Scaling analysis in the numerical renormalization group study of the sub-Ohmic spin-boson model

The spin-boson model has nontrivial quantum phase transitions in the sub-Ohmic regime. For the bath spectra exponent 0 <= s < 1/2, the bosonic numerical renormalization group (BNRG) study of the exponents beta and delta are hampered by the boson-state truncation, which leads to artificial interacting exponents instead of the correct Gaussian ones. In this paper, guided by a mean-field calculation, we study the order-parameter function m(tau = alpha - alpha(c), epsilon, Delta) using BNRG. Scaling analysis with respect to the boson-state truncation N-b, the logarithmic discretization parameter Lambda, and the tunneling strength Delta are carried out. Truncation-induced multiple-power behaviors are observed close to the critical point, with artificial values of beta and delta. They cross over to classical behaviors with exponents beta = 1/2 and delta = 3 on the intermediate scales of tau and epsilon, respectively. We also find tau/Lambda(1-s) and epsilon/Lambda scalings in the function m(tau, epsilon, Lambda). The role of boson-state truncation as a scaling variable in the BNRG result for 0 <= s < 1/2 is identified and its interplay with the logarithmic discretization revealed. Relevance to the validity of quantum-to-classical mapping in other impurity models is discussed.