Symmetry and the critical phase of the two-bath spin-boson model: Ground-state properties

A generalized trial wave function termed as the "multi-D1 Ansatz" has been developed to study the ground state of the spin-boson model with simultaneous diagonal and off-diagonal coupling in the sub-Ohmic regime. Ground-state properties including the energy and the spin polarization are investigated, and the results are consistent with those from the exact diagonalization and density matrix renormalization group approaches for the cases involving two oscillators and two baths described by a continuous spectral density function. Breakdown of the rotational and parity symmetries along the continuous quantum phase transition separating the localized phase from the critical phase has been uncovered. Moreover, the phase boundary is determined accurately with the corresponding symmetry parameters of the rotational and parity symmetries. A critical value of the spectral exponent s* = 0.49(1) is predicted in the weak coupling limit, which is in agreement with the mean-field prediction of 1/2, but much smaller than the earlier literature estimate of 0.75(1).