Dynamic universality class of Model C from the functional renormalization group

We establish new scaling properties for the universality class of Model C, which describes relaxational critical dynamics of a nonconserved order parameter coupled to a conserved scalar density. We find an anomalous diffusion phase, which satisfies weak dynamic scaling while the conserved density diffuses only asymptotically. The properties of the phase diagram for the dynamic critical behavior include a significantly extended weak scaling region, together with a strong and a decoupled scaling regime. These calculations are done directly in $2\le d\le 4$ space dimensions within the framework of the nonperturbative functional renormalization group. The scaling exponents characterizing the different phases are determined along with subleading indices featuring the stability properties.

Figure 1

Phase diagram for Model C as a function of dimension $d$ and the number of field components $N$ from the functional renormalization group. For comparison to the $\epsilon$ expansion, thick lines near $d=4$ denote $\mathcal{O}\left(\epsilon \right)$ results;[7] thin/dashed lines denote the $\mathcal{O}\left({\epsilon }^2\right)$ results according to Ref. 10.

Figure 2

(Top) Dynamic critical exponents $z$ and $z_{\mathcal{E}}$ as a function of $N$ at fixed dimension $d=3.75$. The different regions I–IV are clearly visible, along with the locking phenomenon ($z=z_{\mathcal{E}}$) in II and the distinct values in the independent scaling regimes I and III. (Bottom) The subleading exponents ${\theta }_{\kappa }$ and ${\theta }_{\gamma }$ indicate the stability of the fixed point solutions.

Figure 3

Fixed point values for the mode coupling $\gamma$ and kinetic parameter $\kappa$ as a function $N$ at $d=3.75$. The asymptotic result for $\kappa$ in the limit $d\to 4^{-}$ is shown for comparison. In that case region III extends over the range $2<N<4$.