Resilience and fragility of two-dimensional active crystals

The Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory of melting is a much-admired result that revealed that equilibrium 2D crystals can melt into in two successive KT-like transitions leading to liquid state. A well-known result of KTHNY is to predict an upper bound  $\frac{1}{3}$ to the spin wave exponent  $\eta$ governing the algebraic decay of the two-point correlation function of positional order in the crystal phase. Whereas this result is often used in equilibrium to decide to decide whether a crystal has melted, it has been claimed (or assumed) to apply also for crystals made of active units, which are intrinsically out of equilibrium.

I will show that such active crystals are not subjected to the KTHNY bound $\eta< \frac{1}{3}$. They can be either more resilient to fluctuations without melting ($\eta> \frac{1}{3}$), or more fragile and break for $\eta< \frac{1}{3}$. These results are rationalized within linear elastic theory in terms of two well-defined effective temperatures governing respectively large scale positional deformations and short scale bond-order fluctuations.