Tuning nonequilibrium phases with odd forces: From crystalline order to vortex structures in systems with competing interactions

Nonequilibrium chiral active systems exhibit diverse collective phenomena arising from the interplay between driving forces and interparticle interactions. We numerically investigate a two-dimensional system of particles with competing long-range repulsion and short-range attraction, driven by nonconservative odd forces. We identify five distinct nonequilibrium phases: a crystalline state with long-range translational and orientational order; a triangular velocity configuration exhibiting velocity-space ordering; a nematic phase featuring orientational alignment without polar order; a bubble phase with vortical flows along void boundaries; and a rotating cluster phase showing dynamic restructuring. The transitions between these phases are systematically controlled by varying the attraction strength and the odd-force intensity, revealing a rich phase diagram. Our results underscore the crucial role of odd forces in steering the structural and dynamical evolution of active matter with competing interactions.