Scaling Analysis in the Numerical Renormalization Group Study of the Sub-Ohmic Spin-Boson Model

Abstract

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 $\tau /{\Delta }^{1-s}$ and $\epsilon /\Delta$ scalings in the function $m(\tau ,\epsilon ,\Delta )$. 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.

15 pages, 14 figures, published version