Unconventional fusion and braiding of topological defects in a lattice model

We examine non-Abelian topological defects in an Abelian lattice model in two dimensions. We first construct an exact solvable lattice model that exhibits coexisting and intertwined topological and classical orders. The anyon types of quasiparticle excitations are permuted by lattice symmetry operations like translations, rotations, and reflections. The global anyon permutation symmetry has a group structure of $S_3$, the permutation group of three elements. Topological crystalline defects—dislocations and disclinations—change the anyon type of an orbiting quasiparticle. They exhibit multichannel order-dependent fusion rules and projective braiding operations. Their braiding and exchange statistics breaks modular invariance and violates the conventional spin-statistics theorem. We develop a framework to characterize these unconventional properties that originate from the semiclassical nature of defects.

Figure 1

Trivial twistless defects that preserve local tricoloring $\left(YRB\right)$ bipartite $\left(•,\circ \right)$ order. (a) Negatively curved disclination with Frank angle $-{120}^{\circ }$. (b) Positively curved disclination with Frank angle $+{120}^{\circ }$.

Figure 2

Plaquette stabilizers ${\widehat{P}}_{•}$ and ${\widehat{P}}_{\circ }$ as tensor products of rotors $\sigma ,\tau$.

Figure 3

Contractible Wilson loop on a honeycomb lattice. (a) Product of yellow plaquette operators ${\widehat{P}}_{•}^Y$ and inverses of red plaquette operators ${\left({\widehat{P}}_{•}^R\right)}^{-1}$ inside a domain $\Omega$ leaving a string of $\sigma$ and $\tau$ rotor operators $W{\left(\partial \Omega \right)}_{•}^B$ (on highlighted $•$ and $\circ$ sites) along the boundary $\partial \Omega$ that connects a series of blue plaquettes. (b) Schematics of the boundaries $W{\left(\partial \Omega \right)}_{•}^B$, where the arrows denote the rotors $\sigma ,\tau$ or ${\sigma }^{-1},{\tau }^{-1}$ along the strings in the lattice.

Figure 4

Color splitting of Wilson string through tricolor, trivalent sources and drains. Product of rotor operators are taken over highlighted $•$ and $\circ$ vertices.

Figure 5

Product of rotor operators along a Wilson string $W_{•}^{\chi }=\prod {\sigma }_{v_{•}}{\tau }_{v_{\circ }}$ and $W_{\circ }^{\chi }=\prod {\tau }_{v_{•}}{\sigma }_{v_{\circ }}$ for $\chi =Y,R,B$. $(\sigma ,\tau )\to ({\sigma }^{-1},{\tau }^{-1})$ upon arrow reversal.

Figure 6

Primitive cycles on a genus $g=3$ surface.

Figure 7

Intersection of Wilson strings of opposite $•,\circ$ types. (a) Abelian ${\mathbb{Z}}_k$ phases $e^{\pm 2\pi i/k}$ accumulated by passing a $\circ$-Wilson string in front of an intersecting $•$ one of different color. Intersections are highlighted at bold vertices. (b) Schematics showing intersection number $\langle *,*\rangle =-1,0,+1$ between different directed Wilson strings.

Figure 8

(a) Color branch cut along a zig-zag edge where same color hexagons share edges and vertices. (b) Wilson strings and Abelian anyon changes color across a threefold branch cut. (c) A closed Wilson string ${\Sigma }_1$ composed of $k/3$ copies of the open path that closes at the unicolor drain.

Figure 9

Abelian anyon excitations (highlighted hexagons with signs) connected by open Wilson strings. Products of rotors are taken over highlighted vertices along strings with appropriate signs according to the arrow rule in Fig. 3.

Figure 10

(a) Abelian $•$ anyons (hexagons) connected by a Jordan-Wigner $\circ$ string. Anyons are probed by Wilson loops ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$, which detect a ${R_{•}}^{-1}$ and a ${Y_{•}}^{-1}=R_{•}\times B_{•}$ inside, respectively. (b) Diagrammatic representation of a composite anyon so that a $•$ anyon is attached with a $\circ$ string and vice versa.

Figure 11

$•$- or $\circ$-type Abelian anyons (black or white dots) on an integer mod $k$ triangular lattice. The boundary of the hexagon is identified so that opposite dots along boundary lines represent the same anyon. The vacuum is represented by the origin and the three fundamental anyons $Y,R$, and $B=Y^{-1}{R}^{-1}$ are represented by the primitive lattice vectors ${\mathbf{e}}_Y,{\mathbf{e}}_R,{\mathbf{e}}_B=-{\mathbf{e}}_Y-{\mathbf{e}}_R$ for $•$ type or ${\mathbf{f}}^Y,{\mathbf{f}}^R,{\mathbf{f}}^B=-{\mathbf{f}}^Y-{\mathbf{f}}^R$ for $\circ$ type.

Figure 12

(a) Basis choice of splitting space $V_{[\mathbf{a}+\mathbf{b}]}^{\left[\mathbf{a}\right]\left[\mathbf{b}\right]}$. The black and gray lines are Jordan-Wigner $•$ and $\circ$ strings, respectively. (b) The $R$ symbol of exchanging Abelian anyons $\left[\mathbf{a}\right]$ and $\left[\mathbf{b}\right]$ under the basis choice for splitting space $V_{[\mathbf{a}+\mathbf{b}]}^{\left[\mathbf{a}\right]\left[\mathbf{b}\right]}$ in (a). The phase is obtained by passing the gray line in front of the black one at the intersection.

Figure 13

Topological spin of the Abelian anyon ${\left(Y_{•}\right)}^{y_1}{\left(R_{•}\right)}^{r_1}{\left(Y_{\circ }\right)}^{y_2}{\left(R_{\circ }\right)}^{r_2}$ evaluated by ${360}^{\circ }$ rotation.

Figure 14

Twisting Abelian anyons (colored hexagons) around twist defects (black crosses and colored pentagon). (a) Cyclic color permutation, $\chi \in \{Y,R,B\}=\{0,1,2\}$, of a threefold twist defect $[1/3]$ and its antipartner $\left[\overline{1/3}\right]$. (b) Color and sublattice transposition of a twofold twist defect ${[1/2]}_{\chi }$ with color $\chi$. (c) Color and sublattice twisting of the two threefold and three twofold twist defects.

Figure 15

Disclination twist defects (shaded plaquettes and vertices). Color or sublattice frustrations (question marks) give rise to branch cuts (wavy or dashed lines) that alter anyon types (colored lines) across. (a) A threefold twist defect generated by a $+{120}^{\circ }$ disclination centered at a tetravalent vertex. Sublattice types $•,\circ$ are not affected by the defect. (b) An anti-threefold twist defect generated by a $-{120}^{\circ }$ disclination centered at a bivalent vertex and permutes colors in the opposite direction around. (c) A pair of twofold twist defects generated by a $\pm {60}^{\circ }$-disclination dipole. Each violates both tricoloring and bipartite structure.

Figure 16

A triplet of dislocation threefold twist defects connected by two color permutation branch cuts (wavy lines).

Figure 17

Tetravalent and bivalent vertices $u_4,u_2$ and their nearest plaquettes.

Figure 18

Heptagon and pentagon plaquettes.

Figure 19

Energy levels of $H_{5,7}$ as a function of phase variable $\varphi$. Level crossings of ground states are shown in dashed circles. Colors label $k$ different discrete values for $p\left(\varphi \right)$.

Figure 20

Prototypes of nonlocal closed Wilson loops around twist defects. (a) A closed string ${\mathcal{A}}^{\chi ,•/\circ }$, either entirely $•$ type or $\circ$ type, enclosing two threefold twist defects $[1/3]$ (black crosses). It can be equivalently represented by a loop (left) or a closed branched path with a tricolored source and drain (right) [119]. Wilson strings change colors $\chi \in \{Y,R,B\}=\{0,1,2\}$ across threefold branch cuts (wavy lines). (b) A Wilson loop ${\mathcal{B}}^{\chi ,•/\circ }$ surrounding a threefold twist defect $[1/3]$ and its antiparticle $\left[\overline{1/3}\right]$. (c) A Wilson loop ${\mathcal{Z}}^{\chi }$ containing two twofold twist defects of the same color ${[1/2]}_{\chi }$ (colored pentagons). Up to an Abelian phase, the loop can be flipped inside out and changes color and sublattice types according to Fig. 14.

Figure 21

Local Wilson loops that circle a single defect. (a) Species identification of a twofold twist defect by small Wilson loops ${\Theta }_{\chi }$ that circle twice. The $•$ string is chosen by convention to sit below a $\circ$ one at the intersection. (b) A local Wilson loop, ${\Sigma }_{•}$ or ${\Sigma }_{\circ }$, surrounding a single threefold twist defect $k/3$ times. It consists of a $k$-valent unicolor source and $k/3$ tricolored drains or is equivalently constructed by dragging the color permutation invariant Abelian anyon ${\overline{\kappa }}_{•/\circ }={\left(Y_{•/\circ }\right)}^{-k/3}{\left(R_{•/\circ }\right)}^{k/3}$ (see Fig. 11) around the defect. It exists only when $k$ is divisible by 3.

Figure 22

Wilson loop ${\mathcal{A}}_{i,i+1}^{\chi ,•}$ and ${\mathcal{A}}_{i,i+1}^{\chi ,\circ }$ between the $i\mathrm{th}$ and $(i+1)\mathrm{th}$ threefold twist defect. The sublattice type $•,\circ$ and color $\chi =Y,R$ is view at the base point (gray square) on the loop.

Figure 23

Intersection (brown dots) between nearest and next-nearest $•$- and $\circ$-type Wilson loops.

Figure 24

Eigenvalue lattice for the $•$-Wilson operators for each $i=1,2,...,N-2$. $({\mathcal{A}}_{i,i+1}^{Y,•},{\mathcal{A}}_{i,i+1}^{R,•})=(e^{2\pi i{y}_i/k},e^{2\pi i{r}_i/k})$ at the lattice point ${\mathbf{a}}_i=y_i{\mathbf{e}}_1+r_i{\mathbf{e}}_2$. (a) Lattice for $k$ not divisible by 3. (b) Lattice for $k$ divisible by 3 with shifted origin ${\mathbf{s}}_0$ that depends on the ${\mathbb{Z}}_3$ value of (3.34). The thickened sublattice contains the eigenvalues allowed by (3.34).

Figure 25

Wilson operator ${\mathcal{Z}}_{i,i+1}^{\chi }$ with color $\chi =Y,R,B$ and $•$-sublattice type at the based point chosen at the gray square. Local Wilson loop ${\Theta }_i^{\chi }=w^{l_i^{\chi }},w=e^{2\pi i/k}$, about the $i\mathrm{th}$ defect with $\chi$-colored and $•$-sublattice starting point.

Figure 26

Two local Wilson operators ${\Theta }_i^{\chi }$ and ${\Theta }_{i+1}^{\chi }$ can be deformed, joined, and unlinked into a pair of nonlocal Wilson loops of type ${\chi }_{•}$ and ${\chi }_{\circ }^{\prime }$ related by twofold symmetry transformation induced by defects.

Figure 27

Intersection (brown dots) between adjacent Wilson operators ${\mathcal{Z}}_{2j-1,2j}^{\chi }$ and ${\mathcal{Z}}_{2j,2j+1}^{{\chi }^{\prime }}$ about blue twofold defects ${[1/2]}_B$. Only different colors and sublattice-type strings intersect.

Figure 28

Abelian anyon channel $\left[{\mathbf{a}}_j\right]=[{\mathbf{a}}_{•}^j,{\mathbf{a}}_{\circ }^j]$ of the $j\mathrm{th}$ pair of twofold defects measured by Wilson loops ${\mathcal{Z}}_{2j-1,2j}^{\chi }$.

Figure 29

Viewing curve or open Wilson strings (directed dashed lines) ${\widehat{\lambda }}_i$ around the $i\mathrm{th}$ twist defect starting and ending at a fixed viewpoint (gray square). (a) Series rearrangement of twist defects without passing defects across viewing curves. (b) Redefining the viewing curves $({\widehat{\lambda }}_6,{\widehat{\lambda }}_7)\leftrightarrow ({\widehat{\lambda }}_6^{\prime },{\widehat{\lambda }}_7^{\prime })$ in the same system by passing ${\widehat{\lambda }}_7$ across the sixth defect, thereby changing the color of the seventh one.

Figure 30

Relation between species labels of the fusion ${[1/2]}_{\chi ,{\mathbf{l}}_2}\times {[1/2]}_{\chi \pm 1,{\mathbf{l}}_1}={[1/3]}_{\mathbf{s}}$ or ${\left[\overline{1/3}\right]}_{\mathbf{s}}$ for $k$ divisible by 3. (a) The left-hand side equals ${\Theta }_Y^{k/3}{\Theta }_R^{-k/3}=w^{l_Y-l_R},w=e^{2\pi i/k}$. $\chi =Y,R,B$ is the color of the twofold defect. (b) The right-hand side equals ${\Sigma }_{•}^{-1}{\Sigma }_{\circ }=e^{i\frac{2\pi }{3}(s_{\circ }-s_{•})}$, where ${\kappa }_{•}={\left(Y_{•}\right)}^{k/3}{\left(R_{•}\right)}^{-k/3}$ and ${\overline{\kappa }}_{\circ }={\left(Y_{\circ }\right)}^{-k/3}{\left(R_{\circ }\right)}^{k/3}$.

Figure 31

Representation of objects. Solid black and gray lines are $•$ and $\circ$ strings. Wavy and dashed lines are threefold and twofold branch cuts. $\mathbf{s}=(s_{•},s_{\circ })$ labels $[1/3]$ and $\left[\overline{1/3}\right]$ for $k$ divisible by 3, for $s_{•},s_{\circ }=-1,0,1$. $\mathbf{l}=(l_Y,l_R)$ labels ${[1/2]}_{\chi }$, for $\chi =Y,R,B$, so that $\mathbf{l}-{\mathbf{l}}_0\left(\chi \right)\in {\mathbb{Z}}_k\oplus {\mathbb{Z}}_k$, where ${\mathbf{l}}_0\left(\chi \right)$ is the self-reciprocal species label in (4.25).

Figure 32

Wilson string configurations for splitting states in Eq. (4.38). Crosses and pentagons represent threefold and twofold defects. Threefold defects are equipped with a species label $\mathbf{s}$ only when $k$ is a multiple of 3. Solid black and gray lines are Wilson $•$ and $\circ$ strings, respectively. All $\circ$ strings act after and sit above all $•$ ones except ${\mathcal{B}}_{\circ }^{\mathbf{m}}$ in (e). All closed black $•$ strings act trivially on ${|\text{GS}\rangle }_0$ in Eq. (4.37), while closed gray $\circ$ strings are nontrivial on ${|\text{GS}\rangle }_0$ unless contractible. The phase $\phi \left(\mathbf{n}\right)$ in (g) is given by Eqs. (4.39) and (4.40) for $w=e^{2\pi i/k}$.

Figure 33

Basis transformation of defect states by composition of fundamental $F$ moves. ${\lambda }_i$ are objects, $x_i,y_i$ are admissible fusion channels, and ${\mu }_i,{\nu }_i,{\zeta }_i$ label splitting space degeneracies.

Figure 34

Gluing a conjugate pair of twofold defects on a punctured sphere into a torus with a branch cut (blue dashed line) along the longitudinal cycle.

Figure 35

Dehn twists. (a) Double Dehn twist $T$ along the horizontal direction that leaves the branch cut invariant. (b) $T$ action on a vertical Wilson loop $W\left({\mathbf{c}}_{•}\right)$. (c) Dehn twist $S$ along the vertical direction and its action on a double Wilson loop $\Theta \left({\mathbf{a}}_{\circ }\right)$.

Figure 36

Braiding operation $B$ generated as a sequence of $F$ and $R$ moves. ${\lambda }_i$ are defects of the same $S_3$ type. Ground states are labeled by the intermediate channels $x_i,y,z$ and possible vertex degeneracies ${\mu }_i,{\nu }_i$.

Figure 37

Yang-Baxter identity.

Figure 38

Contraction of the large braid $\mathcal{M}$ in (5.47) on a sphere. Color definition is cyclicly permutated after passing the braiding path across the fixed viewpoint.

Figure 39

Three threefold defects on a sphere. The covering space has six sheets labeled by color and sublattice type. They are glued by identifying similar edges. The result is a pair of disjoint tori distinguished by $•,\circ$.

Figure 40

Six twofold defects, two of each color, on a sphere. The covering space has genus 4. However, there are four loops that are noncontractible in the cover but trivial in the base space: the sum of the small dashed loops, the longer solid line, and ${60}^{\circ }$ rotations of these two.

Figure 41

Triangular lattice representing the integer mod $k$ powers of $\circ$-Wilson operators ${\left({\mathcal{A}}_{i,i+1}^{Y,\circ }\right)}^{n_{2i-1}}{\left({\mathcal{A}}_{i,i+1}^{R,\circ }\right)}^{n_{2i}}$ by lattice points ${\mathbf{n}}_i=n_{2i-1}{\mathbf{e}}_1^{\prime }+n_{2i}{\mathbf{e}}_2^{\prime }$. When $k$ is divisible by 3, the long lattice vector $\mathbf{K}$ labels the good quantum number ${\Sigma }_{i,i+1}^{\circ }$ in (3.34) and divides the lattice into three bands (colored by black, green, and purple circles). $\circ$-Wilson operators on different bands related by the $\mathbf{K}$ vector are identical up to a ${\mathbb{Z}}_3$ phase determined by the eigenvalue of ${\Sigma }_{i,i+1}^{\circ }$.

Figure 42

A pentagon identity.

Figure 43

The Whitehead link between defect world line (red) and anyon trajectory (black/gray) represented by the quadratic phase $\Theta (-{\mathbf{a}}_{\circ })\Theta \left({\mathbf{a}}_{\circ }\right)=w^{{\mathbf{a}}_{\circ }^{T}i{\sigma }_y{\Lambda }_{\chi }{\mathbf{a}}_{\circ }}$ in (D3).