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

While winding a particlelike excitation around a looplike excitation yields the celebrated Aharonov-Bohm phase, we find a distinctive braiding phase in the absence of such mutual winding. In this Letter, we propose an exotic particle-loop-loop braiding process, dubbed the 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.