Quantum phase transitions in the spin-boson model without the counterrotating terms

Yan-Zhi Wang, Shu He, Liwei Duan, Qing-Hu Chen

Published 2018-12-10Version 1

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, the mean-field criticality is demonstrated for all bath exponents $0<s<1$, in contrast to the full spin-boson model. For large value of the bath exponent, a few first-order QPTs 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. The previous picture for many physical phenomena based on the conserved total excitations in open quantum systems might be not valid at least for sub-ohmic baths at the strong coupling.