Eur. Phys. J. B (2013) 86: 141
https://doi.org/10.1140/epjb/e2013-31100-y

Regular Article

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

Department of Physics, Renmin University of China, Beijing 100872, P.R. China

^a e-mail: 36889309@qq.com

Received: 6 December 2012
Received in final form: 11 February 2013
Published online: 8 April 2013

Abstract

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_c, 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_c as the mean-field approximation. The critical exponents are classical: β = 1/2, δ = 3, γ = 1, x = 1/2, y_t^∗ = 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^∗, and propose a chain Hamiltonian possibly suitable for implementing NRG without boson state truncation error.