Title: Active pattern formation emergent from single-species nonreciprocity

Abstract: Nonreciprocal forces violating Newton's third law are common in a plethora of nonequilibrium situations ranging from predator-prey systems to the swarming of birds and effective colloidal interactions under flow. While many recent studies have focused on two species with nonreciprocal coupling, much less is examined for the basic single-component system breaking the actio and reactio equality of force within the same species. Here, we systematically derive the fundamental field theory of single-species nonreciprocal interactions from microscopic dynamics, leading to a generic framework termed Active Model N (N denoting nonreciprocity). We explore the rich dynamics of pattern formation in this intrinsic nonreciprocal system and the emergence of self-traveling states with persistent variation and flowing of active branched patterns. One particular new characteristic pattern is an interwoven self-knitting "yarn" structure with a unique feature of simultaneous development of micro- and bulk phase separations. The growth dynamics of a "ball-of-wool" active droplet towards these self-knitted yarn or branched states exhibits a crossover between different scaling behaviors. The mechanism underlying this distinct class of active phase separation is attributed to the interplay between force nonreciprocity and competition. Our predictions can be applied to various biological and artificial active matter systems controlled by single-species nonreciprocity.