherent-states
  A variational approach based on the multi-coherent-state ansatz with asymmetric parameters is employed to study the ground state of the spin-boson model. Without any artificial approximations except for the finite number of the coherent states, we find the robust Gaussian critical behavior in the whole sub-Ohmic bath regime. The converged critical coupling strength can be estimated with the $1/N$ scaling, where $N $ is the number of the coherent states. It is strongly demonstrated the breakdown of the well-known quantum-to-classical mapping for $1/2<s<1$. In addition, the entanglement entropy displays more steep jump around the critical points for the Ohmic bath than the sub-Ohmic bath. 
Liwei Duan, Shu He, and Qing-Hu Chen
cond-mat.quant-gas cond-mat.str-el