Abstract 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 $J(\omega)=\pi\sum_k c_k^2=\dfrac{\pi}{2}\alpha \omega^{ s}\omega_{\rm c}^{ 1-s}$, 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 $s<s_{\rm c}$ while a continuous transition always occurs when $s=s_{\rm c}$. At $T=0$, we have $s_{\rm c}=1$, while for $T\not=0$, $s_{\rm c}=2$ which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.
Received: 14 January 2008
Revised: 27 February 2008
Accepted manuscript online:
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