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 $x_\lambda\approx 0.847\,\epsilon$, $x_{\bar\eta}\approx 0.153\,\epsilon$, 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: $z\approx 3.08$.