title: 'Quantum phase transitions in the spin-boson model without the counterrotating terms'
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Introduction
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The spin-boson model describes a qubit (two-level system) coupled to a quantum environment represented by a continuous bath of bosonic modes. It is a paradigmatic model in many fields, ranging from quantum optics [@Scully]to condensed matter physics [@Leggett] to open quantum systems [Breuer,weiss1]{}. The Hamiltonian is given by $$H_{SB}=\frac{\Delta }{2}\sigma _{z}+\sum_{k}\omega _{k}a_{k}^{\dag
}a_{k}+\sum_{k}g_{k}\left( a_{k}^{\dag }+a_{k}\right) \sigma _{x},
\label{spinboson}$$where $\sigma _{i}$ ($i=x,y,z$) is the Pauli matrices, $\Delta $ is the qubit frequency, $a_{k}$ ($a_{k}^{\dag }$) is the bosonic annihilation (creation) operator which can annihilate (create) a boson with frequency $%
\omega _{k}$, and $g_{k}$ denotes the coupling strength between the qubit and the bosonic bath, which is usually characterized by the power-law spectral density $J(\omega )$, $$J(\omega )=\pi \sum_{k}g_{k}^{2}\delta (\omega -\omega _{k})=2\pi \alpha
\omega _{c}^{1-s}\omega ^{s}\Theta (\omega _{c}-\omega ),$$where $\alpha $ are a dimensionless coupling constant, $\omega _{c}$ is the cutoff frequency, and $\Theta (\omega _{c}-\omega )$ is the Heaviside step function. The bath exponent $s$ classifies the reservoir into super-Ohmic $%
\left( s>1\right) $, Ohmic $\left( s=1\right) $, and sub-Ohmic $\left(
s<1\right) $ types. In many theoretical studies, due to the weak coupling strength in 