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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 $0 \leqslant s<1/2$, the bosonic numerical renormalization group (BNRG) study of the exponents $β$ and $δ$ 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 $m(τ=α-α_c, ε, Δ)$ using BNRG. Scaling analysis with respect to the boson state truncation $N_{b}$, the logarithmic discretization parameter $Λ$, and the tunneling strength $Δ$ are carried out. Truncation-induced multiple-power behaviors are observed close to the critical point, with artificial values of $β$ and $δ$. They cross over to classical behaviors with exponents $β=1/2$ and $δ=3$ on the intermediate scales of $τ$ and $ε$, respectively. We also find $τ/Δ^{1-s}$ and $ε/Δ$ scalings in the function $m(τ, ε, Δ)$. The role of boson state truncation as a scaling variable in the BNRG result for $0 \leqslant s<1/2$ 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.