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

## TL;DR

This paper investigates quantum phase transitions in the spin-boson model without counterrotating terms, revealing the presence of both first- and second-order transitions and challenging previous understandings under the rotating-wave approximation.

## Contribution

It provides a numerically exact analysis of the spin-boson model without counterrotating terms, discovering new types of quantum phase transitions and their dependence on bath exponents.

## Key findings

- Second-order QPT observed in sub-Ohmic bath
- First-order QPTs appear before critical points
- Multiple first-order QPTs in Ohmic bath at strong coupling

## 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.