  The existing studies of the temperature effect of the spin-boson model show
that the tunneling splitting will increase with temperature, no matter how weak
the couplings are between the bath and the spin. However, the small polaron
theory said that in the weak coupling and low temperature regime, the tunneling
is, in fact, dominated by the diagonal transitions whereas this diagonal
contribution to the tunneling should be faded away with increasing temperature.
Taking advantage of the analysis originated from the Feynman-Vernon's influence
functional theory, the influence on the tunneling by the phonon bath can be
expressed as a product of the effects of every single phonon mode, which can be
studied by numerical exact diagonalization. We find that, in the weak coupling
and low temperature regime, all the spin-single-mode systems exhibit the same
decreasing dependence of tunneling on increasing temperature. In turn, with the
conclusion of influence functional theory, the tunneling of the spin-boson
model should decrease and it is independent of the bath structure. In the
strong coupling regime, however, the temperature effect reverses from
suppressing to enhancing the tunneling with the increase of temperature.
Discrepancies between the old theories and the small polaron theory are also
explained.
