Universal Scaling and Critical Exponents of the Anisotropic Quantum Rabi Model

We investigate the quantum phase transition of the anisotropic quantum Rabi model, in which the rotating and counterrotating terms are allowed to have different coupling strengths. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches, we extract the phase diagram, scaling functions, and critical exponents, which determine the universality class at finite anisotropy (identical to the isotropic limit). We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are extended to the few-body quantum phase transitions with $N>1$ spins, where we expose the same effective parameters, scaling properties, and phase diagram. Thus, a stronger form of universality is established, valid from $N=1$ up to the thermodynamic limit.

Figure 1

(a) Phase diagram, where the blue (red) line indicates a second-order (first-order) QPT. (b) Energy gap of the normal phase ($\xi <1$) at $\lambda =0,0.1,0.2,\dots ,1$ (bottom to top), in units of $N\mathrm{\Omega }/\eta$. The corresponding critical exponents are $\alpha =1/2$ ($\lambda \ne 0$) and $\alpha =1$ ($\lambda =0$). For $\xi >1$, we plot the order parameter $x_{0,\pm }^2$ at $\lambda \ne 0$ (solid curve) and $\lambda =0$ (dotted curve, which is $1/2$ of the former). Panels (a) and (b) are both valid for arbitrary $N$.

Figure 2

(a) Effective mass renormalization as a function of $\xi$. For $\xi <1$, $M={\eta }^2(1-{\xi }^{\prime }{)}^{-1}$, which in the absence of the superradiant phase would diverge at $\xi =(1+\lambda )/(1-\lambda )$ (dashed curves). Instead, $M=M_{\lambda }$ for $\xi >1$. (b) Universal scaling function for $⟨{\tilde{x}}^2⟩$. Symbols are obtained by a solution of the full anisotropic QRM, with $\eta =2^{20}$ and different values of $\lambda$. All numerical data collapse into the single function $X_1(v)$ (solid curve), obtained from Eq. (9).

Figure 3

Universal scaling functions at different values of $\lambda$ and $N$. (a) Data collapse of numerical results, directly computed from $H$ or the anisotropic Dicke model ($N>1$). The solid curve is $P_1(v)$, obtained from Eq. (9). (b) Scaling function for $⟨x^2⟩=⟨p^2⟩$ at $\lambda =0$. The numerical values (symbols) are in agreement with Eq. (13) (solid line).