Ground state properties of sub-Ohmic spin-boson model with simultaneous   diagonal and off-diagonal coupling
  By employing the variational approach, density matrix renormalization group (DMRG), exact diagonalization as well as symmetry and mean-field analyses, the ground state properties of the two-bath spin boson model with simultaneous diagonal and off-diagonal coupling are systematically studied in the sub-Ohmic regime. A novel quantum phase transition from a doubly degenerate "localized phase" to the other doubly degenerate "delocalized phase" is uncovered. Via the multi-D1 ansatz as the variational wave function, transition points are determined accurately, consistent with the results from DMRG and exact diagonalization. An effective spatial dimension $d_{eff} = 2.37(6)$ is then estimated, which is found to be compatible with the mean-field prediction. Furthermore, the quantum phase transition is inferred to be of first order for the baths described by a continuous spectral density function. In the case of single mode, however, the transition is softened. 
Nengji Zhou, Lipeng Chen, Yang Zhao, Dima Mozyrsky, Vladimir Chernyak   and Yang Zhao
cond-mat.stat-mech cond-mat.quant-gas cond-mat.str-el quant-ph