Asymptotic safety in the sine-Gordon model

In the framework of the functional renormalization group method, it is shown that the phase structure of the two-dimensional sine-Gordon model possesses a nontrivial UV fixed point which makes the model asymptotically safe. The fixed point exhibits strong singularity similar to the scaling found in the vicinity of the infrared fixed point. The singularity signals the upper energy-scale limit to the validity of the model. We argue that the sine-Gordon model with a momentum-dependent wave-function renormalization is in a dual connection with the massive sine-Gordon model.

Figure 1

The phase structure of the sine-Gordon model is presented. At $\tilde{u}=0$, we have a line of fixed points. At $\tilde{u}=1$ and $z\to 0$ ($z\to \infty$), we have an infrared (ultraviolet) non-Gaussian fixed point, respectively.

Figure 2

The evolution of the couplings $\tilde{u}$ and $z$ is shown. In the IR and the UV limits, the flows abruptly grow up or fall down at the same scale $k$.

Figure 3

The UV tendencies are plotted for the ZSG model with the choice ${\tilde{z}}_1(\mathrm{\Lambda })=0.0$. The coupling ${\tilde{z}}_1$ tends to infinity except when the singularity prevents it.