Braiding statistics of loop excitations in three dimensions

TL;DR

It is found that different short-range entangled bosonic states with the same (Z(N))(K) symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.

Abstract

While it is well known that three dimensional quantum many-body systems can support non-trivial braiding statistics between particle-like and loop-like excitations, or between two loop-like excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop $α$ around another loop $β$, while both are linked to a third loop $γ$. We study this three-loop braiding in the context of $(\mathbb{Z}_N)^K$ gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with $(\mathbb{Z}_N)^K$ symmetry. We find that different short-range entangled bosonic states with the same $(\mathbb{Z}_N)^K$ symmetry (i.e. different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.

Figures (10)

• Figure 1: (a) Three-loop braiding process. The gray curves show the paths of two points on the moving loop $\alpha$. (b) A top view of the braiding process within the plane that $\gamma$ lies in. (c) A torus $\Omega_\alpha$ is swept out by $\alpha$ during the braiding. Loop $\beta$ (dashed circle) is enclosed by $\Omega_\alpha$.
• Figure 2: (a) Braiding of two loops $\alpha, \beta$. (b) If $\alpha, \beta$ are neutral, the two-loop process can be smoothly deformed into a process in which $\alpha$ is braided around the vacuum.
• Figure 3: Two ways to fuse loops together.
• Figure 4: Braiding processes associated with equations (\ref{['linear2']}) [panel (a)] and (\ref{['linear3']}) [panel (b)]. Here, $\phi_\gamma= \frac{2\pi}{N} c$, while $\phi_{\gamma_1}= \frac{2\pi}{N} c_1$ and $\phi_{\gamma_2}= \frac{2\pi}{N} c_2$.
• Figure 5: Computing three-loop statistics from 2D braiding.