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Cyclic and Coherent States in Flocks with Topological Distance

A simple model of the two dimensional collective motion of a group of mobile agents have been studied. Like birds, these agents travel in open free space where each of them interacts with the first $n$ neighbors determined by the topological distance with a free boundary condition. Using the same prescription for interactions used in the Vicsek model with scalar noise it has been observed that the flock, in absence of the noise, arrives at a number of interesting stationary states. In the `single sink state' the entire flock maintains perfect cohesion and coherence. In the `cyclic state' every agent executes a uniform circular motion, and the entire flock executes a pulsating dynamics i.e., expands and contracts periodically between a minimum and a maximum size of the flock. When refreshing rate of the interaction zone is the fastest, the entire flock gets fragmented into smaller clusters of different sizes. On introduction of scalar noise a crossover is observed when the agents cross over from a ballistic motion to a diffusive motion. Expectedly the crossover time is dependent on the strength of the noise $η$ and diverges as $η\to 0$. In simpler version the translational degrees of freedom of the agents are suppressed but their angular motion are retained. Here agents are the spins, placed at the sites of a square lattice with periodic boundary condition. Every spin interacts with its $n$ = 2, 3 or 4 nearest neighbors. In the stationary state the entire spin pattern moves as a whole when interactions are anisotropic with $n$ = 2 and 3; but it is completely frozen when the interaction is isotropic with $n=4$. These spin configurations have vortex-antivortex pairs whose density increases as the noise $η$ increases and follows an excellent finite-size scaling analysis.