Dynamic universality class of model H with frustrated diffusion: $\epsilon$ expansion

We study a variation of the dynamic universality class of model H in a spatial dimension of $d=4-\epsilon$, by frustrating charge diffusion and momentum density fluctuations along $d_T=1$ or $d_T=2$ dimensions, while keeping the same dynamics of model H in the other $d_L=d-d_T$ dimensions. The case of $d_T=2$ describes the QCD critical point in a background magnetic field. We find that these models belong to a different dynamical universality class due to extended conservation laws compared to the model H, although the static universality class remains the same as the 3-dimensional Ising model. We compute the dynamic critical exponents of these models in first order of $\epsilon$-expansion to find that $x_{\lambda }\approx 0.847\epsilon$, $x_{\overline{\eta }}\approx 0.153\epsilon$, and $z=4-x_{\lambda }\approx 3.15$ when $\epsilon =1$ and $d_T=2$. For $d_T=1$ the results are numerically similar to the model H values: $z\approx 3.08$.

Figure 1

The two real-time diagrams for renormalization of ${\lambda }_{\parallel }$.

Figure 2

The real-time diagram for renormalization of ${\overline{\eta }}_{\perp }$ and ${\overline{\eta }}_{\parallel }$.

Figure 3

The real-time diagrams that are relevant for $\delta g$. It turns out that there is no correction to $\delta g$ from these diagrams.