the spin-boson model in the RWA display the second-order QPT? Are some extreme conditions for model parameters required to realize the second-order QPT? As we know from the literature, the spin-boson Hamiltonian with the RWA is usually treated in sub-space with fixed total excitations $N$. In this way the second-order QPT is absolutely excluded. The constrained condition for the given sub-space should be relaxed to detect more physical phenomena.

In the sub-Ohmic spin-boson model without the RWA, the second-order QPT from the delocalized phase, where spin has the equal probability in the two states, to localized phase, in which spin prefers to stay in one of the two states, has been studied extensively [@Bulla; @QMC; @ED; @Zhang; @guo2012critical; @Frenzel; @CRduan]. Many advanced numerical approaches in quantum many-particle physics have been applied and extended to this model, such as the numerical renormalization group [@Bulla], quantum Monte Carlo simulations [QMC]{}, sparse polynomial space approach [@ED], exact diagonalization in terms of shift bosons [@Zhang], and various matrix product state approaches [@guo2012critical; @Frenzel]. Some analytical approaches based on the polaronic unitary transformation, also known as the Silbey-Harris ansatz  [@Silbey], have been also developed for this model [zheng,Chin,mD1,multi\_CS2,yzhao,Blunden,ZH,cao,HeShu1]{}. The single-coherent-states ansatz [@Silbey] was improved by 