Title: Low-temperature marginal ferromagnetism explains anomalous scale-free correlations in natural flocks,
Abstract: We introduce a new ferromagnetic model capable of reproducing one of the most
intriguing properties of collective behaviour in starling flocks, namely the
fact that strong collective order of the system coexists with scale-free
correlations of the modulus of the microscopic degrees of freedom, that is the
birds' speeds. The key idea of the new theory is that the single-particle
potential needed to bound the modulus of the microscopic degrees of freedom
around a finite value, is marginal, that is has zero curvature. We study the
model by using mean-field approximation and Monte Carlo simulations in three
dimensions, complemented by finite-size scaling analysis. While at the standard
critical temperature, $T_c$, the properties of the marginal model are exactly
the same as a normal ferromagnet with continuous symmetry-breaking, our results
show that a novel zero-temperature critical point emerges, so that in its
deeply ordered phase the marginal model develops divergent susceptibility and
correlation length of the modulus of the microscopic degrees of freedom, in
complete analogy with experimental data on natural flocks of starlings.