  The spin-boson model is a paradigm for studying decoherence, relaxation,
entanglement and other effects that arise in a quantum system coupled to
environmental degrees of freedom. At zero temperature, a
localization-delocalization phase transition is known to exist in the sub-Ohmic
regime, where the standard density matrix renormalization group algorithm is
inadequate due to the divergence in the number of low-frequency modes. This
limitation is circumvented in this work by symmetrically optimizing the phonon
basis and introducing an order parameter accounting for the U(1) symmetry for a
two-bath spin-boson model, by which we are able to determine the classification
and criticality of the phase transition explicitly. Compared with variational
results, the critical phase is characterized by spontaneous vanishing of boson
displacements in both the baths, resulting in an accurate phase diagram with
three model parameters.
