# Braiding with Borromean Rings in (3+1)-Dimensional Spacetime

## TL;DR

This paper introduces a novel braiding process called Borromean-Rings braiding, where a particle interacts with two loops forming a Brunnian link, resulting in a unique topological phase distinct from the Aharonov-Bohm effect.

## Contribution

It proposes the concept of Borromean-Rings braiding, derives its explicit phase expression using Milnor's triple linking number, and constructs TQFT models to realize this exotic braiding.

## Key findings

- Derived explicit braiding phase in terms of Milnor's triple linking number.
- Identified a new topological phase distinct from Aharonov-Bohm.
- Constructed TQFT models with an $AAB$-type topological term.

## Abstract

While winding a particle-like excitation around a loop-like excitation yields the celebrated Aharonov-Bohm phase, we find a distinctive braiding phase in the absence of such mutual winding. In this work, we propose an exotic particle-loop-loop braiding process, dubbed the \emph{Borromean-Rings braiding}. In the process, a particle moves around two unlinked loops, such that its trajectory and the two loops form the Borromean-Rings or more general Brunnian links. As the particle trajectory does not wind with any of the loops, the resulting braiding phase is fundamentally different from the Aharonov-Bohm phase. We derive an explicit expression for the braiding phase in terms of the underlying Milnor's triple linking number. We also propose Topological Quantum Field Theories consisting of an $AAB$-type topological term which realize the braiding statistics.