simply adding other coherent states on an equal footing [@mD1] and by superpositions of two degenerate single coherent states [@ZH], which are generally termed as the multi-coherent-states (MCS) ansatz. Actually, the MCS in the single-mode model was proposed much earlier by Ren and Chen [@tcs].

In this paper, we will extend the variational matrix product state (VMPS) approach  [@guo2012critical] to study the spin-boson model in the RWA for all values of the bath exponents. The MCS variational approach and exact diagonalization within truncated Hilbert space are also employed to provide independent checks in different regimes. The paper is organized as follows. In Sec. II, we briefly introduce the generalized spin-boson model. Some methodologies including the VMPS, the MCS variational approaches, and exact diagonalization in truncated Hilbert space are described. The rich phase transitions revealed by the VMPS method are presented in Sec. III, where the MCS variational approaches and the exact diagonalization are also applied to provide further evidence. The quantum criticality based on VMPS studies is also analyzed. Finally, conclusions are drawn in Sec. IV.

Generalized Model Hamiltonian and methodologies
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Based on Hamiltonian (\[spinboson\]), the generalized spin-boson Hamiltonian can be written as ($\hbar =1$)

$$\hat{H_{SB}} =\frac{\Delta }{2}\sigma _{z}+\frac{\epsilon 