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

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 beta and delta 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 tau = alpha - alpha (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(alpha = alpha (c) ) is found to be a GHF of I mu and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.