Exact results and instabilities in the harmonic approximation of active crystals

Condensates of active particles, such as cells, form almost-crystalline lattices that play a central role in many biological systems. Typically, their properties have been determined merely by analogy to the rather trivial one-dimensional case, leaving a gap between experimentally accessible observables and suitable theoretical models. Within a harmonic approximation, we characterize analytically a two-dimensional triangular lattice of active particles that interact with their nearest neighbors through a general pair potential, obtaining exact expressions for the correlators. We study this "active crystal" as a means of characterizing active matter in the dense phase. Our treatment correctly approximates arbitrary pair potentials, rather than demanding an unphysical non-singular bilinear form. We retain "off-diagonal" terms that are routinely neglected despite quantifying the anisotropy of the particles' local potential. From the exact expressions for the correlation matrices, we derive exact results that shed light on the presence (or absence) of crystalline order. We further calculate the mean-squared particle separation, energy, entropy production rate, and the onset of a pressure-induced instability resulting in the breakdown of the harmonic approximation. The entropy production rate is found to have a general form that is valid for generic active particles and lattice geometries, while resembling that of non-interacting "active modes."