Functional Renormalization Group Approach to the Sine-Gordon Model

The renormalization group flow is presented for the two-dimensional sine–Gordon model within the framework of the functional renormalization group method by including the wave-function renormalization constant. The Kosterlitz–Thouless–Berezinski type phase structure is recovered as the interpolating scaling law between two competing IR attractive area of the global renormalization group flow.

Figure 1

Phase diagram of the SG model. The solid, dashed, and dotted lines show the RG trajectories for the linearized, exact in ${\tilde{u}}_1$, and full solutions, respectively. The wide solid line depicts the separatrix.

Figure 2

The inverse of the turning value ${\tilde{u}}_1^{*}$ of the fundamental amplitude is plotted versus the correlation length $\xi \sim 1/k^{*}$ for the linearized, exact in ${\tilde{u}}_1$ and full solutions, denoted by different point types. The slope of the solid line is $\sqrt{8/3{\pi }^2}\approx 0.52$ according to Eq. (11). In the inset, the turning value ${\tilde{u}}_1^{*}$ is plotted against the reduced temperature $t$. The slope of the fitted dashed line is 0.5.

Figure 3

The phase structure of the SG model given by the numerical solution of Eqs. (12, 13).