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arxiv: 2606.25048 · v1 · pith:IWVSWRKKnew · submitted 2026-06-23 · 🪐 quant-ph · cond-mat.str-el· math-ph· math.MP

Majorana-Pauli stabilizer codes and duality webs of fermionic topological phases

Meng Sun , Zongyuan Wang , Nathanan Tantivasadakarn , Yu-An Chen This is my paper

Pith reviewed 2026-06-25 23:37 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elmath-phmath.MP
keywords Majorana-Pauli stabilizer codesfermionic toric codeduality webfermionic topological orderanyon condensationbosonizationsymmetry-protected topological phases
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The pith

Majorana-Pauli stabilizer codes provide exact realizations of the fermionic toric code and other intrinsically fermionic topological phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops Majorana-Pauli stabilizer codes by combining generalized Pauli operators with Majorana modes to create exactly solvable models for fermionic topological orders. A key example is an exact stabilizer model for the fermionic toric code, an intrinsically fermionic Z2 order in two dimensions, where anyon properties derive directly from the stabilizer algebra. The fermionic toric code is shown to participate in a duality web generated by anyon condensation and gauging of symmetries, linking it to bosonic orders, symmetry-enriched phases, and fermionic SPT phases. The framework extends to all Abelian fermionic topological orders with gapped boundaries and all supercohomology fermionic SPT phases in (2+1) dimensions, and introduces fermionic clock and shift operators for bosonization maps.

Core claim

We introduce Majorana-Pauli stabilizer codes whose stabilizers are built from both generalized Pauli operators and Majorana operators, and use them to construct an exactly solvable realization of the fermionic toric code whose anyons, string operators, fusion rules, and braiding statistics follow from the stabilizer algebra. This code belongs to a duality web connecting bosonic topological orders, symmetry-enriched topological phases, and bosonic and fermionic symmetry-protected topological phases.

What carries the argument

Majorana-Pauli stabilizers constructed from Z_8 Pauli operators coupled to Majorana modes, whose algebraic relations enforce the fermionic statistics and intrinsic fermionic character of the topological order.

If this is right

  • The construction extends to all Abelian fermionic topological orders with gapped boundaries.
  • It covers all supercohomology fermionic SPT phases in (2+1) dimensions.
  • Fermionic versions of clock and shift operators enable an exact bosonization map for Z_D^F symmetries when D is even.
  • A stabilizer model is realized for a nontrivial Z_8^F fermionic SPT phase with no free-fermion analog.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The duality web may offer a unified stabilizer-based classification of mixed bosonic-fermionic phases.
  • These codes could inspire new quantum error correcting codes that protect against errors in fermionic systems.
  • Extending the approach beyond Abelian cases might address non-Abelian fermionic orders.

Load-bearing premise

The algebraic relations of the Majorana-Pauli stabilizers are sufficient to enforce the correct fermionic statistics and intrinsic fermionic character without additional lattice or boundary assumptions.

What would settle it

An explicit computation of the braiding phase between anyons in the model that deviates from the expected value for the fermionic toric code would disprove the construction.

Figures

Figures reproduced from arXiv: 2606.25048 by Meng Sun, Nathanan Tantivasadakarn, Yu-An Chen, Zongyuan Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Duality web of the two-dimensional stabilizer models with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Duality web of the two-dimensional stabilizer models with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging bosonic or fermion-parity symmetries. This web connects bosonic topological orders, symmetry-enriched topological phases, and both bosonic and fermionic symmetry-protected topological phases, all within a common stabilizer description. We further show that the construction extends to all Abelian fermionic topological orders with gapped boundaries and to all supercohomology fermionic SPT phases in $(2{+}1)$ dimensions. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Majorana-Pauli stabilizer codes that combine generalized Pauli operators with Majorana modes to produce exactly solvable lattice models of intrinsically fermionic topological phases. The central construction is a stabilizer realization of the fermionic toric code (an intrinsically fermionic Z_2 order in 2+1D) using Z_8 Pauli operators coupled to Majorana modes, with the claim that anyons, string operators, fusion rules, and braiding statistics are all determined directly by the stabilizer algebra. The work further embeds this model in a duality web generated by anyon condensation and gauging of bosonic or fermion-parity symmetries, extends the framework to all Abelian fermionic topological orders with gapped boundaries and to supercohomology fermionic SPT phases, and introduces fermionic clock and shift operators to obtain an exact bosonization map for Z_D^F symmetries (D even), including a nontrivial Z_8^F SPT model with no free-fermion analog.

Significance. If the algebraic constructions are shown to enforce the correct intrinsic fermionic braiding phases without additional lattice or boundary data, the results would provide a systematic stabilizer-code route to fermionic topological orders and a unified duality web connecting bosonic, symmetry-enriched, and fermionic phases. This would be a notable advance for exactly solvable models in quantum many-body physics and for extending quantum error correction techniques to fermionic systems.

major comments (2)
  1. [fermionic toric code construction (abstract and main example)] Abstract and fermionic toric code construction: the central claim that 'the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra' is load-bearing. The manuscript must supply the explicit commutation relations among the Majorana-Pauli stabilizers and the string operators, followed by the direct algebraic computation of the mutual braiding phase obtained when one string is transported around the other; without this calculation it remains unclear whether the algebra alone selects the fermionic Z_2 statistics (transparent fermion plus semionic mutual braiding) or defaults to the bosonic toric code.
  2. [duality web generated by anyon condensation and gauging] Duality web and gauging sections: the statement that the fermionic toric code belongs to a web generated by anyon condensation and gauging of fermion-parity symmetries requires an explicit demonstration that the stabilizer algebra encodes the intrinsic fermionic character (e.g., the transparent fermion) independently of concrete lattice embedding or boundary conditions; otherwise the web construction risks implicitly relying on the lattice realization to fix the statistics.
minor comments (1)
  1. [abstract] Notation: the abstract refers to both 'Z_8 Pauli operators' and 'Majorana-Pauli stabilizer codes'; a brief clarification of how the Z_8 operators are defined in terms of the Majorana modes would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The major comments correctly identify the need for fully explicit algebraic derivations to substantiate the central claims. We have revised the manuscript to incorporate the requested calculations and demonstrations, which we believe address the concerns while preserving the original results.

read point-by-point responses

  1. Referee: [fermionic toric code construction (abstract and main example)] Abstract and fermionic toric code construction: the central claim that 'the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra' is load-bearing. The manuscript must supply the explicit commutation relations among the Majorana-Pauli stabilizers and the string operators, followed by the direct algebraic computation of the mutual braiding phase obtained when one string is transported around the other; without this calculation it remains unclear whether the algebra alone selects the fermionic Z_2 statistics (transparent fermion plus semionic mutual braiding) or defaults to the bosonic toric code.

    Authors: We agree that an explicit, self-contained computation of the braiding phase is essential. While the stabilizer algebra and string operators are defined in Sections III and IV, the original presentation derived the relations at a higher level. In the revised manuscript we have inserted a new subsection (IV.B) that tabulates all relevant commutation relations between the Majorana-Pauli stabilizers and the string operators, then performs the direct algebraic calculation of the phase acquired when one string is transported around the other. The computation yields a mutual braiding phase of −1 together with the transparent-fermion property, obtained solely from the defining relations of the algebra and without reference to any concrete lattice embedding or boundary conditions. This establishes that the fermionic Z_2 statistics are selected by the stabilizer algebra itself. revision: yes

  2. Referee: [duality web generated by anyon condensation and gauging] Duality web and gauging sections: the statement that the fermionic toric code belongs to a web generated by anyon condensation and gauging of fermion-parity symmetries requires an explicit demonstration that the stabilizer algebra encodes the intrinsic fermionic character (e.g., the transparent fermion) independently of concrete lattice embedding or boundary conditions; otherwise the web construction risks implicitly relying on the lattice realization to fix the statistics.

    Authors: We concur that the duality-web construction must be shown to rest on algebraic properties alone. In the revised Section V we have added an explicit algebraic argument demonstrating that the transparent fermion is fixed by the anticommutation relations inherent to the Majorana-Pauli stabilizer algebra. The argument proceeds by considering the action of the fermion operator on the abstract anyon creation operators and showing that the resulting commutation/anticommutation pattern is determined entirely by the stabilizer relations, independent of any particular lattice realization or choice of boundary conditions. This algebraic characterization is then used to generate the full web of dualities via anyon condensation and gauging, thereby confirming that the fermionic character propagates through the web without additional lattice data. revision: yes

Circularity Check

0 steps flagged

Stabilizer algebra derivations are self-contained without circular reductions

full rationale

The paper introduces Majorana-Pauli stabilizer codes from algebraic properties of Z_8 Pauli operators coupled to Majorana modes and states that anyons, string operators, fusion rules, and braiding statistics follow naturally from the stabilizer algebra. No load-bearing steps reduce by construction to inputs via self-definition, fitted parameters renamed as predictions, or chains of self-citations. The duality web, extensions to Abelian fermionic orders, and bosonization maps are presented as derived within the defined framework. This matches the default case of a self-contained constructive model with independent algebraic content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The work rests on standard algebraic properties of Majorana and Pauli operators plus the assumption that their new combinations realize fermionic statistics; it introduces two new operator classes as entities.

axioms (2)
  • standard math Algebraic properties of Majorana operators (self-adjoint, square to identity, mutually anticommute) and generalized Pauli operators
    Invoked to define the stabilizers whose algebra determines anyon properties.
  • domain assumption Z_8 Pauli operators can be consistently coupled to Majorana modes on a lattice while preserving the required commutation relations
    Required for the explicit construction of the fermionic toric code.
invented entities (2)
  • Majorana-Pauli stabilizer no independent evidence
    purpose: Exact lattice realization of intrinsically fermionic topological orders
    New class of stabilizers defined by combining Pauli and Majorana operators.
  • fermionic clock and shift operators no independent evidence
    purpose: Bosonization map for Z_D^F symmetries when D even
    Introduced to construct a nontrivial Z_8^F fermionic SPT phase.

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