On the Renormalization Group Flow of Active Flocks

Abstract

In this paper, we study the statistical field-theoretic renormalization of active flocks via the MSRDJ action formulation for stochastic systems, focusing on the Toner-Tu theory of `Malthusian flocks', or polar-ordered, momentum non-conserving active fluids where relaxation times for density fluctuations are so short that they can be eliminated as a hydrodynamic variable. Working in the limit of isotropic diffusion in two spatial dimensions, we compute the renormalization of the couplings and their anomalous dimensions to all orders, facilitated by the Ward identities associated with the non-linear realization of a diagonal rotation/ shift symmetry, and a non-local field-dependent redundancy specific to the MSRDJ representation of this model. We find a range of behavior depending on the parameters of the theory. If $\kappa$ is the diffusion coefficient and $\Delta$ is the variance of the noise, we find a line of fixed points and a marginal vertex instability at $\Delta/\kappa = 2\pi$. This instability separates Gaussian, and symmetry-protected strongly interacting gapless phases, realizing non-equilibrium critical behavior beyond standard Wilson--Fisher criticality, and implies the persistence of long range order when $\Delta/\kappa$ is below the critical value. We revisit and contextualize various claims and counter-claims in the literature in light of our findings, and discuss extensions of our analysis to flocks with anisotropic diffusion, and where density fluctuations are reintroduced.