Genons, twist defects, and projective non-Abelian braiding statistics

TL;DR

The paper develops a unified framework for extrinsic twist defects (genons) in topological phases, showing that braiding these defects realizes adiabatic Dehn twists on an effective high-genus surface with genus $g$ proportional to the number of defects. It provides multiple complementary descriptions—bulk geometry, 1+1D edge CFT, and 1D gating protocols—to compute projective non-Abelian braiding statistics for $Z_2$ and $Z_3$ genons in both Abelian and non-Abelian states, including Ising$\times$Ising and three-component Abelian cases. It demonstrates that genons can carry quantum dimension $d=\sqrt{|m-l|}$ (with specific instances $d=2$ for Majorana-related cases) and, in some setups, enable universal topological quantum computation by exploiting Dehn twists, even when the host state is non-universal. The work also analyzes confinement versus deconfinement via symmetry gauging (orbifold states), revealing that gauging can modify the braid representation and quantum dimension, thereby enriching the landscape of topological quantum computing primitives.

Abstract

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 non-universal for TQC.

Figures (18)

• Figure 1: Worldline of quasiparticle $\gamma_i$ and twist defect, labelled by $g$. Braiding $\gamma_i$ around the twist defect changes the quasiparticle to $\gamma_{g(i)}$. The arrow indicates the time direction.
• Figure 2: A pair of twist defects induces two distinct non-contractible loops, labelled $a$ and $b$. For twist defects in $(mm0)$ states, these loops effectively cross only once, leading to a magnetic algebra for the quasiparticle loop operators. Here, the branch cut indicated by the dash 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/anti-vortex pair.
• Figure 3: $n$ pairs of twist defects induces $2(n-1)$ distinct non-contractible loops, $a_i, b_i$, for $i = 1,\cdots, n-1$. For twist defects in the $(mm0)$ states, the quasiparticle loop operators give rise to $n-1$ copies of the magnetic algebra, leading to $|m|^{n-1}$ topologically degenerate states.
• Figure 4: The $U(1) \times U(1)$ CS theory with $n$ pairs of dislocations on a sphere can be mapped to a $U(1)$ 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 $U(1)$ 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 $M_{n-1}$.barkeshli2010
• Figure 5: (a) A loop that encloses the twist defects labelled $2$ 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 non-contractible 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.