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

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.

9 pages, 7 figures

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• DMRG and periodic boundary conditions: a quantum information perspective
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• Kondo physics and dissipation: A numerical renormalization-group approach to Bose-Fermi Kondo models
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