Braiding statistics and congruent invariance of twist defects in bosonic bilayer fractional quantum Hall states

We describe the braiding statistics of topological twist defects in fractional quantum Hall (FQH) states. In particular, we study Abelian bosonic $\left(mmn\right)$ FQH states with bilayer symmetries. Twist defects are staircases that lift orbiting quasiparticles from one layer to another. These point defects exhibit semiclassical non-Abelian characteristics. We develop a procedure to evaluate consistent basis transformations ($F$ symbols) of the defect fusion theory. Unlike quantum anyonic excitations of a topological phase, the exchange and braiding statistics of twist defects is not characterized by modular transformations, but instead a subcollection known as congruent transformations. We also characterize the projective nature of unitary braiding operations and relate it to the global ${\mathbb{Z}}_2$ bilayer symmetry.

Figure 1

(a) Double Wilson loop operator ${\widehat{\Theta }}_{\mathbf{a}}$ by moving a quasiparticle $\mathbf{a}$ twice around a twofold defect (cross). (b) A linking phase $e^{2\pi i{\mathbf{a}}^T{K}^{-1}{\sigma }_x\mathbf{a}}$ from joining two double Wilson operators ${\widehat{\Theta }}_a$ around a pair of defects. The two unlinked $\mathbf{a}$ and ${\sigma }_x\mathbf{a}$ loops can then be absorbed in the condensate if the defects fuse to the vacuum.

Figure 2

Quasiparticle string configurations for splitting states $|V_{{\psi }_3}^{{\psi }_2{\psi }_1}\rangle$. Branch cuts (wavy lines) switch labels $\mathbf{b}\to {\sigma }_x\mathbf{b}$ of passing quasiparticles and end at twofold defects (crosses). (i) For ${\mathbb{Z}}_n$ toric code where $m=0$, quasiparticle $\mathbf{a}=a_1\mathbf{e}+a_2\mathbf{m}$ is attached with strings $\mathbf{b}=a_2\mathbf{m}$ and $\mathbf{a}-\mathbf{b}=a_1\mathbf{e}$; twofold defect ${\sigma }_{\lambda }$ is attached with string $\mathbf{y}=\lambda \mathbf{e}$. (ii) For $\left(mmn\right)$-FQH states with $n$ odd, quasiparticle $\mathbf{a}$ is attached with strings $\mathbf{b}=\mathbf{a}-\mathbf{b}=2^{-1}\mathbf{a}$, and twofold defect ${\sigma }_{\lambda }$ is attached with string $\mathbf{y}=2^{-1}\lambda \mathbf{t}$, $2^{-1}=(m^2-n^2+1)/2$.

Figure 3

Dehn twists. (a) Double Dehn twist $T=t_x^2$ along the horizontal direction that leaves the branch cut (wavy line) invariant. (b) $T$ action on a vertical Wilson loop ${\widehat{W}}_{\mathbf{a}}$. (c) Dehn twist $S=t_y$ along the vertical (longitudinal) direction and its action on a double Wilson loop ${\widehat{\Theta }}_{\mathbf{a}}$.

Figure 4

Derivation of the $F$ matrix ${\left[F_{{\sigma }_{{\lambda }_1+{\lambda }_2+{\lambda }_3}}^{{\sigma }_{{\lambda }_3}{\sigma }_{{\lambda }_2}{\sigma }_{{\lambda }_1}}\right]}_{\mathbf{a}}^{\mathbf{b}}$. The sum over Wilson loops ${\widehat{W}}_{\mathbf{c}}$ (blue) is restricted to an arbitrary set of $|m-n|$ quasiparticles $\mathbf{c}=z\mathbf{t}+\gamma \mathbf{s}$ with distinct spin components $\gamma$ as ${\widehat{W}}_{\mathbf{c}}|GS\rangle ={\widehat{W}}_{{\sigma }_x\mathbf{c}}^{†}|GS\rangle$ (see Fig. 1). The operator $\frac{1}{\sqrt{|m-n|}}{{\sum }_{\mathbf{c}}}^{\prime }{\widehat{W}}_{\mathbf{c}}$ flips the branch cut configuration from joining ${\sigma }_{{\lambda }_3}\times {\sigma }_{{\lambda }_2}$ to connecting ${\sigma }_{{\lambda }_2}\times {\sigma }_{{\lambda }_1}$.