This work focuses on the investigation of non-reciprocal interactions in the XY Model using the Kuramoto model of synchronization in the overdamped limit. Initially, we provide partial results of the reciprocal XY Model by examining the spatial correlation function and the transition temperature. Through a comparison of simulation and theoretical results, we gain insights into the critical behavior of the model. To extend the analysis, we introduce non-reciprocal interactions using the Kuramoto model in the overdamped regime, which offers a nonlinear mathematical framework for understanding the dynamics of the system. This is particularly relevant as the reciprocal XY Model lacks a Hamiltonian description. By incorporating non-reciprocal interactions, we observe that the system does not undergo a topological phase transition. Instead, a dynamic analysis reveals, under certain initial distribution and conditions, the emergence of waves and their characteristic propagation. We explore these phenomena in both one-dimensional and two-dimensional scenarios, demonstrating that the waves propagate with a linear velocity and exhibit a linear dispersion relation

Treballs Finals de Màster en Física dels Sistemes Complexos i Biofísica, Facultat de Física, Universitat de Barcelona. Curs: 2022-2023. Tutor: Demian Levis Sotomayor