Critical Dynamics of Spin Boson Model

In this work, we study the low-energy properties of the spin-boson model (SBM), which describes the dynamics of a 1/2 spin associated with a thermostat characterized by a power law spectral density, $f(\omega)\propto \omega^s$. The theoretical description is constructed in the Schwinger--Keldysh technique, based on the representation of the 1/2-spin by Majorana spinors. We study the critical dynamics of the system near the quantum phase transition by constructing and analyzing the system of renormalization group equations. Our theoretical approach is more universal, contrary to the one based on quantum-classical mapping, since it is applicable for $s\leq 1$. We show that in both the ohmic case $s=1$, and the weakly subohmic case $s\lesssim 1$, the second order quantum phase transition is observed in the model considered, and the critical magnetization exponent agrees with the exact hyperscaling result, $1/\delta=(1-s)/(1+s)$. Furthermore, we obtain the dependence of the critical value of the spin-boson coupling constant on the temperature of the ohmic bosonic thermal bath.