Dynamics of O(2) excitations in a non-reciprocal medium

We investigate emergent dynamics due to non-reciprocity in the $\mathcal{O}(2)$ model. The lattice XY model, where non-reciprocity stems from vision cone like couplings, can be described by a continuum description in which non-reciprocity translates into a new term depending on the rotational of the orientation field. We argue that non-reciprocity is akin to activity and we highlight the connection between our hydrodynamic equation and the constant density Toner-Tu framework. The active force advects and reshapes patterns, a generic feature found in many non-reciprocal systems. We show how $1d$ excitations in the non-reciprocal $\mathcal{O}(2)$ model can be described by a generalized Burgers equation, derived from our continuum model. We then extend the results to $2d$ perturbations. As such, we establish the first principles of excitation trajectory control in a non-reciprocal $\mathcal{O}(2)$ medium. Concretely, we explain how tuning the degree of non-reciprocity and the orientation of the background medium impacts the time evolution of excitations. We also showcase how initially different excitations lead to very different dynamical behavior. Non-reciprocity also affects the stability of defect-free excitations with non-zero winding numbers and, unlike in its equilibrium $O(2)$ counterpart, enables the system, above a certain threshold, to relax to its ground state.