Odd active solids. Vortices, velocity oscillations and dissipation-free modes

A wide range of physical and biological systems, including colloidal magnets, granular spinners, and starfish embryos, are characterized by strongly rotating units that give rise to odd viscosity and odd elasticity. These active systems can be described using a coarse-grained model in which the pairwise forces between particles include a transverse component compared to standard interactions due to a central potential. These non-potential, additional forces, referred to as odd interactions, do not conserve energy or angular momentum and induce rotational motion. Here, we study a two-dimensional crystal composed of inertial Brownian particles that interact via odd forces and are in thermal contact with their environment. We discover that, in the underdamped regime, the energy injected by odd forces can counteract dissipation due to friction, leading to quasi-dissipation-free excitations with finite frequency and wavelength. In the resulting non-equilibrium steady state, the system exhibits angular momentum and velocity correlations. When the strength of the odd forces exceeds a certain threshold or friction is too low, we show that a chiral active crystal with only harmonic springs becomes linearly unstable due to transverse fluctuations. This instability can be mitigated by introducing nonlinear central interactions, which suppress the divergence of short-wavelength velocity fluctuations and allows us to numerically explore the linearly unstable regime. This is characterized by pronounced temporal oscillations in the velocity featuring the existence of vortex structures and kinetic temperature values larger than the thermal temperature.