Fermionic Toric Code

• The fermionic toric code is an exactly solvable 2+1D lattice model that generalizes Kitaev’s toric code using Z2-graded fusion and fermion parity constraints.
• It features a unique Hamiltonian structure with commuting projectors that enforce closed-string conditions and proper fermionic occupancy.
• Its topological order is characterized by modified anyon braiding statistics, a four-fold degenerate ground state, and tensor network formulations.

The fermionic toric code is an exactly solvable lattice model that generalizes Kitaev’s original toric code to the fermionic setting. It realizes (2+1)D intrinsically fermionic topological order by incorporating $Z_2$-graded fusion rules, nontrivial associativity relations sensitive to fermion parity, and a Hamiltonian structure that depends crucially on the anticommuting nature of fermion operators. This model cannot be fully captured by bosonic string-net or gauge theory constructions, as reflected in its fundamentally distinct locality, braiding statistics, and field-theoretic description.

1. Construction and Hamiltonian Structure

The canonical construction places qubits on lattice edges (typically a honeycomb or trivalent lattice) and spinless fermions on the vertices. The Hilbert space is

$$L_\Gamma^{\mathrm{fTC}} = \bigoplus_{I \subset V} \left( \prod_{v \in I} c_v^\dagger |0_V\rangle \otimes \bigotimes_{e \in E(\Gamma)} \mathbb{C}^2 \right)$$

where $c_v^\dagger$ is the creation operator for a fermion at vertex $v$ and the rest is the standard qubit space of the bosonic toric code (Gu et al., 2013).

The Hamiltonian is a sum of commuting projectors:

$H_{f\mathrm{TC}} = - \sum_{v} Q_v - \sum_{p} Q_p$

where the vertex term $Q_v$ enforces both the closed-string constraint and proper local fermion occupancy:

$$Q_v = \frac{1}{2}\left(1 + \prod_{i \in v} \sigma_i^z\right) \left\{ 1 - \left[ c_v^\dagger c_v - N^f(\{ \sigma_{a \in v}^z \}) \right]^2 \right\}$$

with $N^f$ a function determined by the $Z_2$-graded fusion rule:

$N^f(1,1,0) = 1,\quad N^f = 0\ \text{otherwise}$

The plaquette term flips the spins on the boundary and updates adjacent fermion decorations based on the fusion rules and local associativity relations:

$$Q_p = \frac{1}{2}\left(1 + \widehat{O}\left(\{\sigma_{b \in p}^z\}\right) \prod_{i \in p} \sigma_i^x\right) \prod_{v \in p} Q_v$$

where $\widehat{O}$ involves products of fermion creation/annihilation operators and local phase factors determined by the graded pentagon equations.

2. $Z_2$-Graded Fusion Rules and Associativity

The key difference from the bosonic toric code is the introduction of $Z_2$-graded fusion coefficients. In the bosonic case, $1 \otimes 1 = 0$ (two strings fuse to vacuum). In the fermionic variant,

• $N_{11}^{0, f} = 1$: two strings fusing at a vertex can optionally decorate that vertex with a fermion (odd parity).
• $N_{00}^{0, b} = N_{01}^{1, b} = N_{10}^{1, b} = 1$: as in the bosonic model. All other channels vanish.

The local “F-symbols” (associativity relations) are $Z_2$-graded and potentially pick up a phase $\alpha$ when a fermion is transposed in a fusion tree. Specifically, consistency of the associativity relations under moves that generate a fermion loop constrain $\alpha^2 = -1 \implies \alpha = \pm i$ (Gu et al., 2013). These Grassmann-valued data lead to nontrivial phase factors ($\pm i$) in processes involving fermion exchange, and more generally to “graded pentagon equations.”

3. Ground State Manifold and Topological Order

The ground state is a superposition of all closed string configurations, each decorated according to the $Z_2$-graded fusion rule:

$$|\Psi_{f\mathrm{TC}}\rangle = \sum_X \text{phase}(X) \left( \prod_{v\in X} c_v^\dagger \right) |X\rangle$$

where the sum runs over closed string coverings and $|X\rangle$ denotes the spin configuration (Gu et al., 2013). On a torus, this leads to a four-fold degenerate ground state, as in the bosonic toric code.

The $K$-matrix description is modified from the bosonic case:

$$K^\text{TC} = \begin{pmatrix} 0 & 2 \ 2 & 0 \end{pmatrix}, \qquad K^{f\mathrm{TC}} = \begin{pmatrix} 0 & 2 \ 2 & 1 \end{pmatrix}, \quad K^{\overline{f\mathrm{TC}}} = \begin{pmatrix} 0 & 2 \ 2 & 3 \end{pmatrix}$$

The presence of an odd diagonal entry in $K^{f\mathrm{TC}}$ signals fermionic topological order; no local bosonic system realizes this $K$-matrix (Gu et al., 2013).

4. Anyon Types, Fusion, and Braiding

Both bosonic and fermionic toric codes exhibit four anyon species with abelian braiding, but their statistics differ. In $f$TC:

• The fermion ($\psi$) is a superselection sector with odd parity.
• Two bosonic anyons ($\sigma^{(1)}, \sigma^{(2)}$) correspond to parity-even electric and magnetic sectors.
• Fusion rules:

$$\psi^2 = (\sigma^{(1)})^2 = (\sigma^{(2)})^2 = I,\quad \sigma^{(1)}\sigma^{(2)} = \psi$$

• Mutual statistics:

$$R_{\sigma^{(1)},\sigma^{(2)}} R_{\sigma^{(2)},\sigma^{(1)}} = -1$$

Thus, the $e$ and $m$-like excitations exhibit mutual semionic statistics, while the $\psi$ sector is fermionic.

The modular $S$ and $T$ matrices explicitly encode the distinction via nontrivial $\pm i$ phase factors:

$$T = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & -\alpha^{-1} & 0 \end{pmatrix}, \quad S = \frac{1}{2} \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & 1 & -1 & -1 \ 1 & -1 & \pm i & \mp i \ 1 & -1 & \mp i & \pm i \ \end{pmatrix}$$

with $\alpha = \pm i$ from the associativity relations (Gu et al., 2013).

5. Mapping to Conventional Fermion and Spin Systems

Via the Jordan–Wigner transformation, the fermionic toric code can be mapped to spin or Majorana systems. For example, in the toric honeycomb model,

• Spins are mapped to pairs of Majorana operators ($\psi, \chi$) per site,
• Complex fermions are constructed as combinations:

$$c^\dagger_{\uparrow, i} = \frac{1}{2}(\psi_{i_w} - i \psi_{i_b}), \quad c^\dagger_{\downarrow, i} = \frac{1}{2}(\chi_{i_w} - i \chi_{i_b})$$

• Plaquette operators translate to products of local fermion parity, e.g.,

$W_p = (2n_{\uparrow,i} - 1)(2n_{\uparrow, j} - 1)$

• The Hamiltonian becomes:

$$H = -J_1\sum_{⟨ij⟩_{hd}}(2n_{\uparrow,i} - 1)(2n_{\uparrow,j} - 1) -J_2\sum_{⟨ij⟩_{hd}}(2n_{\downarrow,i}-1)(2n_{\downarrow,j}-1)$$

Extra boundary couplings are introduced to recover the toric code’s fourfold degeneracy on a torus in the fermionic chain mapping (Liang et al., 2010).

6. Tensor Network and Category Theoretic Formulations

The ground state and excitation spectrum can be formulated via fermionic tensor networks, in particular, fPEPS and fermionic MPO-injectivity (Wille et al., 2016, Williamson et al., 2016). Here,

• Local tensors $A$ carry both bosonic and Grassmann-valued fermionic indices.
• The virtual layer supports an fMPO symmetry constructed from tensors $T_\pm(g)$ with a branch-dependent structure and explicit Grassmann parity constraints.
• fMPO-injectivity is enforced via the existence of a (pseudo-)inverse tensor $\widetilde{\mathcal{A}}$ such that $\widetilde{\mathcal{A}} \cdot A = P$, $P$ projecting onto the fMPO-symmetric subspace.
• The topological data, including fusion and associativity ($F$-symbols), is encoded via group cohomology elements ($H^1$, $H^2$, supercohomology $\overline{H}^3$).

These features enable a systematic classification of fermionic topological phases and direct implementation in tensor-network-based simulations (Williamson et al., 2016).

7. Physical Realizations and Applications

The fermionic toric code underpins the classification of 2+1D fermionic topological orders, provides intuition for the physics of fractional quantum Hall states, and serves as a target phase for quantum simulation platforms:

• Majorana island arrays, as in two-dimensional superconducting networks (Terhal et al., 2012), where fourth-order perturbation theory yields an effective toric code Hamiltonian.
• Platform-agnostic approaches, such as superconducting qubit arrays or custom fermion codes on quantum hardware, can implement the toric code Hamiltonian or its fermion-decorated variant (Chien et al., 2020).
• Quantum error correction using Majorana qubit codes that specifically detect and correct for odd- and even-weight errors, leveraging the fermionic code’s unique stabilizer structure (Kundu et al., 2023).

Experimental protocols for state preparation exploit the Clifford circuit structure inherent in toric code-like stabilizer Hamiltonians, with measurement-based "gluing" for arbitrary lattice geometries (Chen et al., 2022).

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Summary Table: Fermionic Toric Code vs. Bosonic Toric Code

Aspect	Bosonic Toric Code	Fermionic Toric Code
Hilbert space	$\bigotimes_{e} \mathbb{C}^2$	Direct sum over fermionic Fock sectors on vertices and qubits on edges
Fusion rules	$1 \otimes 1 = 0$	$N^{0,b}_{11}=0$, $N^{0,f}_{11}=1$, graded by fermion parity
Ground state degeneracy	Four (on torus)	Four (on torus), but structure distinguished by fermionic parity
Effective field theory	$$K = \begin{pmatrix} 0 & 2 \ 2 & 0 \end{pmatrix}$$	$$K = \begin{pmatrix} 0 & 2 \ 2 & 1 \end{pmatrix}$$ (or dual with 3)
Anyon data	Abelian, mutual semions with bosonic $S$, $T$ matrices	Abelian, phases $\pm i$ in braiding; $S$, $T$ matrices encode fermion parity
Locality	Strict tensor product structure	Fock space, non-trivial anti-commutation, non-local in bosonic mapping
Associativity (F-symbols)	Trivial pentagon relation	Graded pentagon, $6j$ symbols with $\alpha = \pm i$ required for consistency

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The fermionic toric code thus exemplifies a class of exactly solvable 2+1D models with intrinsic fermionic topological order, not realizable in purely local bosonic systems. Its defining features—graded fusion, nontrivial associativity, modified $K$-matrix, and unique braiding statistics—establish it as the paradigmatic model for investigating fermionic topologically ordered phases, their classification, and potential application in quantum simulation and error correction.