Spatially patterned phases in a reaction-time-symmetry-broken model of flocking

We introduce a Vicsek-like flocking model with a minimal form of time-delayed orientational interactions, in which the delays occur on a time scale that is well-separated from other time scales in the model. We achieve this by implementing an ``index-ordered'' update rule, mimicking a scenario in which agents have a distribution of times with which they react to information. This model retains the usual disorder-to-order transition common in flocking models, but we show that it also possesses a second transition, deep in the polar flocking phase, to a state with spatially patterned transverse velocities. We characterize this transition and its sensitivity to finite-size effects using the Binder cumulant, and demonstrate -- via direct measurements and by measuring a susceptibility of the phase to particle index permutations -- that the stability of this phase is directly tied to a subtle spatial organization of a slow-relaxing index-order field. These results highlight the potential for even seemingly insignificant temporal asymmetries to fundamentally alter the collective behavior of active matter.