Loop braid groups and integrable models

Abstract

Loop braid groups characterize the exchange of extended objects, namely loops, in three-dimensional space generalizing the notion of braid groups that describe the exchange of point particles in two-dimensional space. Their interest in physics stems from the fact that they capture anyonic statistics in three dimensions which is otherwise known to only exist for point particles on the plane. Here, we explore another direction where the algebraic relations of the loop braid groups can play a role: quantum integrable models. We show that the symmetric loop braid group can naturally give rise to solutions of the Yang-Baxter equation, proving the integrability of certain models through the RTT relation. For certain representations of the symmetric loop braid group, we obtain integrable deformations of the XXX-, XXZ-and XYZ-spin chains. � 2023 World Scientific Publishing Company. All rights reserved.