The quantum Rabi model (QRM), which describes a two-level system coupled to a single electromagnetic mode, provides a basic paradigm of light-matter interactions. Due to recent theoretical and experimental progress, the QRM attracted interest from researchers working on quantum optics, solid state physics, fundamental properties of quantum physics, and mathematical physics. The first analytical solution of the QRM, which is an important breakthrough on the mathematical and physical aspects of this model, was obtained by Braak only in 2011 [1]. While it is well known that at weak coupling the QRM can be approximated by the Jaynes-Cummings (JC) Hamiltonian, recent experimental progress in circuit QED has allowed to reach the ultra-strong and even deep-strong coupling regimes [2,3,4]. Very recently [5,6], it was proved that both the QRM and JC model can realize a new type of quantum phase transition (QPT) which, surprisingly, does not require the thermodynamic limit.

The QPT considered by Refs. [5,6] occurs as a function of coupling strength when the ratio η between the atomic and cavity frequency tends to infinity, and its properties are in fact very similar to the superradiant QPT of the Dicke model. In the latter case, a single cavity mode interacts with many two-level systems (large *N*), while the QRM corresponds to the *N=*1 case. The similarity to thermodynamic QPTs motivates investigating whether familiar concepts like universality classes are still generally applicable at *N=*1. More generally, the relationship of this new type of few-body QPTs and their thermodynamic counterparts has not been clearly understood.

To elucidate these issues, a recent investigation [7] involving Maoxin Liu, Stefano Chesi, Zu-Jian Ying and Hai-Qing Lin from CSRC, Hong-Gang Luo from Lanzhou University, and Xiaosong Chen from CAS-ITP & UCAS, has developed a general analytical approach valid in the limit of large η. The treatment not only derives a perturbative effective Hamiltonian applicable to the critical region (from a fourth-order Schrieffer-Wolff transformation), but extends the perturbative results obtaining a non-perturbative effective mass and mean-field potential. These quantities provide a full description of the low-energy properties at arbitrary coupling strengths (i.e., also away from the critical point).

The treatment of Ref. [7] is not only applicable to the original QRM and JC model at *N=*1, but also allows to include a finite anisotropy and an arbitrary (but finite) value of *N*. As expected, a well-defined universality class is established at arbitrary values of the anisotropy parameter (except the JC model). It was also found that the low-energy description at given anisotropy is independent of *N*, after appropriately rescaling η. Therefore, the few-body QPTs at arbitrary *N* display identical non-universal features, like the critical coupling or the value of the order parameter away from the QPT.

The effective low-energy theory described in this work is in excellent agreement with numerical calculations of the phase diagram, critical exponents, and scaling functions, also presented in Ref. [7]. Other interesting features revealed by this work are a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Besides these theoretical achievements, this work could be relevant for experimental platforms like circuit QED and trapped ion systems, where the appropriate regime might be realized soon [3,4,8,9]. Because of its great potential interest, the article was selected by Physical Review Letters as an “Editors’ Suggestion”.

**Fig. 1:** Phase diagram (a) and order parameter (b) of the QRM with anisotropy. The red (blue) line in panel (a) indicates a first (second) order phase transition. The order of the transition is revealed by the first and second derivative of the ground state energy, respectively shown in panels (d) and (c).

**References: **

[1] D. Braak, Phys. Rev. Lett. **107**, 100401 (2011).

[2] P. Forn-Diaz, et al., Phys. Rev. Lett. 105, 237001 (2010).

[3] F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Nat. Phys. 13, 44 (2017).

[4] P. Forn-Díaz, et al., Nat. Phys.13, 39 (2017).

[5] M.-J. Hwang, R. Puebla, and M. B. Plenio, Phys. Rev. Lett. **115**, 180404 (2015).

[6] M.-J. Hwang and M. B. Plenio, Phys. Rev. Lett. **117**,123602 (2016).

[7] S. Maoxin Liu, Stefano Chesi, Zu-Jian Ying, Xiaosong Chen, Hong-Gang Luo, and Hai-Qing Lin Phys. Rev. Lett. **119**, 220601 (2017) - Editor’s Suggestion. DOI: https://doi.org/10.1103/PhysRevLett.119.220601

[8] R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, Phys. Rev. Lett. 118, 073001 (2017).

Z. Chen, Y. Wang, T. Li, L. Tian, Y. Qiu, K. Inomata, F. Yoshihara, S. Han, F. Nori, J. S. Tsai, and J. Q. You, Phys. Rev. A 96, 012325 (2017).