The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive 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, $J(\omega )\propto {\omega }^s$. Using an $ϵ$ 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 display mean-field transition behavior for $0<s<1/2$, controlled by a noninteracting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.

• Received 25 October 2004

DOI:https://doi.org/10.1103/PhysRevLett.94.070604