Eur. Phys. J. B 78, 127-135 (2010)
https://doi.org/10.1140/epjb/e2010-10426-0

Criticality of the mean-field spin-boson model: boson state truncation and its scaling analysis

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

Corresponding authors: ^a phy.hyh@gmail.com - ^b nhtong@ruc.edu.cn

Received: 28 May 2010
Revised: 6 September 2010
Published online: 28 October 2010

Abstract

The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents β and δ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states N_b. We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in 0 < s < 1/2 and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables τ = α – α_c and x = 1/N_b. The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, m(α = α_c) is found to be a GHF of ϵ and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.