Single Mode Approximation for sub-Ohmic Spin-Boson Model: Adiabatic Limit and Critical Properties

In this work, the quantum phase transition in the sub-Ohmic spin-boson model is studied using a single-mode approximation, by combining the rotating wave transformation and the transformations used in the numerical renormalization group (NRG). Analytical results for the critical coupling strength $\alpha_c$, the magnetic susceptibility $\chi(T)$, and the spin-spin correlation function $C(\omega)$ at finite temperatures are obtained and further confirmed by numerical results. We obtain the same $\alpha_c$ as the mean-field approximation. The critical exponents are classical: $\beta=1/2$, $\delta=3$, $\gamma=1$, $x=1/2$, $y_t^{*}=1/2$, in agreement with the spin-boson model in $0<s<1/2$ regime. $C(\omega)$ has nontrivial behavior reflecting coherent oscillation with temperature dependent damping effects due to the environment. We point out the original NRG has problem with the crossover temperature $T^{*}$, and propose a chain Hamiltonian possibly suitable for implementing NRG without boson state truncation error.