Loop braid groups and integrable models

TL;DR

This paper explores how the algebraic structure of loop braid groups can generate solutions to the Yang--Baxter equation, leading to integrable models related to spin chains in three-dimensional space.

Contribution

It demonstrates that the symmetric loop braid group can produce solutions to the Yang--Baxter equation, establishing integrability of new models including deformations of known spin chains.

Findings

Solutions to the Yang--Baxter equation derived from loop braid groups.

Integrable deformations of XXX, XXZ, and XYZ spin chains.

Connection between algebraic structures of loops and quantum integrability.

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 {\it 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.