A note on localized transition in the spin-boson model by variational calculation

We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as J(\omega)=\pi\sum_k c_k^2=\dfrac\pi2\alpha \omega^ s\omega_\rm c^ 1-s, there exists a universal s_\rm c for both kinds of variational schemes, the localized transition happens only for s\le s_\rm c, moreover, the localized transition is discontinuous for s