Single mode approximation for sub-Ohmic spin-boson model: adiabatic limit and critical properties

In this work, we study the quantum phase transition in the sub-Ohmic spin-boson model using a single-mode approximation. It combines the rotating wave transformation and the transformations used in the numerical renormalization group (NRG). Analytical results for the critical coupling strength α <Subscript> c </Subscript>, the magnetic susceptibility χ(T), and the spin-spin correlation function C(ω) at finite temperatures are obtained and further confirmed by numerical results. We obtain the same α <Subscript> c </Subscript> as the mean-field approximation. The critical exponents are classical: β = 1/2, δ = 3, γ = 1, x = 1/2, y <Subscript> t </Subscript> <Superscript>∗</Superscript>=1/2, in agreement with the spin-boson model in 0 > s > 1/2 regime. C(ω) has nontrivial behavior reflecting coherent oscillation with temperature dependent damping effects due to the environment. We point out that the original NRG has a problem with the crossover temperature T <Superscript>∗</Superscript>, and propose a chain Hamiltonian possibly suitable for implementing NRG without boson state truncation error. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013