Leading birds by their beaks: the response of flocks to external perturbations

We study the asymptotic response of polar ordered active fluids (“flocks”) to small external aligning fields $h$. The longitudinal susceptibility ${\chi }_{{}_{\parallel }}$ diverges, in the thermodynamic limit, like $h^{-\nu }$ as $h \rightarrow 0$\mathrm{.}Infinitesystemsoflinearsize$L$,$\chi_{_\parallel}$saturatestoavalue$\sim L^\gamma$\mathrm{.}Theuniversalexponents$\nu$and$\gamma$dependonlyonthespatialdimensionality$d$, and are related to the dynamical exponent $z$ and the “roughness exponent” $\alpha$ characterizing the unperturbed flock dynamics. Using a well supported conjecture for the values of these two exponents, we obtain $\nu =2\mathrm{/}3$, $\gamma =4\mathrm{/}5$ in $d=2$ and $\nu = 1/4$,$\gamma = 2/5$in$d = 3$. These values are confirmed by our simulations.