Twist defects and projective non-Abelian braiding statistics

It has recently been realized that a general class of non-Abelian defects can be created in conventional topological states by introducing extrinsic defects, such as lattice dislocations or superconductor-ferromagnet domain walls in conventional quantum Hall states or topological insulators. In this paper, we begin by placing these defects within the broader conceptual scheme of extrinsic twist defects associated with symmetries of the topological state. We explicitly study several classes of examples, including $Z_2$ and $Z_3$ twist defects, where the topological state with $N$ twist defects can be mapped to a topological state without twist defects on a genus $g\propto N$ surface. To emphasize this connection we refer to the twist defects as genons. We develop methods to compute the projective non-Abelian braiding statistics of the genons, and we find the braiding is given by adiabatic modular transformations, or Dehn twists, of the topological state on the effective genus $g$ surface. We study the relation between this projective braiding statistics and the ordinary non-Abelian braiding statistics obtained when the genons become deconfined, finite-energy excitations. We find that the braiding is generally different, in contrast to the Majorana case, which opens the possibility for fundamentally novel behavior. We find situations where the genons have quantum dimension 2 and can be used for universal topological quantum computing (TQC), while the host topological state is by itself nonuniversal for TQC.

Figure 1

Worldline of quasiparticle ${\gamma }_i$ and twist defect, labeled by $g$. Braiding ${\gamma }_i$ around the twist defect changes the quasiparticle to ${\gamma }_{g\left(i\right)}$. The arrow indicates the time direction.

Figure 2

A pair of twist defects induces two distinct noncontractible loops, labeled $a$ and $b$. For twist defects in $\left(mm0\right)$ states, these loops effectively cross only once, leading to a magnetic algebra for the quasiparticle loop operators. Here, the branch cut indicated by the dashed line connecting two dislocations is merely a gauge choice, which is similar to supposing that the phase winding of a vortex in a superconductor is all localized to a single cut connecting a vortex/antivortex pair.

Figure 3

$n$ pairs of twist defects induces $2(n-1)$ distinct noncontractible loops, $a_i,b_i$, for $i=1,...,n-1$. For twist defects in the $\left(mm0\right)$ states, the quasiparticle loop operators give rise to $n-1$ copies of the magnetic algebra, leading to ${\left|m\right|}^{n-1}$ topologically degenerate states.

Figure 4

The $\mathrm{U}\left(1\right)\times \mathrm{U}\left(1\right)$ CS theory with $n$ pairs of dislocations on a sphere can be mapped to a $\mathrm{U}\left(1\right)$ CS theory on a genus $g=n-1$ surface, $M_{n-1}$. $M_{n-1}$ consists of two copies of the original space. A new $\mathrm{U}\left(1\right)$ gauge field, $c$, is defined on $M_{n-1}$, such that $c=a$ on the top half and $c=-\tilde{a}$ on the bottom half of $M_{n-1}$ (Ref. 32).

Figure 5

(a) A loop that encloses the twist defects labeled 2 and 3 is mapped to the $b$ cycle of the torus. (b) Effect of a counterclockwise exchange of 1 and 2. By following the effect on the noncontractible loop, we see that in terms of the genus $g$ surface, it has the effect of a Dehn twist along an $a$ cycle. Thus, the original $b$ loop becomes the loop $b+a$ after counterclockwise exchange of 1 and 2. (c) A loop that encloses a pair of twist defects connected by a branch cut is an $a$ cycle of the genus $g$ surface. (d) Effect of a clockwise exchange of 2 and 3. We see that the $a$ loop gets mapped to the $a+b$ loop. Thus, the clockwise exchange of 2 and 3 is equivalent to a Dehn twist along the $b$ loop.

Figure 6

The edge-state understanding of the topological degeneracy. (a) The genons are oriented along a single line, and then the system is cut along the line, yielding gapless counterpropagating edge states along the line. The original topological state is obtained from gluing the the system back together by turning on appropriate interedge tunneling terms. (b) Depiction of the two branches (red and blue) of counterpropagating edge excitations. The arrows between the edge states indicate the kinds of electron tunneling terms that are added. Away from the genons, in the $A$ regions, the usual electron tunneling terms involving tunneling between the same layers, ${\Psi }_{eRI}^{†}{\Psi }_{eLI}+\text{H.c.}$, are added. In the regions including the branch cuts separating the genons, twisted tunneling terms are added: ${\Psi }_{eR1}^{†}{\Psi }_{eL2}+{\Psi }_{eR2}^{†}{\Psi }_{eL1}+\text{H.c.}$

Figure 7

Physical meaning of the zero-mode operators ${\alpha }_i$ and ${\beta }_i$. In subsequent figures, we use the bottom figure, which is more precisely described by the top figure.

Figure 8

Effect of a clockwise braid on the zero modes ${\alpha }_1$ and ${\alpha }_2$. (a) ${\alpha }_1\to {\alpha }_2$. (b) ${\alpha }_2$ gets transformed to a combination of two loop, ${\beta }_1{\beta }_2^{†}$, and ${\alpha }_2$. Therefore, ${\alpha }_2\to {\alpha }_2{\alpha }_1^{†}{\alpha }_2$, where recall that ${\beta }_i={\alpha }_i^{†}$. As explained in the text, there is an ordering ambiguity between ${\alpha }_2$ and ${\alpha }_1^{†}{\alpha }_2$ that is fixed with an additional constraint.

Figure 9

Illustration of 1D protocol for carrying out an effective braiding process. This protocol does not require a continuous motion in both directions and may be easier to implement in many physical realizations, for instance through gating.

Figure 10

Illustration of operations (1), (2), (3), (4), and (4${}^{\prime }$), described in the main text for performing the cnot and $\pi /8$ phase gates. The black dots are $(\sigma ,1)$ topological quasiparticles. The red circles indicate that the topological charge of the two twist defects enclosed in each circle is trivial. The black loops with arrows stand for quasiparticle paths. The shaded region enclosed by a blue circle $C_0^{\prime }$ stands for a measurement which projects the state in loop $C_0^{\prime }$ to the trivial sector $(1,1)$.

Figure 11

Mapping Fig. 10 to single Ising theory on a torus. The loops $C_0$, $C_0^{\prime }$, $C_{g_1{g}_2}$, and $C_{g_2{g}_3}$ are depicted.

Figure 12

Orienting all of the defects along a line and cutting yields three counterpropagating edge modes. Since the defects are $Z_3$, there are now three different regions, $A_i$, $B_i$, and $C_i$, where we introduce different twisted tunneling terms.

Figure 13

Physical meaning of the zero-mode operators ${\alpha }_i$, ${\beta }_i$, and ${\gamma }_i$ for the $Z_3$ twist defect case. In subsequent figures, we use the bottom figure, which is more precisely described by the top figure.

Figure 14

Physical meaning of the quasiparticle Wilson loop operators.

Figure 15

Effect of a clockwise braid on the zero modes on the second and third dislocation. (a) ${\alpha }_2\to {\alpha }_3$; (b) ${\alpha }_3$ gets transformed into the combination of ${\gamma }_3^{†}{\gamma }_2$ and ${\alpha }_3$. Therefore, ${\alpha }_3\to {\gamma }_3^{†}{\gamma }_2{\alpha }_3$. Similar to the $Z_2$ case, the ordering ambiguity or additional phase factor can be fixed with additional constraints.

Figure 16

Illustration of the 1D protocol for carrying out an effective braiding process of the $Z_3$ twist defects. The purple, orange and, gray circles mark up the zero modes located on the corresponding defects.

Figure 17

Illustration of the equivalence between a sphere with three $Z_3$ twist defects and a torus. Panels (a) and (b) shows the sphere with twist defects cut along the branch-cut line connecting defects $1,2,3$. The red, blue, and green lines denote the edge states of the three layers along the cut. After the cutting, the system in (b) is equivalent to three decoupled systems of the three layers shown in (c). Then the three layers are glued along the branch-cut line after a twist. Along line $1-2$ the three layers are glued in the pattern of red to blue (rb), blue to green (bg), and green to red (gr), in which the first and second colors are the layer index of the lower and upper edges, respectively. The result of this gluing is shown in (d), which is equivalent to (e). Then the green and blue layers are glued along the $2-3$ line, shown in panels (f) and (g). Finally, the rest of the edge states in (g) are glued following the rule of red to green, green to blue, which leads to the torus shown in (h) and (i).

Figure 18

A loop around two $Z_3$ twist defects 2 and 3 (the dotted lines) on the sphere (a) is topologically equivalent to a nontrivial loop on the torus shown in (c). This is the same loop that appears in Fig. 15b. The color of the dotted line indicates the layer indices. This equivalence is obtained by following the steps shown in Fig. 17. Panel (b) shows an intermediate steps of the topological deformation which corresponds to Fig. 17d. Other intermediate steps are omitted.