Collective motion is often modeled within the framework of active fluids, where the constituent active particles, when interactions with other particles are switched off, perform normal diffusion at long times. However, in biology, single-particle superdiffusion and fat-tailed displacement statistics are also widespread. The collective properties of interacting systems exhibiting such anomalous diffusive dynamics, which we call active Levy matter, cannot be captured by current active fluid theories. Here, we formulate a hydrodynamic theory of active Levy matter by coarse-graining a microscopic model of aligning polar active particles that perform superdiffusion akin to Levy flights. Applying a linear stability analysis on the hydrodynamic equations at the onset of collective motion, we find that, in contrast to its conventional counterpart, the order-disorder transition can become critical. We then estimate the corresponding critical exponents by finite size scaling analysis of numerical simulations. Our work highlights the novel physics in active matter that integrates both anomalous diffusive motility and inter-particle interactions.
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