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abstract: 'We study the spin-boson model without the counterrotating terms by a numerically exact method based on variational matrix product states. Surprisingly, the second-order quantum phase transition (QPT) is observed for the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order QPTs can also appear before the critical points. With the decrease of the bath exponents, these first-order QPTs disappear successively, while the second-order QPT remains robust. The second-order QPT is further confirmed by multi-coherent-states variational studies, while the first-order QPT is corroborated with the exact diagonalization in the truncated Hilbert space. Extension to the Ohmic bath is also performed, and many first-order QPTs appear successively in a wide coupling regime, in contrast to previous findings. The previous pictures for many physical phenomena for the spin-boson model in the rotating-wave approximation have to be modified at least at the strong coupling.'
address: |
     $^{1}$ Department of Physics and Zhejiang Province Key Laboratory of Quantum Technology and Device, Zhejiang University, Hangzhou 310027, China\
    $^{2}$ Department of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610066, China\
    $^{3}$ Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 
author:
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    Yan-Zhi Wang$^{1}$, Shu He$^{2}$, Liwei Duan$^{1}$, and Qing-Hu Chen$%
    ^{1,3,*}$
