Lattice model for fermionic toric code

The $Z_2$ topological order in $Z_2$ spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.

Figure 1

The honeycomb lattice has a fixed branching structure (branching arrows from down to up). (a) A vertex term is not just a product of ${\sigma }^z$ surrounding the vertex, but also contains a projector for the fermion number on the vertex, which is determined by the so-called ${\mathbb{Z}}_2$-graded fusion rule. A plaquette term only acts on the subspace of the projector $P_p={\prod }_{v\in p}{Q}_v$, and in addition to the ${\sigma }^x$ surrounding the plaquette, a term $\widehat{O}\left(\left\{{\sigma }_{b\in p}^z\right\}\right)$ consisting of a product of fermion creation/annihilation operators is also needed. (b) We map a spin-$\uparrow$ state to the absence of a string and a spin-$\downarrow$ state to the presence of a string. (c) To define the operator $\widehat{O}\left(\left\{{\sigma }_{b\in p}^z\right\}\right)$, we label the links of a plaquette by $i,j,k,l,m,n$ and the vertices of a plaquette by $1,2,3,4,5,6$. In Table 1, we compute the explicit expression of $\widehat{O}\left(\left\{{\sigma }_{b\in p}^z\right\}\right)$ using the fermionic associativity relations.

Figure 2

Examples of the $Q_p$ operator acts on closed string configurations for a hexagon. (a)–(c) correspond to the first three terms in Table 1. We use solid/dotted lines to represent the presence/absence of strings and black dots to represent the occupation of fermions

Figure 3

The fusion rules for the toric code model. Here we use solid/dotted lines to represent presence/absence of (${\mathbb{Z}}_2$) strings.

Figure 4

(a) The only nontrivial associativity relation. (b) An example of pentagon equation that issues the self-consistency of the associativity relations for toric code model.

Figure 5

(a) The branched fusion. The yellow arrow indicates the local direction that induces a branching structure for the links connecting to a vertex. (b) The ${\mathbb{Z}}_2$ fusion rules with a graded structure. The solid dot represents a fermion.

Figure 6

All possible fermionic associativity relations with a fixed choice of a branching structure (branching arrows from up to down). Only the first associativity relation is nontrivial, and all the others only change the shape/length of a string or the position of a fermion. Associativity relations with different branching structures can be computed from the above basic relations and the bending moves.

Figure 7

One of the fermionic pentagon equations that the fermionic associativity relations need to satisfy.

Figure 8

(a) The left-/right-handed bending moves. (b) By applying the bending moves in the upper path and lower path, we have $\beta =-{\beta }^{\prime }$. (c) Fermionic associativity relations with different branching structure can be computed through bending moves. (d) The “loop weight” can be computed by using associativity relations.

Figure 9

An example of evaluating the operator $\widehat{O}\left(\left\{{\sigma }_{b\in p}^z\right\}\right)$ for a given initial decorated closed string configuration.

Figure 10

By applying the Dehn twist and making use of the fermionic associativity relations, we are able to compute the $T$ matrix for the fermionic toric code model.

Figure 11

By applying the ${90}^{\circ }$ rotation and making use of the associativity relations, we are able to compute the $S$ matrix for the fermionic toric code model.

Figure 12

The only five nontrivial fermionic pentagon equations that involve the nontrivial associativity relation in Fig. 6, and only the first relation (a) gives rise to a constraint on the numerical values of $\alpha$. The other cases which only change the shape/length of a string or the position of a fermion automatically satisfy the pentagon equations.