How motility affects Ising transitions

We study a lattice gas model of hard-core particles on a square lattice experiencing nearest neighbour attraction $J$. Each particle has an internal orientation, independent of the others, that point towards one of the four nearest neighbour and it can move to the neighbouring site along that direction with the usual Metropolis rate if the target site is vacant. The internal orientation of the particle can also change to any of the other three with a constant rate $\omega.$ The dynamics of the model in $\omega\to \infty$ reduces to that of the Lattice Gas (LG) which exhibits a phase separation transition at particle density $\rho=\frac12$ and temperature $T=1,$ when the strength of attraction $J$ crosses a threshold value $\ln(1+ \sqrt{2}).$ This transition belongs to Ising universality class. For any finite $\omega>0,$ the particles can be considered as attractive run-and-tumble particles (RTPs) in two dimensions with motility $\omega^{-1}.$ We find that RTPs also exhibit a phase separation transition, but the critical interaction required is $J_c(\omega)$ which increases monotonically with increased motility $\omega^{-1}.$ It appears that the transition belongs to Ising universality class. Surprisingly, in these models, motility impedes cluster formation process necessitating higher interaction to stabilize microscopic clusters. Moreover, MIPS like phases are not found when $J=0.$