Lattice Model for Fermionic Toric Code

TL;DR

The paper constructs an exactly solvable lattice model for a fermionic version of the toric code by introducing a $$-graded fusion rule and fermionic pentagon equations. This framework yields a fixed-point Hamiltonian with commuting projectors and a ground-state structure identical in degeneracy to the bosonic toric code but carrying distinct topological data, captured by the $T$ and $S$ matrices and the $K$-matrix $K^{fTC}=0221$ (and its dual 0223). The authors formulate a complete algebraic machinery using Grassmann-valued 6j symbols, bending moves, and spin-structure-aware branching to realize and analyze the fermionic topological order, including the ground-state wavefunction on the sphere and torus, and the algebraic anyon model. The work provides a general method to classify and construct 2+1D fermionic topological orders and suggests experimental avenues in systems with strong spin-orbit coupling or flat bands where such orders may emerge.

Abstract

The Z_2 topological order in Z_2 spin liquid and in exactly soluble 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 exact soluble 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 soluble models and fermionic topologically ordered states.

Figures (12)

• 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 additional to the $\sigma^x$ surrounding the plaquette, a term $\hat{O}(\{\sigma_{b\in p}^z\})$ 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 $\hat{O}(\{\sigma_{b\in p}^z\})$, 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.\ref{['Hamiltonian']}, we compute the explicit expression of $\hat{O}(\{\sigma_{b\in p}^z\})$ using the fermionic associativity relations.
• Figure 2: Examples of the $Q_p$ operator acts on closed string configurations for a hexagon. (a), (b) and (c) correspond to the first three terms in table \ref{['Hamiltonian']}. We use solid/dot lines to represent 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/dot lines to represent presence/absence of ($\mathbb{Z}_2$) strings.
• Figure 4: (Color online)(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: (Color online) (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.