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

TL;DR

This study investigates the effects of boson state truncation on the numerical renormalization group analysis of the sub-Ohmic spin-boson model, revealing how artificial exponents emerge and crossover to classical behavior near criticality.

Contribution

The paper introduces a detailed scaling analysis of boson state truncation effects in BNRG for the sub-Ohmic spin-boson model, clarifying the origin of artificial exponents and their crossover to classical values.

Findings

Artificial exponents due to truncation cross over to classical exponents

Identification of boson truncation as a key scaling variable

Discovery of specific scaling relations involving $ au$, $ abla$, and $ ilde{ au}$

Abstract

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 $\beta$ 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 $0 \leqslant s<1/2$ 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.