Microscopic definitions of anyon data

We present microscopic definitions of both the $F$ symbol and $R$ symbol—two pieces of algebraic data that characterize anyon excitations in $\left(2+1\right)$-dimensional systems. An important feature of our definitions is that they are operational; that is, they provide concrete procedures for computing these quantities from microscopic models. In fact, our definitions, together with known results, provide a way to extract a complete set of anyon data from a microscopic model, at least in principle. We illustrate our definitions by computing the $F$ symbol and $R$ symbol in several exactly solvable lattice models and edge theories. We also show that our definitions of the $F$ symbol and $R$ symbol satisfy the pentagon and hexagon equations, thereby providing a microscopic derivation of these fundamental constraints.

Figure 1

Two processes involving two identical Abelian anyons moving along three paths 1,2,3. In the first process $\left(M_1{M}_2{M}_3\right)$, the anyon on the left travels to point $A$ and the anyon on the right travels to point $B$, and vice versa in the second process $\left(M_3{M}_2{M}_1\right)$. The final states of the two processes differ by $R(a,a)$, the exchange statistics of the anyons.

Figure 2

Space-time diagrams for two processes in which an anyon $abc$ splits into anyons $a,b,c$. The horizontal and vertical axes denote the space and time directions while the lines denote anyon world lines. The final states produced by these processes, $|1\rangle ,|2\rangle$, are equal up to the $U\left(1\right)$ phase $F(a,b,c)$.

Figure 3

The pentagon identity: Consistency requires that the product of the three $F$ symbols on the upper path is equal to the product of the two $F$ symbols on the lower path.

Figure 4

The two processes that are compared in the microscopic definition of the $F$ symbol.

Figure 5

The two operators $A_v$ and $B_p$ in the toric code Hamiltonian.

Figure 6

Movement and splitting operators for $e$ anyons. The splitting operator for $e$ is $S(e,e)={\sigma }_{1,2}^z$. The movement operator that moves $e$ from vertex 3 to 4 is $M_{4,3}^e={\sigma }_{3,4}^z$.

Figure 7

The two operators $Q_v$ and $B_p$ in the doubled semion Hamiltonian: $Q_v$ is a product of ${\sigma }^x$ operators acting on the three blue links adjacent to the vertex $v$, while $B_p$ involves a product of ${\sigma }^z$ operators on the six red links adjacent to the plaquette $p$ and a product of $i^{(1-{\sigma }^x)/2}$ on the six pink “legs” of the plaquette $p$.

Figure 8

(a) Definition of semion creation operator $a_n^{†}$: The operator $a_n^{†}$ acts on all $(m,U),(m,L)$, and $(m,R)$ links with $m<n$. (b) Movement and splitting operators for semions: The splitting operator $S(s,s)$ acts on the blue links near 1 and 2 according to Eq. (4.22), while the movement operator $M_{43}^s$ acts on the blue links near 3 and 4 according to Eq. (4.20).

Figure 9

Space-time diagrams for two processes in which an anyon $ab$ splits into anyons $a,b$. The final states $|1\rangle ,|2\rangle$ are equal up to a $U\left(1\right)$ phase, $R(a,b)$.

Figure 10

The second hexagon equation (5.3). Consistency requires that the product of the $U\left(1\right)$ phases along the upper path is equal to the product along the lower path.

Figure 11

(a) To define $R$, we include an additional set of anyon states $\left\{\right|a_Y\rangle \}$ located at a point $Y$ that is not on the $x$ axis. We assume that $Y$ has a negative $y$ coordinate and an $x$ coordinate between 1 and 2. (b) The two processes that are compared in the microscopic definition of the $R$ symbol.

Figure 12

$R$ symbol calculation for the toric code model. Vertices and plaquettes are labeled by $n$ and $\widehat{n}$, respectively. The ${\sigma }^z$ operators applied for $M^e$ or $S(e,m)$ are colored in red. The ${\sigma }^z$ operators applied for $M^m,S(m,e)$ are colored in blue.

Figure 13

To compute $R$ from a chiral boson edge theory, we choose $Y$ to be the point 3 located on the edge of the topological phase. This choice is acceptable because we can think of the movement operator $M_{Y1}$ as being supported along an arc which only touches the edge at points 1 and 3.

Figure 14

Two processes in which an anyon $d$ splits into anyons $a,b,c$. The inner product of the final states $|1,e,\mu ,\nu \rangle ,|2,f,\kappa ,\lambda \rangle$ defines the non-Abelian $F$ symbol.

Figure 15

The two microscopic processes that define the non-Abelian $F$ symbol.

Figure 16

Two processes in which an anyon $c$ splits into anyons $a,b$. The inner product of the final states $|1,\nu \rangle ,|2,\mu \rangle$ defines the non-Abelian $R$ symbol.

Figure 17

The two processes that are compared in the microscopic definition of the non-Abelian $R$ symbol.

Figure 18

Five microscopic processes used to derive the pentagon identity.

Figure 19

Graphical proof of $|1\rangle =F(a,b,c\left)\right|2\rangle$: The two processes are related by a microscopic $F$ move.

Figure 20

Graphical proof of $|2\rangle =F(a,bc,d\left)\right|3\rangle$. The first equality follows from topological invariance, while the second equality follows from a microscopic $F$ move.

Figure 21

Graphical proof of $|3\rangle =F(b,c,d\left)\right|5\rangle$. The first equality follows from topological invariance, while the second follows from a microscopic $F$ move.

Figure 22

Graphical proof of $|1\rangle =F(b,c,d\left)\right|4\rangle$. The first equality follows from topological invariance, while the second follows from a microscopic $F$ move.

Figure 23

Graphical proof of $|4\rangle =F(b,c,d\left)\right|5\rangle$. The first and third equalities follow from topological invariance, while the second follows from a microscopic $F$ move.

Figure 24

(a) Graphical representation of the identity (A11). (b) Schematic representation of the two unitary operators $U_L,U_R$ which are supported in the regions $x\le 0$ and $x\ge 3$, respectively, and satisfy $U_L|i\rangle =S_L|i\rangle$ and $U_R|i\rangle =S_R|i\rangle$.

Figure 25

Graphical derivation of the first hexagon equation (5.3).

Figure 26

Six microscopic processes used to derive the first hexagon equation.

Figure 27

Graphical representation of the identity that is needed to relate states $|4\rangle$ and $|5\rangle$ in Fig. 25.

Figure 28

(a) The relation $|\mathrm{\Psi }\rangle =|{\mathrm{\Psi }}^{\prime }\rangle$ is the first step in our proof of the identity in Fig. 27. (b) Regions of support of the operators $U_1^a,U_1^{b,c}$ used in proof of $|\mathrm{\Psi }\rangle =|{\mathrm{\Psi }}^{\prime }\rangle$.

Figure 29

(a) The relation $|{\mathrm{\Psi }}^{\prime }\rangle =|{\mathrm{\Psi }}^{″}\rangle$ is the second step in our proof of the identity in Fig. 27. (b) Regions of support of the operators $U_2^a,U_3^{b,c}$ used in proof of $|{\mathrm{\Psi }}^{\prime }\rangle =|{\mathrm{\Psi }}^{″}\rangle$.