Coherent--incoherent transition in the spin--boson model with a super-ohmic bath

The temperature effect on tunnelling splitting in the spin--boson model with a super-ohmic bath is studied by the small polaron theory. The coherent--incoherent transition temperature is calculated and its dependence on dissipation strength and bare tunnelling splitting is analysed. In additional to the traditional transition point described in textbooks, a new kind of transition is found in the low dissipation region, showing different temperature dependence in the transition. The relation to the corresponding transition in the polaron--phonon system is also discussed.