3D Fermionic Toric Code

• 3D Fermionic Toric Code is a 3+1D Z2 topological order characterized by emergent fermionic point charges and bosonic loop fluxes.
• Its cubic lattice model employs deformed toric code operators that yield fermionic self-statistics and nontrivial charge–loop braiding.
• Recent studies integrate higher-form symmetry, symmetry enrichment, and Floquet protocols to achieve finite-temperature topological order and robust error correction.

The 3D fermionic toric code is a $3{+}1$-dimensional $\mathbb Z_2$ topological order in which the deconfined point excitation is an emergent fermion and the flux sector is loop-like rather than anyonic. In its canonical form it is a variant of the usual 3D toric code, defined on a cubic lattice with a single qubit on each edge and commuting vertex and plaquette terms, but with operator algebra chosen so that the point-like charge has fermionic self-statistics rather than bosonic self-statistics (Zhou et al., 4 Mar 2025). The phase has also been described as the ordinary fermionic $\mathbb Z_2$ gauge theory with fermionic charges and bosonic $\mathbb Z_2$ flux loops, distinguishing it from an anomalous variant with fermionic loops (Fidkowski et al., 2021). Recent work places the 3D fermionic toric code at the intersection of higher-form symmetry, finite-temperature topological order, symmetry enrichment, and Floquet quantum error correction (Zhou et al., 4 Mar 2025, Cheng, 2015, Watanabe et al., 13 Feb 2026).

1. Canonical formulation as a 3D $\mathbb Z_2$ gauge theory

A standard presentation uses a cubic lattice with periodic boundary conditions, i.e. a 3-torus, and a single qubit on each edge. The Hamiltonian has the same commuting-projector structure as the conventional 3D toric code,

$H = - \sum_v A_v - \sum_p B_p,$

with mutually commuting vertex terms $A_v$ and plaquette terms $B_p$ (Zhou et al., 4 Mar 2025). The stabilizers satisfy the same global relations as in the bosonic 3D toric code,

$\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$

where the second product is over the six plaquettes of an elementary cube $c$ (Zhou et al., 4 Mar 2025).

The shared stabilizer skeleton obscures the essential distinction. The local form of $\mathbb Z_2$0, $\mathbb Z_2$1, and of the short string operators is described as a deformation of the usual 3D toric-code operators, arranged so that the point-like violations of $\mathbb Z_2$2 are fermions rather than bosons (Zhou et al., 4 Mar 2025). A short string operator $\mathbb Z_2$3 on an edge $\mathbb Z_2$4 anticommutes with the two adjacent vertex terms and commutes with all other stabilizers, so products of $\mathbb Z_2$5 along a path create and move point-like excitations at the endpoints. Flux excitations arise from violations of $\mathbb Z_2$6, and membrane operators

$\mathbb Z_2$7

create loop excitations on the boundary of a surface $\mathbb Z_2$8 (Zhou et al., 4 Mar 2025).

This formulation makes the 3D fermionic toric code a qubit Hamiltonian with intrinsically fermionic emergent content. The microscopic Hilbert space is bosonic, but the deconfined charge sector is fermionic, and this distinction controls both the braiding structure and the higher-form symmetry of the phase (Zhou et al., 4 Mar 2025).

2. Excitations, braiding, and loop self-statistics

The excitation content consists of point-like charges and loop-like fluxes. Vertex violations $\mathbb Z_2$9 are point-like and are created in pairs at the ends of string operators built from the short string pieces $\mathbb Z_2$0. Plaquette violations $\mathbb Z_2$1 form closed loops on the dual lattice because $\mathbb Z_2$2 forbids isolated endpoints of flux lines (Zhou et al., 4 Mar 2025). As in ordinary $\mathbb Z_2$3 gauge theory, taking a charge around a flux loop yields a minus sign, so the charge–loop mutual statistics is nontrivial (Zhou et al., 4 Mar 2025).

A crucial clarification is that the ordinary 3D fermionic toric code does not assign fermionic self-statistics to the flux loop. The phase analyzed as the ordinary 3D fermionic toric code is the FcBl phase: fermionic charge, bosonic loop. By contrast, a distinct FcFl phase has fermionic charges and fermionic loops, and that phase is anomalous rather than a stand-alone $\mathbb Z_2$4-dimensional bosonic lattice topological order (Fidkowski et al., 2021).

Phase	Point excitation	Flux loop
BcBl	bosonic charge	bosonic loop
FcBl	fermionic charge	bosonic loop
FcFl	fermionic charge	fermionic loop

The distinction between FcBl and FcFl is diagnosed by a loop self-statistics invariant $\mathbb Z_2$5, defined from a 36-step membrane process on a tetrahedral geometry. The ordinary 3D fermionic toric code has $\mathbb Z_2$6, while the anomalous fermionic-loop phase has $\mathbb Z_2$7 (Fidkowski et al., 2021). The same work argues that FcFl can only exist at the boundary of a non-trivial $\mathbb Z_2$8-dimensional invertible bosonic phase with action

$\mathbb Z_2$9

so the phrase “3D fermionic toric code” conventionally refers to FcBl, not to the anomalous fermionic-loop theory (Fidkowski et al., 2021).

This corrects a common compression of terminology. In $\mathbb Z_2$0 dimensions, “fermionic” may refer either to the charge sector or to loop self-statistics; the ordinary fermionic toric code is fermionic in the first sense and bosonic in the second (Fidkowski et al., 2021).

3. Anomalous 2-form symmetry and finite-temperature quantum topological order

The 3D fermionic toric code has a distinguished higher-form symmetry generated by closed Wilson loops of the emergent fermion. For a contractible closed path $\mathbb Z_2$1, the associated string operator $\mathbb Z_2$2 commutes with the Hamiltonian and detects the parity of linked flux loops; in the loop-less sector it acts trivially, so these operators define a $\mathbb Z_2$3-form symmetry (Zhou et al., 4 Mar 2025). The key property is that this $\mathbb Z_2$4-form symmetry is anomalous when viewed in a purely bosonic $\mathbb Z_2$5-dimensional setting. The anomaly disappears if physical fermions are available, because physical fermions can be bound to the emergent fermions and condensed (Zhou et al., 4 Mar 2025).

This anomalous $\mathbb Z_2$6-form symmetry is the mechanism behind the phase’s finite-temperature behavior. A 2025 analysis identified the 3D fermionic toric code as the first explicit example of a three-dimensional local Hamiltonian system whose equilibrium thermal states exhibit quantum topological order at sufficiently small but nonzero temperature (Zhou et al., 4 Mar 2025). The argument constructs a quasi-local channel that removes all loop excitations shorter than $\mathbb Z_2$7, yielding a cleaned state close to a loop-less state $\mathbb Z_2$8. The fidelity bound is

$\mathbb Z_2$9

so for

$\mathbb Z_2$0

the cleaned thermal state approaches $\mathbb Z_2$1 faster than any inverse polynomial in $\mathbb Z_2$2 (Zhou et al., 4 Mar 2025).

The loop-less state realizes the anomalous $\mathbb Z_2$3-form symmetry as a strong symmetry, and results on anomalous higher-form symmetries then imply that no short-range entangled mixed state can approximate it. Pulling this conclusion back through the quasi-local channel gives the main theorem: below $\mathbb Z_2$4, the fidelity between the thermal state $\mathbb Z_2$5 and any short-range entangled state decays as $\mathbb Z_2$6, equivalently the trace distance approaches $\mathbb Z_2$7 up to $\mathbb Z_2$8 corrections (Zhou et al., 4 Mar 2025).

The contrast with the bosonic 3D toric code is sharp. There the corresponding $\mathbb Z_2$9-form symmetry is anomaly-free, and Hastings’ finite-temperature short-range-entanglement result applies. In the fermionic toric code, low-temperature thermal states remain long-range entangled precisely because the fermionic Wilson-loop symmetry is anomalous (Zhou et al., 4 Mar 2025).

4. Symmetry enrichment, loop fractionalization, and fermionic SPT connections

When a global symmetry $H = - \sum_v A_v - \sum_p B_p,$0 is imposed, the 3D fermionic toric code becomes a symmetry-enriched $H = - \sum_v A_v - \sum_p B_p,$1 topological order with fermionic charges. The point-like charge $H = - \sum_v A_v - \sum_p B_p,$2 carries symmetry fractionalization classified by

$H = - \sum_v A_v - \sum_p B_p,$3

exactly as in two-dimensional anyon fractionalization. Flux loops $H = - \sum_v A_v - \sum_p B_p,$4 introduce two additional layers of structure: loop–membrane intersection fractionalization

$H = - \sum_v A_v - \sum_p B_p,$5

and, when $H = - \sum_v A_v - \sum_p B_p,$6, an intrinsic loop fractionalization class

$H = - \sum_v A_v - \sum_p B_p,$7

interpreted as the edge data of a $H = - \sum_v A_v - \sum_p B_p,$8-dimensional SPT phase living on the loop (Cheng, 2015).

This framework is directly tied to fermionic symmetry-protected phases. Gauging fermion parity in a $H = - \sum_v A_v - \sum_p B_p,$9-dimensional fermionic SPT produces a $A_v$0 gauge theory with fermionic gauge charges, i.e. a 3D fermionic toric code. In that interpretation, the Gu–Wen supercohomology datum

$A_v$1

is proposed to be the loop fractionalization class $A_v$2 of the fermion-parity flux loop in the gauged theory (Cheng, 2015).

The classification is only partial because $A_v$3 and $A_v$4 are not fully independent when $A_v$5; the paper emphasizes equivalences and ambiguities in that regime (Cheng, 2015). Even so, several structural consequences follow. For $A_v$6, candidate nontrivial loop fractionalization patterns in the fermionic toric code are argued to be anomalous, leading to the conclusion that there is no nontrivial $A_v$7-dimensional $A_v$8 fermionic SPT of that type. For $A_v$9, by contrast, a $B_p$0-dimensional layer construction provides evidence for a nontrivial interacting $B_p$1-dimensional fermionic SPT whose gauged form is a symmetry-enriched fermionic toric code with nontrivial $B_p$2 on the flux loops (Cheng, 2015).

Within this viewpoint, the 3D fermionic toric code is not merely “$B_p$3 gauge theory with a fermionic charge.” It is also the natural gauged endpoint of a class of fermionic SPT constructions, with loop fractionalization providing the bridge between higher-form topological order and fermionic symmetry protection (Cheng, 2015).

5. Floquet realizations, logical structure, and monitored dynamics

The 3D fermionic toric code has a particularly important operational realization in Floquet quantum error correction. A 2023 construction introduced a 3D Floquet fermionic toric code by extending the 2D honeycomb Floquet-code framework to a trivalent lattice in three dimensions, with instantaneous stabilizer codes having the same topological order as the 3D fermionic toric code (Dua et al., 2023). That work framed the construction in terms of condensation of topological excitations and “rewinding” of measurement schedules, using periodic sequences of non-commuting local measurements to engineer desired instantaneous stabilizer groups (Dua et al., 2023).

A subsequent 2026 construction made this program explicit for the full logical code space. It identifies a 3D Kekulé–Kitaev lattice, a tricoordinated and 3-edge-colored geometry in which deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains (Watanabe et al., 13 Feb 2026). This loop property prevents sequential color measurements from collapsing logical information. On a 3-torus, the corresponding $B_p$4D fermionic toric-code phase encodes three logical qubits, with round-independent inner logical line operators and round-dependent membrane representatives for the complementary logicals (Watanabe et al., 13 Feb 2026).

The basic backbone is the three-color cycle

$B_p$5

implemented using only two-body Pauli measurements. However, on the 3D Kekulé–Kitaev lattice a simple 3-round color cycle does not expose the full plaquette-syndrome set, because some plaquettes involve all three colors and some syndrome information remains in the $B_p$6 sector (Watanabe et al., 13 Feb 2026). The full schedule is therefore extended to

$B_p$7

which reconstructs all plaquette stabilizer eigenvalues without disturbing the logical subspace (Watanabe et al., 13 Feb 2026). The same work reports that the relevant logical-operator group remains invariant over the entire 10-round sequence, so the protocol implements a trivial logical automorphism while preserving all three logical qubits (Watanabe et al., 13 Feb 2026).

The same lattice geometry also supports a family of monitored Kitaev models with random measurements of the non-commuting bond parities. In the measurement-probability simplex $B_p$8, the corner regions near $B_p$9, $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$0, and $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$1 exhibit area-law entanglement, while a central critical region shows

$\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$2

in $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$3 (Watanabe et al., 13 Feb 2026). The absence of edge critical points on the Kekulé–Kitaev lattice is again tied to the finite-loop property. This suggests that Floquet protocols preserving logical information are closely linked to trajectories in measurement space that remain within area-law topological regimes rather than crossing measurement-induced criticality (Watanabe et al., 13 Feb 2026).

6. Formal frameworks, antecedents, and research frontiers

The 3D fermionic toric code inherits much of its conceptual vocabulary from $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$4-dimensional fermionic topological order. The original fermionic toric-code lattice model was introduced in 2013 as an exactly soluble $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$5-dimensional fermionic version of the toric code, built from $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$6-graded fusion rules and fermionic associativity data with $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$7, and described at low energies by spin Chern–Simons theories with

$\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$8

(Gu et al., 2013). In $\prod_v A_v = 1, \qquad \prod_{p \in c} B_p = 1,$9 dimensions the excitation content changes from anyons to point charges and flux loops, but the same theme persists: fermionic locality is stricter than bosonic locality, and intrinsically fermionic topological data cannot be reduced to a bosonic toric-code deformation (Gu et al., 2013).

Two formal developments sharpen this point. First, fermionic MPO-injective tensor networks provide an exact PEPS/fPEPS description of the $c$0-dimensional fermionic toric code and of fermionic twisted quantum doubles, but their explicit extension to $c$1D remains open; the natural $c$2D analogue would require a higher-dimensional version of fermionic MPO symmetry, i.e. membrane-level graded virtual symmetries (Wille et al., 2016). Second, Majorana–Pauli stabilizer codes furnish an exact stabilizer realization of the $c$3-dimensional intrinsically fermionic toric code using $c$4 Pauli operators coupled to Majorana modes, and organize it within a duality web generated by anyon condensation and gauging of bosonic or fermion-parity symmetries (Sun et al., 23 Jun 2026). A plausible implication is that comparable hybrid stabilizer descriptions in $c$5 dimensions would have to encode loop condensation and higher-form braiding directly at the stabilizer-algebra level rather than by a straightforward lift of ordinary Pauli stabilizer codes.

Current frontiers therefore separate into three partially connected directions. One is the field-theoretic and symmetry-based direction, centered on anomalous $c$6-form symmetry and finite-temperature long-range entanglement (Zhou et al., 4 Mar 2025). A second is the symmetry-enriched and higher-categorical direction, in which loop fractionalization and anomaly constraints organize possible fermionic $c$7 gauge theories (Cheng, 2015). The third is the operational and coding-theoretic direction, where Floquet protocols and monitored dynamics produce instantaneous 3D fermionic toric-code order using only two-body measurements while preserving a nontrivial logical subspace (Dua et al., 2023, Watanabe et al., 13 Feb 2026).

Taken together, these developments establish the 3D fermionic toric code as a central example of a $c$8-dimensional spin topological order: simple enough to admit explicit lattice, Floquet, and symmetry-based descriptions, yet rich enough to exhibit anomalous higher-form symmetry, boundary-only variants, and low-temperature quantum topological order unavailable in the bosonic 3D toric code.