Criticality of the Mean-Field Spin-Boson Model: Boson State Truncation and Its Scaling Analysis

TL;DR

This paper investigates how boson state truncation affects the critical behavior in the mean-field spin-boson model, revealing a scaling form for magnetization and the impact of truncation on critical exponents.

Contribution

It provides a detailed scaling analysis of boson state truncation effects in the mean-field spin-boson model, clarifying its influence on critical exponents and universal functions.

Findings

Boson truncation is a relevant operator for 0<s<1/2, altering critical exponents.

Magnetization near criticality follows a generalized homogeneous function with double-power form.

Truncation effects vanish for s>1/2, indicating different regimes of influence.

Abstract

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 $\epsilon$ and $x$. In the regime $s>1/2$, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.