  The effective theories for many quantum phase transitions can be mapped onto
those of classical transitions. Here we show that such a mapping fails for the
sub-ohmic spin-boson model which describes a two-level system coupled to a
bosonic bath with power-law spectral density, J(omega) ~ omega^s. Using an
epsilon expansion we prove that this model has a quantum transition controlled
by an interacting fixed point at small s, and support this by numerical
calculations. In contrast, the corresponding classical long-range Ising model
is known to have an upper-critical dimension at s = 1/2, with mean-field
transition behavior controlled by a non-interacting fixed point for 0 < s <
1/2. The failure of the quantum-classical mapping is argued to arise from the
long-ranged interaction in imaginary time in the quantum model.
