Artifact of the phonon-induced localization by variational calculations in the spin-boson model

We present energy and free energy analyses on all variational schemes used in the spin-boson model at both T=0 and $T\neq0$. It is found that all the variational schemes have fail points, at where the variational schemes fail to provide a lower energy (or a lower free energy at $T\neq0$) than the displaced-oscillator ground state and therefore the variational ground state becomes unstable, which results in a transition from a variational ground state to a displaced oscillator ground state when the fail point is reached. Such transitions are always misidentied as crossover from a delocalized to localized phases in variational calculations, leading to an artifact of phonon-induced localization. Physics origin of the fail points and explanations for different transition behaviors with different spectral functions are found by studying the fail points of the variational schemes in the single mode case.