Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks

As the constituent particles of a flock are polar and in a driven state, their interactions must, in general, be fore-aft asymmetric and nonreciprocal. Within a model that explicitly retains the classical spin angular momentum field of the particles we show that the resulting asymmetric contribution to interparticle torques, if large enough, leads to a buckling instability of the flock. More precisely, this asymmetry also yields a natural mechanism for a difference between the speed of advection of polarization information along the flock and the speed of the flock itself, concretely establishing that the absence of detailed balance, and not merely the breaking of Galilean invariance, is crucial for this distinction. To highlight this we construct a model of asymmetrically interacting spins fixed to lattice points and demonstrate that the speed of advection of polarization remains nonzero. We delineate the conditions on parameters and wave number for the existence of the buckling instability. Our theory should be consequential for interpreting the behavior of real animal groups as well as experimental studies of artificial flocks composed of polar motile rods on substrates.