Generalized universality in the massive sine-Gordon model

A nontrivial interplay of the UV and IR scaling laws, a generalization of the universality is demonstrated in the framework of the massive sine-Gordon model, as a result of a detailed study of the global behavior of the renormalization group flow and the phase structure.

Figure 1

RG flow of ${\tilde{u}}_1,\dots ,{\tilde{u}}_4$ for ${\beta }^2=12\pi$ and $M^2={10}^{-9}{\Lambda }^2$ (solid line) or $M^2=0$ (dense dotted line). The plot also shows the asymptotic IR scaling $\sim k^{-2}$ (rare dotted line). The massive and the massless flows depart at the scale $M$, indicated by a dashed vertical line, placed at the intersection of the dotted lines.

Figure 2

RG flow of ${\tilde{u}}_1,\dots ,{\tilde{u}}_4$ for ${\beta }^2=4\pi$ and $M^2={10}^{-9}{\Lambda }^2$ (solid line) or $M^2=0$ (dotted line). The scales of the spinodal instability $k_{\mathrm{SI}}/\Lambda$ and the mass $M/\Lambda$ are indicated. The inset shows RG flows of ${\tilde{u}}_1$ for several different initial values at $M^2={10}^{-9}{\Lambda }^2$.

Figure 3

The region with spinodal instability. Different symbols show the boundary obtained by the numerical computation with 10 coupling constants and the lines show the results obtained by taking into account a single Fourier mode only.

Figure 4

The (1, 1) element sensitivity matrix is computed numerically for various initial conditions for ${\beta }^2=4\pi$ as the function of $k/\Lambda$. The trajectories of molecular and the ionized phases are shown by dashed and solid lines, respectively.

Figure 5

RG flow of ${\tilde{u}}_1,\dots ,{\tilde{u}}_4$ for $1/z_{\Lambda }=12\pi$ and $M^2={10}^{-9}{\Lambda }^2$. The plot shows the flows obtained from the LPA ($z_{\lambda }=z_{\Lambda }$) and from beyond the LPA approximations but the flows coincide. The evolution is similar to the WH-RG evolution obtained in the previous section.

Figure 6

Internal-space RG evolution of the coupling ${\tilde{u}}_1$ for several different initial values at $M^2={10}^{-9}{\Lambda }^2$ for constant $z_{\lambda }=z_{\Lambda }$ (solid lines) and for running $z_{\lambda }$ (dashed lines). In the vicinity of the phase boundary the flows depart slightly.

Figure 7

The scale ${\lambda }_c$, where the singularity appears, as the function of the initial value of the first Fourier amplitude $u_{1c}$ and $1/\pi z_{\Lambda }$ for $M^2={10}^{-4}$.