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Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code

T0 review · 0 major / 4 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Even with every higher-form symmetry broken, the 3D toric code keeps a sharp finite-temperature topological phase, labeled by a decoded Wilson-loop order parameter that no quasi-local channel can fake.

desk verdict Solid, carefully scoped paper that actually delivers a channel-invariant finite-T order parameter for the 3D toric code after all exact higher-form symmetries are broken. read the letter →

arxiv 2607.00134 v2 pith:V4V46SZ6 submitted 2026-06-30 cond-mat.str-el cond-mat.othercond-mat.stat-mech

Haruki Watanabe This is my paper · ORCID
classification cond-mat.str-elcond-mat.othercond-mat.stat-mech
keywords 3Dtoriccodefinite-temperaturetopologicalorderentanglemententropydecodedWilsonloopquasi-localchannelsBianchiidentitymixed-statephasesquantumMonteCarlo
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether two finite-temperature Gibbs states can be topologically distinct when the Hamiltonian itself breaks every exact higher-form symmetry that would normally protect the order. In the three-dimensional Z2 toric code with a generic magnetic field, the answer is yes. The topological entanglement entropy remains locked at ln 2 throughout the ordered phase and drops to zero across a thermal transition whose location is only shifted by the fields. That plateau, however, can be manufactured from a trivial product state by a constant-depth channel, so it is not a true mixed-state invariant. The author therefore constructs a decoded Wilson-loop correlator f_W that equals 1 deep in the topological phase and 0 in the trivial phase in the thermodynamic limit, and that is rigorously pinned to zero on every quasi-local image of a product state. The protection is geometric: the Bianchi identity forces magnetic flux to form closed loops, so the transition cannot terminate even though no exact symmetry remains. Large-scale quantum Monte Carlo maps the entire field-temperature phase diagram and confirms a single 3D-Ising critical surface. The result supplies a concrete, channel-invariant order parameter for Gibbs-state topology and places geometry-protected transitions on equal footing with spontaneous symmetry breaking.

What carries the argument

The decoded Wilson-loop correlation f_W — the connected correlator of two non-contractible Wilson loops evaluated after a restricted flux-cleaning (error-correction) channel. Light-cone bounds force f_W = 0 on every quasi-local image of a product state, while positive loop tension plus near-degenerate holonomy sectors force f_W o 1 inside the deconfined phase.

What would settle it

A large-scale quantum Monte Carlo measurement of f_W (or of the topological entanglement entropy) deep inside the claimed topological region that fails to approach 1 (respectively ln 2) as linear size increases, or that shows the thermal boundary terminating at a finite field rather than remaining sharp down to the known zero-temperature critical fields.

Watch

Extended reading notes

Core claim

In the 3D Z2 toric code subject to a generic magnetic field that explicitly breaks every higher-form symmetry, the topological phase at finite temperature remains sharply separated from the trivial phase. The topological entanglement entropy stays quantized at ln 2 until the thermal transition, but that value is not a quasi-local-channel invariant. The decoded Wilson-loop correlation f_W is such an invariant: it quantizes to 1 in the topological phase and 0 in the trivial phase as system size goes to infinity, so no quasi-local channel can carry a product state into the topological Gibbs state.

Load-bearing premise

The all-orders proof that the entanglement plateau survives the fields rests on the unperturbed thermal ensemble still having a finite correlation length below its own critical temperature; if that clustering fails, the geometric cancellation that protects the plateau no longer holds order by order.

Editorial extensions

If this is right

  • Finite-temperature topological order can be diagnosed by a single channel-invariant number even when every microscopic higher-form symmetry is broken.
  • Geometry (the Bianchi identity) alone can protect a non-terminating thermal phase boundary of 3D-Ising type.
  • Bulk topological entanglement entropy is insufficient to certify mixed-state phases; a recoverability order parameter such as f_W is required.
  • The same construction supplies a classical Monte Carlo order parameter for the Fradkin–Shenker gauge–Higgs model that resolves all three of its phase boundaries.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same decoded-loop idea should furnish channel invariants for other discrete gauge theories and for the fermionic 3D toric code, where genuine long-range entanglement is expected to survive.
  • Once f_W is accepted as the mixed-state label, one can systematically ask which other Gibbs-state diagnostics (Markov length, negativity, etc.) remain constant along thermal Lindbladian paths but jump under general quasi-local channels.
  • Geometry-protected transitions may appear in continuum models whose only exact constraint is a Bianchi or flatness identity, offering a route to finite-temperature topology without lattice gauge structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies the finite-temperature phase structure of the 3D Z2 toric code in a generic magnetic field that explicitly breaks every exact higher-form symmetry. Combining an all-orders perturbative argument (SM C) with large-scale continuous-time worldline QMC, it shows that the topological entanglement entropy remains quantized at γ = ln 2 throughout the deconfined phase and collapses to zero across a 3D-Ising thermal transition. Because γ can be manufactured from a product state by a constant-depth channel, the authors introduce the decoded Wilson-loop correlation f_W (the connected correlator of Wilson loops after restricted error correction). They argue that f_W → 1 in the topological phase and f_W → 0 in the trivial phase as L → ∞, and that light-cone factorization pins f_W = 0 on every quasi-local-channel image of a product state, making f_W a genuine mixed-state topological invariant. Specific-heat finite-size scaling recovers the duality-derived T_c and 3D Ising exponents; conventional Wilson, membrane and Fredenhagen–Marcu diagnostics are shown to be incomplete.

Significance. If the claims hold, the work supplies a concrete, QMC-accessible order parameter that distinguishes Gibbs phases of the 3D toric code without relying on exact higher-form symmetries, and that is invariant under the quasi-local channels that define mixed-state equivalence. The geometric (Bianchi-identity) protection mechanism, the transparent reporting of the fixed-aperture limitation of γ, the Peierls argument for the field-free limit of f_W, and the recovery of the known duality T_c are genuine strengths. The construction of f_W as a decode-then-read, channel-invariant diagnostic is a useful addition to the emerging toolkit for mixed-state topological order.

minor comments (4)
  1. Sec. III E and Fig. 11: the fixed-aperture under-resolution of γ at (0.75,0.15) is reported transparently, but a short quantitative estimate of the expected O(e^{-ℓ/ξ}) bias (using the measured defect density as a proxy for ξ) would help the reader judge how far the production point sits from the asymptotic regime.
  2. Sec. IV A–C: the distinction between the analytic depth-D sweep decoder and the numerical 2D-cut MWPM realization is clear in the text, but a single sentence in the caption of Fig. 7 stating which decoder is used for the plotted data would remove residual ambiguity.
  3. SM C, Step 5: the finite-ξ clustering assumption for the unperturbed glued-replica ensemble is correctly scoped as the sole physical input; a parenthetical pointer to the polymer-expansion regime (low T or the exact lines) already present later in the SM could be moved earlier for readers who stop at the main-text sketch.
  4. Table I and Sec. V: the classical-versus-quantum (bosonic/fermionic) distinction is important; a one-sentence reminder that γ itself remains classical for the bosonic code (while γ_N is the genuine quantum diagnostic) would prevent misreading of the bosonic plateau as long-range entanglement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: f_W and the γ plateau are established by independent QMC, external duality/Reiss–Schmidt benchmarks, a model-independent light-cone argument, and a Peierls bound, not by definitional or fitted self-reference.

full rationale

The paper’s load-bearing claims do not reduce to their inputs by construction. The γ = ln 2 plateau is anchored at the solvable point by the Castelnovo–Chamon formulas (external), extended by an all-orders cumulant argument whose sole physical input is finite-ξ clustering of the unperturbed ensemble (standard off-criticality, not the target result), and confirmed by QMC against the Wegner-duality T_c. The paper itself then shows γ is not a quasi-local-channel invariant by an explicit constant-depth construction, so the central invariant is f_W. f_W is defined operationally (decode-then-read connected correlator); its dichotomy is argued by a Peierls bound on dilute flux (unconditional for the field-free code below T_0 = 2/ln 5) plus large-scale QMC at generic fields, and its channel invariance follows from the Lieb–Robinson light-cone factorization on product-state images—model-independent once decoder depth D ≪ L/3 is fixed. Phase boundaries are cross-checked by specific-heat FSS (3D Ising), χ_zz, and the external Reiss–Schmidt zero-T critical fields. No parameter is fitted and re-labeled as a prediction; no uniqueness theorem or ansatz is imported from overlapping-author work as a load-bearing premise; precedents for decoded/decorated loops are cited as related but not as the present claim. The mild shared use of the same worldline ensemble for γ and f_W is ordinary multi-observable sampling, not circularity. Score 0 is therefore appropriate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The load-bearing content is geometric (Bianchi identity) plus standard clustering of the unperturbed ensemble and the light-cone bound for channels. No free parameters are fitted to produce the quantization of f_W or γ; critical fields and Tc are taken from prior literature or measured. The only invented entity is the order parameter f_W itself.

assumptions (4)
  • domain assumption Bianchi identity ∏_{p∈∂c} B̂_p ≡ 1 for every elementary cube (exact operator identity).
    Invoked throughout as the kinematic reason flux remains closed loops; protects both the γ plateau and the decoder structure (Secs. I, III D, IV A).
  • domain assumption Unperturbed (hx=hz=0) glued-replica ensemble clusters with finite ξ(T) for T < T_c^{(0,0)}.
    Sole physical input of the all-orders stability proof (SM C, Step 5, Eq. (C15)); standard off criticality but not elementary.
  • standard math Light-cone / Lieb–Robinson bound for quasi-local channels of finite range.
    Used to prove that f_W vanishes on every quasi-local image of a product state (Sec. IV B).
  • domain assumption Trivial-phase Gibbs states of the model lie in the two-way quasi-local-channel orbit of product states.
    Physically expected input needed to conclude that f_W=0 labels the entire trivial class (Sec. IV B); corroborated by strong-coupling limits but not proved for the whole phase.
invented entities (1)
  • decoded Wilson-loop correlation f_W
    purpose: Provide a quasi-local-channel-invariant order parameter that quantizes to 1 (topological) versus 0 (trivial) and is directly measurable by QMC.
    Defined in Sec. IV A as the connected correlator of Wilson loops after a restricted flux-cleaning channel; no independent experimental handle outside the model is given.

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Pith. "Pith review of Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code." pith.science (2026). https://pith.science/paper/V4V46SZ6

@misc{pith2026260700134,
  author       = {Pith},
  title        = {Pith review of: Phase distinction of Gibbs states without symmetry breaking: topological invariants of the 3D toric code},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4V46SZ6}},
  note         = {Machine review of arXiv:2607.00134}
}
abstract

We study the finite-temperature topological order of the three-dimensional $\mathbb{Z}_2$ toric code in a generic magnetic field, where every higher-form symmetry is explicitly broken and can at most be emergent. We show perturbatively, and confirm by large-scale quantum Monte Carlo at fields up to half the zero-temperature critical values, that the topological entanglement entropy stays quantized at $\gamma = \ln 2$ throughout the topological phase -- at finite temperature and under the symmetry-breaking field alike -- and collapses to $0$ across the thermal transition, a quantization protected geometrically by the Bianchi identity rather than by any exact symmetry of the system. The plateau $\gamma = \ln 2$ is, however, not invariant under quasi-local channels: a constant-depth channel can generate this identical quantized value from a trivial product state. We therefore introduce the decoded Wilson-loop correlation $f_W$ -- the connected correlator of Wilson loops read out after error correction -- which quantizes to $1$ in the topological phase and $0$ in the trivial phase as $L\to\infty$. Unlike $\gamma$, $f_W$ is a quasi-local-channel invariant: it is pinned to $0$ on every quasi-local-channel image of a product state and to $1$ in the topological phase, so no quasi-local channel carries the trivial phase to the topological one, and a fortiori no two-way equivalence connects them -- a robust topological invariant of the mixed state.

Figures

Figures reproduced from arXiv: 2607.00134 by the authors.

Figure 1
Figure 1. FIG. 1. The three tiers of states and their phase-equivalence [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figures from the paper (16 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Electric ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Bulk thermodynamic response on the three coordinate planes ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Four-quadrant partition [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Four-quadrant partition [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Topological entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Topological entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Finite-size series of the decoded Wilson-loop correlation [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Finite-size series of the decoded Wilson-loop correlation [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The decoded Wilson-loop correlation [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The decoded Wilson-loop correlation [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The full three-dimensional ( [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The full three-dimensional ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Specific-heat scaling at ( [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Specific-heat scaling at ( [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Finite-size scaling of the topological-entanglement [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Finite-size scaling of the topological-entanglement [PITH_FULL_IMAGE:figures/full_fig_p036_11.png]

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Reference graph

Works this paper leans on

37 extracted references · cited by 1 Pith paper

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    (B3) is a statement about theuniversalpart ofS (n)

    Subleading R´ enyi-dependence does not affect γtopo Eq. (B3) is a statement about theuniversalpart ofS (n). The non-universal area-law coefficientα n is in general a non-trivial function ofn; in our finite-L= 8 measurement this affects the abso- lute magnitudes of the four constituent entropies S(2)(A1), S(2)(A2), S(2)(A3), S(2)(A4) but cancels iden- tica...

  2. [2]

    (B3) was proved in Ref

    Caveat 1: chiral topological order Eq. (B3) was proved in Ref. [69] for non-chiral phases (including allZ N gauge theories such as the toric code). For chiral topological order, the entanglement spec- trum may carry additional R´ enyi-dependent universal data [69]; this regime is not realized by the model studied here, so the equivalence (B3) suffices on ...

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    [69, 70] is aground-statestate- ment

    Caveat 2: extension to finite temperature The projector structure ofρ A—the boundary parity constraint that reduces the rank of the flat spectrum by D—that underpins Refs. [69, 70] is aground-statestate- ment. At finite temperature,ρ(T) =Z −1e−H/T is a ther- mal mixture of all eigenstates, and Eq. (B3) does not by itself fixγ (n)(T) atT >0. For the presen...

  4. [4]

    A single four-region combi- nation [Eq

    Normalization: the per-sector valueγand the fullT= 0entropy We report the topological entanglement entropy asγ, the universal subleading constant of a single smooth en- tangling boundary,S(A) =α|∂A|−γ+· · ·, in the Levin– Wen normalization [66, 67]. A single four-region combi- nation [Eq. (6)] extracts one such constant, γ= lnD= ln 2,(B5) withD= P i d2 i ...

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    (3) at (h x, hz) = (0,0) (couplingsJ e,J m kept general; the main text setsJ e = Jm = 1) and the symmetric eight-bipartition topolog- ical combination of CC [their Eq

    Lemma: R´ enyi-ntopological entropy at hx =h z = 0 For the Hamiltonian of Eq. (3) at (h x, hz) = (0,0) (couplingsJ e,J m kept general; the main text setsJ e = Jm = 1) and the symmetric eight-bipartition topolog- ical combination of CC [their Eq. (3.14)], the R´ enyi-n topological entanglement entropy obeys, for everyinte- gerR´ enyi indexn≥2 (and, via the...

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    Replica factorization at fixedn Working in theσ x product basis{|α⟩}, witha v(α) =Q l∈star(v) αl the vertex eigenvalues andG A the group of plaquette products acting trivially outsideA, CC derive [their Eqs. (4.20)–(4.22)] the exact factorization, valid for every integern≥2 and any bipartition, TrA ρ n A =Z (P) (n)Z (S)(n), Z (P) (1) =Z (S)(1) = 1, (B7) Z...

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    Charge (vertex) sector atn= 2: closed form Z(S)(2) is the collision probability of the classical marginalp A and can be evaluated in closed form for an arbitraryregionA. Expandinge βJeav = cosh(βJ e) [1 + te av] witht e ≡tanh(βJ e) and using Q v∈S av(α) =Q l∈δS αl (withδSthe set of links with exactly one endpoint in the vertex subsetS, distinct from the e...

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    This is verified empirically, with the crossover scale supplied by Eq

    Remark: the annular partition of the main text TheC 4-symmetric annular partition used in the main text couples to the flux (membrane) bit with unit coeffi- cient and to the charge bit with coefficient zero. This is verified empirically, with the crossover scale supplied by Eq. (B11): atL= 8 the measured plateau remains ln 2, flat, down toT= 0.1< T ∗(8)≈0...

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    (D20) for the bipartitions that host a collective mem- brane operation, and Eq

    Flux (plaquette) sector atn= 2 CC evaluateZ (P) (n) for general integern[their Eq. (D20) for the bipartitions that host a collective mem- brane operation, and Eq. (D34) for the remaining bipar- titions], and only then taken→1. Specializing their expressions ton= 2 requires thr...

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    Generalnand remarks For general integernthe even-parity twist sum gives, in place of Eq. (B13), the factor Tn(r) = 1 2 (1 +r) n + (1−r) n − → ( 2 n−1, r→1 (T < T c), 1, r→0 (T > T c), (B14) and the R´ enyi normalization 1 1−n ln(2n−1) =−ln 2 con- verts the replica degeneracy i...

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    Step 1: worldline representation and positivity Split ˆH= ˆHd + ˆHod into itsσ z-diagonal and off- diagonal parts, ˆHd =−J m P p ˆBp −h zP l ˆσz l and ˆHod = −Je P v ˆAv −h xP l ˆσx l , with diagonal energyE d(s) = −Jm P p bp(s)−h zP l sl. Iterating the Duhamel iden- titye −β ...

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    For each slicesletD(s) ={p:b p(s) =−1}be the flux-defect set; under lattice dualityDis a set of dual links, aZ 2 1- chain ˜D

    Step 2: kinematic closure of the flux sector This is the geometric heart of the stability. For each slicesletD(s) ={p:b p(s) =−1}be the flux-defect set; under lattice dualityDis a set of dual links, aZ 2 1- chain ˜D. (i) Closure. The Bianchi identityQ p∈∂c ˆBp =11 (each of a c...

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    (C6) Here TrH⊗H runs over the doubled space, and Tr A and Tr ¯A over the factors of the originalH=H A ⊗ H ¯A

    Step 3: replica geometry The R´ enyi-2 entropy measured in our QMC is S(2)(A) =−ln⟨SWAP A⟩(SM D 1), and all forms below evaluate to the same purity, ⟨SWAPA⟩ ≡Tr H⊗H (ρ⊗ρ) SWAP A = TrA (Tr ¯Aρ)2 = TrA ρ2 A = Z2(A) Z2 . (C6) Here TrH⊗H runs over the doubled space, and Tr A and T...

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    Connected

    Step 4: linked-cluster expansion The all-orders cancellation of Proposition 1 is built from three ingredients across Steps 4–6: a linked-cluster expansion (this step), an exponential clustering bound (Step 5), and a matched-boundary cancellation assem- bled in Step 6. Treat th...

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    Step 5: strict locality and exponential clustering Because ˆH0 is a sum of commuting stabilizers, imaginary-time evolution doesnotspread supports [Eq. (9)]: ˆσ z l (τ) = ˆσ z l e2τ Je( ˆAv1+ ˆAv2) and ˆσx l (τ) = ˆσx l e2τ Jm P p∋l ˆBp haveτ-independentsupport—star(v 1)∪ star(...

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    III A)—so the combination is the conditional mu- tual informationI(b:d|ac) =−S(A 1) +S(A2) +S(A3)− S(A4), which fixes the sign vectorσ= (−,+,+,−)

    Step 6: matched-boundary cancellation and assembly The four regions are nested unions of the annular quadrants—A1 =abcd,A 2 =acd,A 3 =abc,A 4 =ac (Sec. III A)—so the combination is the conditional mu- tual informationI(b:d|ac) =−S(A 1) +S(A2) +S(A3)− S(A4), which fixes the sig...

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    For its boundary-local dressing to cancel in the matched-boundary combination [Eq

    Zero temperature (2 ln 2) AtT= 0 the gap can be brought to bear: ˆH0 is a commuting-projector Hamiltonian obeying local topo- logical order, so the Bravyi–Hastings–Michalakis theo- rem [104] keeps the gap open for small fields (the pCUT phase diagram [63] extends this across t...

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    The lineh x = 0(Proposition 2) Here every ˆBp commutes with ˆH, so the Gibbs state block-diagonalizes over flux sectors,ρ= P b P(b)ρ b (P(b) the statistical weight andρ b the normalized Gibbs state of flux sectorb), withbranging—by Step 2(ii)—over closed null-homologous loops....

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    The lineh z = 0(electric one-form symmetry) For any closed dual surface Σ the membrane ˆM(Σ) =Q l⊥Σ ˆσx l (the closed-surface counterpart of the logical membrane ¯Ma of Sec. IV) commutes with ˆHat arbitrary hx: the only nontrivial check, [ ˆM(Σ), ˆBp], carries the sign (−1)|∂p...

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    Generic fields: the dressed twist With no exact symmetry surviving, the global object is thereplicatwist factor er(hx, hz;T)≡ Z A,− Z A,+ ,(C17) the ratio of the glued (replica) partition functions of the annular region with and without the relative twist of the magnetic coupl...

  13. [21]

    Proposition 1 at generic fields is an all-orders, term-by-term result resting on asinglephys- ical input—the finite-ξclustering bound of Eq

    Exact status of the stability analysis The kinematic flux closure of Step 2 is unconditionally rigorous, as is the existence of the exact one-form sym- metry on theh z = 0 line; the pinning ofγ (2) = ln 2 along that line rests on the solvable-point anchor plus the all- orders ...

  14. [22]

    The diagonal (classical-marginal) estimator Ath x ̸= 0 the production estimator measures theσ z-diagonal (classical-marginal) R´ enyi-2 entropy −lnP s pA(s)2 (Sec. III E). Proposition 1 extends to it verbatim. The object P s pA(s)2 is the matched sector of the glued replica tr...

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    Chain-trick R´ enyi-2 estimator The topological entanglement entropy is measured with the same continuous-time worldline quantum Monte Carlo framework as the specific heat (SM A). The R´ enyi- 2 entropyS (2)(A) =−ln⟨SWAP A⟩is obtained by the chain-trick replica protocol—includ...

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    6(a)] is supplemented by a companion FSS campaign at the plateau temperatureT= 0.5, withL∈ {10,12,14}, using the identical chain-trick pipeline and production parameters of Sec

    Finite-size scaling and partition cross-checks The main-text plateau evidence atL= 8 [Fig. 6(a)] is supplemented by a companion FSS campaign at the plateau temperatureT= 0.5, withL∈ {10,12,14}, using the identical chain-trick pipeline and production parameters of Sec. III E (w...

  17. [25]

    The square-lattice cluster, via the Kitaev–Preskill tripartite combination Dress the paramagnet|+⟩ ⊗N (one qubit per site) with a controlled phase on every nearest-neighbor edge, |θ⟩= ˆUθ |+⟩⊗N , ˆUθ = Y ⟨i,j⟩ e iθˆniˆnj ,ˆn= 1−ˆσz 2 , (E1) a strictly local, depth-one (mutuall...

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    The two-qubit plaquette cluster, via the Levin–Wen annulus Place two qubitsa v, bv on each site and linka v by a controlled phase to the fourb’s of the unit plaquette Pv ={v, v+ ˆx, v+ ˆy, v+ ˆx+ ˆy}, |θ⟩= ˆUθ |+⟩⊗2N , ˆUθ = Y v Y w∈Pv e iθˆna v ˆnb w ,(E3) again depth-one and...

  19. [27]

    The bare topo- logical entanglement entropy is thereforenotan FDLU invariant—not even constant along one FDLU orbit, let alone under the broader quasi-local channels of Sec

    The bare TEE is not an FDLU invariant In both families any two members are related by a depth-one local unitary ( ˆUθ ˆU † θ′ is again a product of con- trolled phases), so each family lies in a single FDLU class (the trivial phase); yet the boundary-cancelingγsweeps continuou...

  20. [28]

    [37]):β g (the classical gauge coupling, not the inverse temperature) tracks the magnetic couplings (Jm, hx, T) andKthe electric fieldh z

    Model PlaceZ 2 gauge variablesz l =±1 on the links andZ 2 matter variabless v =±1 on the sites of a periodic cubic lattice; with gauge couplingβ g and matter couplingK the Gibbs measure is [36] P[z, s] = 1 Z exp h βg X p bp +K X ⟨vv ′⟩ sv zvv ′ sv′ i , bp = Y l∈∂p zl, (F1) the...

  21. [29]

    II B), which likewise van- ishes]: either way the bare holonomy bit is destroyed as L→ ∞

    Why decoding is needed For two parallel noncontractible loopsC 1, C2 bound- ing a cylinderS cyl, the bare correlator is the net flux throughS cyl; a closed flux loop pierces it an even num- ber of times unless its linking parity with∂S cyl =C 1 ∪C2 is odd, so only such linking...

  22. [30]

    One object, three boundaries Combining the two pieces, fW ≃(1−2P fail)−m 2 a − →    1,deconfined (κ >0, m a = 0), 0,confined (via two-point ath x c , T c), 0,Higgs (via one-point ath z c), (F4) valid in the dilute-flux regime where the decoder is re- liable (⟨fWa⟩ ≃m a); ...

  23. [31]

    Dictionary and channel status By Wegner duality [40] the pure gauge theory (K=

  24. [32]

    (F3) is the thermal-ensemble analog of the random- plaquette/accuracy-threshold problem [91, 106]

    maps to the 3D Ising model: ∆F log becomes the dual Ising correlation-length penalty (line tension), 1− 2Pfail the expectation value of the dual disorder pa- rameter, andm a the matter-induced explicit breaking of the corresponding symmetry, so the decoding (de- confinement) t...

  25. [33]

    Its value on the trivial class is fixedmodel- independentlyby the light-cone (channel-invariance) ar- gument of Sec

    The connected correlator The diagnostic is theconnecteddecoded correlation fW =⟨fW1fW2⟩ − ⟨fWa⟩2,(G1) an exact identity in two directly measurable expecta- tions. Its value on the trivial class is fixedmodel- independentlyby the light-cone (channel-invariance) ar- gument of Se...

  26. [34]

    Confined/Higgs

    Two control parameters a. Flux-loop tensionκ(two-point) The flux marginal is a gas of closed dual loops; we say it has positive tension if Pr[ζ⊂flux]≤e −κ|ζ| for every dual loopζ. Positive tension means the gas is dilute: loops of length≳L/3—comparable to the loop sepa- ration...

  27. [35]

    Deep-phase Peierls theorem The honest analytic deliverable is the deconfined two- point limit.Suppose theτ= 0flux marginal has positive Peierls tension,Pr[ζ⊂flux]≤e −κ|ζ| withκ >ln 5. Then for two noncontractible loops at separation≥L/3, the cycle-wise restricted cleaner defin...

  28. [36]

    What is rigorous, conditional, and open (i) Exact:the connected definition (G1); the slav- ing identity (23) and the Stokes compensation (a fully bounding recovery returns fW1fW2 ≡11, so the informa- tive class is carried by the flux the restricted decoder leaves unresolved); ...

  29. [37]

    That is a misla- bel here

    Governing statistics: 3D-Ising interface, not Nishimori It is tempting to mapP fail to the random-bond Ising model on the Nishimori line via the Dennis–Kitaev– Landahl–Preskill correspondence [91]. That is a misla- bel here. The flux is drawn from thethermalGibbs ensemble, wit...

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Reviewed July 12, 2026 · model on record in the stance chip above.