Domain walls from SPT-sewing

Aleksander Kubica; Isaac H. Kim; Yabo Li; Zijian Song

REVIEW 3 major objections 5 minor 2 cited by

Domain walls from SPT-sewing

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Gauging a lower-dimensional symmetry-protected state sewn along the seam between two bulks constructs gapped domain walls of quantum-double gauge theories, and in two-dimensional Abelian models this one construction produces every…

desk verdict A solid, inventive construction paper with a real completeness theorem for Abelian quantum doubles; the 'all' claim has a proof gap in the gauging correspondence, but the 3d anchoring walls stand on their own. read the letter →

arxiv 2411.11967 v1 pith:RTGNR5MT submitted 2024-11-18 quant-ph cond-mat.str-el

Yabo Li , Zijian Song , Aleksander Kubica , Isaac H. Kim This is my paper · ORCID

classification quant-phcond-mat.str-el MSC 81T4581P68

keywords domainwallssymmetry-protectedtopologicalordergaugingquantumdoubletoriccodenon-invertiblesymmetryanchoring

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

The paper introduces a single construction—SPT-sewing—for making gapped domain walls in topologically ordered systems: take two bulks with facing boundaries, decorate the seam with a lower-dimensional symmetry-protected topological state, and gauge the whole system. The central result is that in two-dimensional Abelian quantum doubles this construction is complete for invertible walls: every invertible domain wall of the $G$ quantum double arises from gauging a 1d SPT with $G\times G\times G$ symmetry. For non-Abelian groups the paper constructs invertible walls of the $S_3$ quantum double from a 1d SPT with non-invertible $S_3\times\mathrm{Rep}(S_3)\times S_3$ symmetry and conjectures the same statement for every finite group. In the 3d toric code the method creates anchoring domain walls, which convert point-like electric charges into semi-loop-like magnetic excitations anchored on the wall, in both Abelian and non-Abelian variants. If correct, the paper turns the classification of quantum-double domain walls into a question of writing down SPT topological actions.

What carries the argument

The carrying object is the SPT-sewing map: two $G$-symmetric product states are placed on lattices with facing boundaries, the boundary degrees of freedom are entangled by a lower-dimensional SPT fixed-point state written as a topological action built from group cohomology, and the gauging map sends the symmetric vertex operators to the star operators of the quantum double. The folded-system picture is the workhorse: folding the two bulks along the seam turns the sewn wall into a gapped boundary of the $G\times G$ (or $G\times G\times G$) quantum double, so the wall's action on anyons is read from the anyons condensed on that boundary. Invertibility of the wall is controlled by the type-II part of the 2-cocycle $\nu=\omega_1\eta\omega_2$: the wall is invertible exactly when $\eta$ is non-degenerate. For the completeness proof, the canonical decomposition of Clifford unitaries into Hadamard, CNOT, CZ, and permutation gates supplies a normal form for braided autoequivalences of the $\mathbb{Z}_2^n$ quantum double, which is then matched term-by-term to $G\times G\times G$ topological actions.

What would settle it

Enumerate all braided autoequivalences of the $\mathbb{Z}_2\times\mathbb{Z}_2$ quantum double (there are 72) and write down the $G\times G\times G$ topological action for each one; if any of the 72 anyon maps cannot be reproduced by the paper's SPT-sewing formula, Theorem 1 is false.

Watch

Extended reading notes

Core claim

The paper's claim is that gauging a lower-dimensional SPT placed on the seam between two bulks produces the same gapped domain walls that were previously found case-by-case, and that in the 2d Abelian quantum double $D(G)$ this procedure is exhaustive for invertible walls. An invertible wall is a braided autoequivalence of the anyon model; the proof shows that every such map, expressed in a canonical form, can be reproduced by a topological action built from background gauge fields for three copies of $G$, and then constructs the SPT-sewn wall whose folded boundary condenses exactly the corresponding anyons. For non-Abelian $G$, the paper shows that the standard $G\times G$ symmetry is insufficient: for $S_3$ the $S_3\times S_3$ sewn wall is non-invertible, while sewing a 1d SPT with the non-invertible $S_3\times\mathrm{Rep}(S_3)\times S_3$ symmetry yields the trivial wall and the invertible wall exchanging the $C$ and $F$ anyons. In three dimensions, gauging 2d SPTs embedded in the trivial $\mathbb{Z}_2$ product state produces the type-I double-semion wall and the type-II and type-III anchoring walls; on the latter, point charges pass through and re-emerge as semi-loops of magnetic flux anchored on the wall, with Abelian anyons in type-II and non-Abelian anyons in type-III.

Load-bearing premise

The proof that SPT-sewing covers every wall rests on the previously established classification saying all domain walls come from gauging a one-dimensional phase with some unbroken symmetry and cocycle; if that classification is incomplete, the theorem's coverage is incomplete.

Editorial extensions

If this is right

In any 2d Abelian quantum double, the classification of invertible domain walls reduces to writing down $G\times G\times G$ topological actions, so every braided autoequivalence of the anyons is realized by some choice of cocycle.
The $S_3$ construction produces the $C\leftrightarrow F$ wall even though $H^2(S_3,U(1))$ is trivial, showing that non-invertible symmetry is essential for some invertible walls of non-Abelian models.
The 3d toric code has gapped domain walls that convert point charges into anchored semi-loops; type-II walls host an Abelian anyon model and type-III walls host a non-Abelian anyon model.
Because gauging can be implemented with constant-depth adaptive circuits for solvable groups, the domain walls built by SPT-sewing are expected to be preparable in constant depth on quantum hardware.
Anchoring walls generalize electric-magnetic exchange to three dimensions and are not captured by the Lagrangian-subgroup classification of boundaries.

Reading between the lines

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

If Theorem 1 extends as conjectured, invertible domain walls in every finite-group quantum double would be generated by a single canonical symmetry, $G\times\mathrm{Rep}(G)\times G$; a natural test is to compute the wall group of the $D_4$ or $A_4$ quantum double and compare it with the SPT-sewing output.
Anchoring walls suggest a new kind of defect resource: a wall that converts point-like logical operators into loop-like operators could mediate between quantum error-correcting codes with different error models, and its energy barrier deserves testing as an ingredient for self-correcting quantum memories.
The same sewing idea can be applied to string-net or Walker-Wang bulks by choosing the seam SPT from the appropriate generalized symmetry, with the 3d construction here serving as the first concrete pattern.
The non-Abelian conjecture could be made numerically falsifiable by checking whether every braided autoequivalence of the $S_3$ quantum double, not just the two constructed walls, is realized by some $S_3\times\mathrm{Rep}(S_3)\times S_3$ SPT.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This one is worth a serious referee. The new thing is SPT-sewing: entangle two disconnected lattices with a lower-dimensional SPT, then gauge. That differs from the gauged-SPT defects in [49], and it buys you a lot: a complete constructive classification of invertible domain walls for any finite Abelian group G (Theorem 1), plus concrete non-Abelian and 3d constructions.

The Abelian proof is the strongest part. For Z2^n it goes through the Clifford group structure, and for general Abelian G it uses Kunneth and partition functions; the mapping from topological actions to anyon maps is explicit. The S3 section is honest: the S3 x S3-invariant wall is non-invertible, and the invertible C<->F wall requires the non-invertible Rep(S3) symmetry. They write down explicit stabilizers and a ribbon operator F_CF that commutes with them. The 3d anchoring walls are completely explicit stabilizer models with ribbon operators; I checked the type-II wall's Z4 anyon dictionary and it works. These are not existence claims, they come with commuting Hamiltonians.

The soft spot is exactly where the reader puts it. Theorem 2, which underlies the 'all' in Theorem 1, is not proved in the paper. It cites Beigi-Shor-Whalen for the wall Hamiltonians and asserts that gauging fixed-point K-SPT states realizes exactly those walls. There is no explicit dictionary from the [7] data to the gauging construction, and the claim that same-phase symmetric states map to same-type walls is also asserted. So the completeness theorem inherits an external classification plus an unverified correspondence. The delta-function-to-anyon-map step in Appendix B.2 is sketched as well. This is a real proof gap, not a counterexample; I see no circularity.

Minor soft spots: the nontriviality of |SPT2> is argued by projecting to a known (Z3 x Z3) x Z2 SPT rather than by classifying the full non-invertible SPT phase, and the 3d type-III wall's non-Abelian fusion is described at the level of examples. But the authors label these as evidence and leave classification open, which is fair.

For whom: people working on exactly solvable topological models, gauging and SPT, and fault-tolerant boundaries. I would take it to a reading group and I would cite it. Send it to peer review. The referee should ask for a fuller proof of Theorem 2 or an explicit reduction to [7], and for a clean statement about how gauging behaves under symmetric circuits. With that, the paper is solid.

Referee Report

3 major / 5 minor

Summary. The paper introduces SPT-sewing, a method for constructing gapped domain walls in quantum double models by entangling two close boundaries with a lower-dimensional symmetry-protected topological state and then applying the gauging map. The main formal result is Theorem 1, which states that for a finite Abelian group G every invertible domain wall of the G quantum double arises from SPT-sewing with G x G x G symmetry; the proof relies on Theorem 2, which asserts that all domain walls of a G quantum double are obtained by gauging 1d phases with unbroken subgroup K and cocycle nu. For the non-Abelian example G = S3, the paper constructs an invertible C <-> F domain wall by gauging an S3 x Rep(S3) x S3 SPT, and states Conjecture 1 that this construction is general. In the 3d toric code, the paper constructs type-I, type-II, and type-III domain walls by gauging 2d SPTs, including double-semion walls and two types of anchoring walls that convert point-like excitations into semi-loop excitations anchored on the wall; explicit stabilizer Hamiltonians and ribbon operators are provided.

Significance. If Theorem 1 is fully established, it gives a constructive completeness statement for invertible domain walls in Abelian quantum doubles: every such wall is a gauged 1d SPT. The 3d anchoring walls are a new class of explicitly solvable domain walls, and the paper provides detailed stabilizer and ribbon-operator calculations for them, so their existence is robust even if the completeness proof has gaps. The S3 example is a useful model-building step toward the non-Abelian conjecture, and the paper honestly frames the classification of non-invertible SPTs as open. The explicit lattice constructions, the use of the gauging map, and the concrete anyon maps are valuable contributions, and the connection to Clifford decompositions for Z2^n is an interesting technical tool.

major comments (3)

[Section IV.C and Appendix B.2, Theorem 2] Theorem 1 says 'all invertible domain walls', and its support is Theorem 2, but Theorem 2 is not proved to the same standard as the rest of the paper. The proof states that Ref. [7] gives the domain wall Hamiltonians and asserts that these correspond to gauging fixed-point K-SPT states, without providing an explicit dictionary from the Beigi-Shor-Whalen classification to the gauging construction. It also asserts that two symmetric states in the same phase yield domain walls related by a shallow circuit after gauging, without checking that the gauging map preserves this equivalence. Because this completeness step is load-bearing for Theorem 1, the proof should either supply the dictionary or state Theorem 1 as conditional on a precise completeness statement for Theorem 2.
[Appendix B.2, Eqs. (B19)-(B22)] The reduction from an arbitrary invertible (K,nu) domain wall to a projected G x G x G SPT is the core of the Abelian theorem, but the steps 'it is straightforward to show that each delta function gives rise to an anyon map' and 'we can always write the topological action given from nu as a bilinear form' are asserted rather than derived. In particular, the treatment explicitly handles (K,0) cases and then states that the general cocycle case can be brought to the same form; this requires a proof that the p-group decomposition and the diagonal gauge-field change do not alter the anyon map. Without this, the claim that all braided autoequivalences are covered is incomplete.
[Section V.D and Appendix A.5] The nontriviality of the S3 x Rep(S3) x S3 SPT |SPT2> is inferred from the projection to a (Z3 x Z3) semi-direct Z2 SPT, while the paper itself notes that no classification of non-invertible SPTs is available. The explicit stabilizers and the F^CF ribbon operator in Appendix E.2 are self-contained and support the claim that the constructed wall exchanges C and F, but the wording that gauging a genuinely nontrivial non-invertible SPT produces this wall goes beyond what is established. I recommend presenting this as direct evidence for Conjecture 1 with the classification issue stated explicitly.

minor comments (5)

[Appendix A.5, first paragraph] There is a repeated article in 'the the diagonal group'; please fix this typo.
[Section V.C, Eq. (44)] The phrase 'the action of the domain wall on these anyons is A <-> A, B <-> D, E <-> E' is ambiguous because the first and last entries are not exchanges; clarify that A and E pass through while B and D are swapped.
[Appendix D.3] There is a typo 'decompostion', and after checking the (11) and (12) matrix entries the text says the other entries can be obtained similarly without showing them; please spell out that the remaining cases are identical or provide the calculation.
[Section VII] The simplified models are said to produce the same physics as the constructions of Section VI, but no locality-preserving equivalence to the triangular-lattice models is given; a brief argument or reference would make this claim precise.
[Tables I and IV] The 'Domain wall' column mixes rows obtained from different mechanisms, such as gauged-SPT defects, SPT-sewing, and symmetry-breaking limits; adding a column that labels the mechanism would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Abelian completeness theorem rests on external classification theorems and explicit gauging calculations, not on the paper's own target.

full rationale

The derivation chain is not circular. Theorem 1 for Abelian groups is built from (i) the external Clifford canonical-form theorem of Ref. [72], (ii) the external domain-wall Hamiltonian classification of Ref. [7], and (iii) an explicit gauging/partition-function dictionary developed in Appendix B. The closest point to a circularity concern is the proof of Theorem 2 in Appendix B.2, which states: 'In Ref. [7], the domain wall Hamiltonian in a G quantum double on square lattice are given, which corresponds to gauging the fixed-point K SPT states broken from the G × G symmetry.' This is a citation to an external classification plus an asserted correspondence, and the word 'all' in Theorem 1 inherits this as a completeness gap. It is not circular: Ref. [7] is not the present authors' work, and the asserted correspondence is not a restatement of Theorem 1. Self-citations such as Ref. [35] supply a partition-function formula for projected 1d phases in the proof; that formula does not assume the target domain-wall classification and is independently checkable. The non-Abelian C↔F wall is verified by explicit stabilizers (Eq. E12) and by showing the ribbon operator F_CF (Eq. E13) commutes with them; |SPT2⟩'s nontriviality is supported by projection to the independently constructed (Z3×Z3)⋊Z2 SPT (Appendix A.5 and Appendix E.2). The 3d anchoring walls are explicit stabilizer models (Section VII and Appendix F) with explicit ribbon operators; no parameter is fitted and renamed a prediction. The paper itself flags the open parts: Conjecture 1 is labeled a conjecture with proof left to future work, and the classification of anchoring walls is 'left for future work' (Section VIII). These are limitations, not circular reductions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

No fitted parameters; all constructions are analytical. The paper relies on standard gauge-theory and SPT classifications, and on external results for Clifford circuits and domain wall classification.

assumptions (5)

domain assumption Quantum double models and their ground states are obtained from gauging symmetric product states (Section III).
The gauging map used throughout is assumed to map global-symmetry states to gauge-theory ground states, following Yoshida [61].
domain assumption Gapped domain walls in quantum doubles are classified by pairs (K, ν) and their gauged versions (Ref. [7]).
Theorem 2, and therefore the Abelian completeness theorem, relies on this external classification being exhaustive.
domain assumption Every braided autoequivalence of anyons in Zn2 quantum double corresponds to a Clifford group element with the canonical form of Bravyi and Maslov [72].
Used in Lemma 2 to enumerate all anyon maps; the paper applies the theorem 'with some adjustments' whose details are not fully shown.
standard math Group cohomology classifies bosonic SPT phases and the stated cocycles describe the relevant SPT states.
Standard classification of SPT phases by H^{d+1}(G, U(1)) is used throughout.
domain assumption The S3 × Rep(S3) × S3 states are genuine SPTs under the non-invertible symmetry.
Nontriviality is argued via projection to a known (Z3 × Z3) ⋊ Z2 SPT, not via a general classification of non-invertible SPTs.

invented entities (2)

Anchoring domain walls (type-II and type-III) in the 3d toric code
purpose: Domain walls that transform point-like excitations into semi-loop-like excitations anchored on the wall
The paper constructs explicit lattice Hamiltonians realizing these walls and analyzes their anyon content, so they are a new theoretical construct with internal consistency but no external falsifiable prediction beyond the models themselves.
G × Rep(G) × G non-invertible SPT states
purpose: Input states for SPT-sewing that, after gauging, produce invertible domain walls such as the C ↔ F wall in S3 quantum double
The states are explicitly written (e.g., |SPT2⟩ in Eq. (52)) and their stabilizers are given; nontriviality is argued via the projected (Z3 × Z3) ⋊ Z2 SPT, not via an external classification.

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Cite this review

Pith. "Pith review of Domain walls from SPT-sewing." pith.science (2026). https://pith.science/paper/RTGNR5MT

@misc{pith2026241111967,
  author       = {Pith},
  title        = {Pith review of: Domain walls from SPT-sewing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RTGNR5MT}},
  note         = {Machine review of arXiv:2411.11967}
}

abstract

We introduce a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible $G\times \text{Rep}(G)\times G$ symmetry and invertible domain walls in the quantum double associated with the group $G$. We prove this correspondence when $G$ is Abelian and provide evidence for the general case by studying the quantum double model for $G=S_3$. We also use our method to construct \emph{anchoring domain walls}, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.

Figures

Figures reproduced from arXiv: 2411.11967 by the authors.

**Figure 2.** (a) The action of the operators A g v and Bp. (b) Gauging maps any state to a zero-flux state. IV. DOMAIN WALLS IN THE 2D ABELIAN QUANTUM DOUBLE In this section, we generalize SPT-sewing from Section II to the 2d quantum double model for any Abelian group G. The main result of this section is that all the domain walls can be obtained from G × G × G SPT-sewing. A. Gauged-SPT domain walls To study the general SPT-sewi… view at source ↗

**Figure 3.** (a) After folding the system along the domain [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Ribbon ξ starts from a site s0 and ends at a site s1. The ribbon operator F (h,g) ξ on this ribbon is composed of some shift operators on the edges and a Kronecker delta. they have quantum dimensions dC = dF = 2. Correspondingly, their ribbon operators creating C and …

**Figure 5.** Figure 5: An S3 × Rep(S3) × S3 SPT is defined by a Hamiltonian composed of these commuting stabilizers; see Eq (35) for the definition of the operators. In this figure, m and l are site labels for odd and even sites, and a, b, and c are the labels of each layer. The term gcm cor…

**Figure 6.** Figure 6: The C ↔ F domain wall in the S3 quantum double is obtained from the e ↔ m domain wall in the Z3 quantum double via gauging the charge conjugation symmetry. state with a Z2 charge conjugation symmetry. The domain wall becomes an invertible domain wall in this Z3 quantum…

**Figure 7.** Figure 7: (a) Lattice structure used in the construction of the 3d models. Each triangular lattice is perpendicular [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

**Figure 8.** Figure 8: The excitations of the type-I domain wall. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Excitations of the anchoring domain wall [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Excitations of the anchoring domain wall [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

**Figure 11.** Figure 11: Excitations of the anchoring domain wall in [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]

**Figure 12.** Figure 12: (a) Lattice used in the construction of type-II domain wall. We assign two qubits per site, each one being [PITH_FULL_IMAGE:figures/full_fig_p056_12.png]

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Works this paper leans on

46 extracted references · 35 canonical work pages · cited by 2 Pith papers

[1]

× G to a U (1) value 4

Group cohomology An n-cochain is a map from group elements in Gn = G × ... × G to a U (1) value 4. We denote it as cn(g1, g2, ..., gn). The set of n-cochain forms a group Cn(G, U(1))5, and the group elements have the following multiplication rule, (c · c′)n(g1, g2, ..., gn) = cn(g1, g2, ..., gn) · c′ n(g1, g2, ..., gn). (A1) The coboundary operator δ is a...

work page
[2]

Cup product Instead of the algebraic interpretation of group cohomology we introduced above, there is also a geometric interpretation. In this interpretation, Cn(G, M) is the group of n-cochains satisfying the following condition, Cn(G, M) := {νn|gνn(g0, g1, ..., gn) = νn(gg0, gg1, ..., ggn)}, (A14) in which νn is a map such that νn : Gn+1 → M . The relat...

work page
[3]

T opological action and SPT state In this subsection, we provide a brief introduction to the topological action, and its corresponding fixed- point SPT wavefunctions. For a (d + 1)-dimensional G-SPT state, the topological action is given by Stop [M, A] = 2πi Z M L[A], (A21) in which A is a background gauge field that is a G-valued 1-cocycle, and M is the ...

work page
[4]

Consider a lattice L with vertices v ∈ V

G-SPT phases in 2 + 1D In this subsection, we construct the 2 + 1D G-SPT Hamiltonian and ground state wavefuncion. Consider a lattice L with vertices v ∈ V. We assign one qudit per vertex. The qudit state is given by |g⟩, g ∈ G. The trivial G-symmetric ground state can be written as, |Ψ⟩ =   1 |G| X g∈G |g⟩   ⊗N . (A35) The global symmetry operator is...

work page
[5]

Consider we have a 1 + 1D S3 × S3 state, and the Z2 × Z2 subgroup breaks in the the diagonal group Z2

1 + 1D (Z3 × Z3) ⋊ Z2 SPT phases As an illustrative example, and also for the benefit of later discussion, let us consider constructing the lattice model for the 1 + 1D (Z3 × Z3) ⋊ Z2 SPT. Consider we have a 1 + 1D S3 × S3 state, and the Z2 × Z2 subgroup breaks in the the diagonal group Z2. The corresponding nontrivial SPTs are classified by the nontrivia...

work page
[6]

Hd (G × G′, M) = dM k=0 Hk G, Hd−k (G′, M)

K¨ unneth formula and the classification of 1d domain walls When M is Abelian, finitely-generated, and a trivial G × G′ module, we have the following K¨ unneth formula [81]. Hd (G × G′, M) = dM k=0 Hk G, Hd−k (G′, M) . (A53) In 1+1d, the K¨ unneth formula can help us understand the domain wall classifications. Let us look at one example. When d = 2, M = U...

work page
[37]

[72], with some adjustments, any action on the Pauli operators can given by the conjugation of the following canonical unitary operator U = ΩΠΩ′

Proof of Lemma 2 According to the theorem 1 in Ref. [72], with some adjustments, any action on the Pauli operators can given by the conjugation of the following canonical unitary operator U = ΩΠΩ′. The three unitary operators are Ω = nY i,j=1 CZ Γi,j i,j , Ω′ = nY i,j=1 CZ Γ′ i,j i,j , Π =( Y i̸=j CX ∆i,j i,j )S( nY i=1 H hi i )( Y i̸=j CX ∆′ i,j i,j ), (...

work page
[38]

diagonal

Proof of Theorem 1 In this section, we prove that G × G × G SPT-sewing gives rise to all the invertible domain walls in a 2d G quantum double for any Abelian group G. An Abelian group is always isomorphic to a product of cyclic group, G = Zn1 × Zn2 × · · ·Znq . Thus the most general form of topological action that characterizes 1d G × G × G SPT is given b...

work page

Show all 46 references

[39]

Gauging is a bijective, isometric duality map from wavefunctions with global symmetries to wavefunctions with gauge (local) symmetries [61]

Gauging map In this subsection, we provide a brief discussion of the gauging map for ground state wavefunctions and operators in systems with a finite symmetry group G. Gauging is a bijective, isometric duality map from wavefunctions with global symmetries to wavefunctions wit...
[40]

The S3 group is the semidirect product of Z3 and Z2, S3 = Z3 ⋊ Z2

Gauging S3 global symmetry In this subsection, we take the S3 symmetry as an example to show how to sequentially gauge the Z3 and Z2 subgroups, and finally obtain the ground state and Hamiltonian of the S3 quantum double. The S3 group is the semidirect product of Z3 and Z2, S3...
[41]

F c2,e ξ F c2,t ξ F c,t ξ F c,e ξ ! + F c2,c ξ F c2,tc2 ξ F c,tc ξ F c,c2 ξ ! + F c2,c2 ξ F c2,tc ξ F c,tc2 ξ F c,c ξ !# , (D25) F ([c],ω) ξ = 1 3

Ribbon operators of S3 quantum double Given a ribbon ξ, one can define a ribbon operator F (h,g) ξ such that it commutes with all the stabilizers except for the stabilizers at the endpoints [1]. The ribbon operator of quantum doubles is defined in Fig. 4 44 and the multiplicat...
[42]

We can sew the two smooth boundaries together by applying the SPT entangler, which corresponds to the nontrivial 2-cocycles ω2 ∈ H 2(S3 × S3, U(1))

S3 × S3 SPT-sewn domain wall Consider we have two S3 Ising models with smooth boundaries in the symmetric phase. We can sew the two smooth boundaries together by applying the SPT entangler, which corresponds to the nontrivial 2-cocycles ω2 ∈ H 2(S3 × S3, U(1)). We have H 2(S3 ...
[43]

The ( Z3 × Z3) ⋊ Z2 SPT is calculated in Appendix A 5

C ↔ F domain wall In this subsection, we construct the C ↔ F domain wall by gauging the ( Z3 × Z3) ⋊ Z2 SPT on a codimension-1 submanifold of the 2d S3 Ising model. The ( Z3 × Z3) ⋊ Z2 SPT is calculated in Appendix A 5. Unlike in Appendix A 5, where we assume there is one qubi...
[44]

The SPT on this sub- manifold is called the type-I SPT, which is equivalent to the Levin-Gu SPT [89]

Type-I domain wall We begin by constructing a nontrivial Z2 SPT on a codimension-1 submanifold. The SPT on this sub- manifold is called the type-I SPT, which is equivalent to the Levin-Gu SPT [89]. The corresponding type-I 3-cocycle is given by [61, 70] ωI (g1, g2, g3) = exp p...
[45]

Restricted to the domain wall, the model is effectively a Z2 ×Z2 twisted quantum double (TQD), which is also equivalent to theZ4 quantum double (QD) [92]

Type-II domain wall In this subsection, we discuss the 2 + 1D Z2 × Z2 SPT-sewn domain wall. Restricted to the domain wall, the model is effectively a Z2 ×Z2 twisted quantum double (TQD), which is also equivalent to theZ4 quantum double (QD) [92]. We consider a lattice construc...
[46]

The procedure is briefly summarized as follows: Consider that we have three copies of the lattice displayed in Eq

Type-III domain wall In this subsection, we discuss the Z2 × Z2 × Z2 SPT-sewn domain wall between two 3+1D toric codes and one 3+1D toric code. The procedure is briefly summarized as follows: Consider that we have three copies of the lattice displayed in Eq. (F1). For each cop...
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Pith tools

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

[1] [1]

× G to a U (1) value 4

Group cohomology An n-cochain is a map from group elements in Gn = G × ... × G to a U (1) value 4. We denote it as cn(g1, g2, ..., gn). The set of n-cochain forms a group Cn(G, U(1))5, and the group elements have the following multiplication rule, (c · c′)n(g1, g2, ..., gn) = cn(g1, g2, ..., gn) · c′ n(g1, g2, ..., gn). (A1) The coboundary operator δ is a...

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[2] [2]

Cup product Instead of the algebraic interpretation of group cohomology we introduced above, there is also a geometric interpretation. In this interpretation, Cn(G, M) is the group of n-cochains satisfying the following condition, Cn(G, M) := {νn|gνn(g0, g1, ..., gn) = νn(gg0, gg1, ..., ggn)}, (A14) in which νn is a map such that νn : Gn+1 → M . The relat...

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[3] [3]

T opological action and SPT state In this subsection, we provide a brief introduction to the topological action, and its corresponding fixed- point SPT wavefunctions. For a (d + 1)-dimensional G-SPT state, the topological action is given by Stop [M, A] = 2πi Z M L[A], (A21) in which A is a background gauge field that is a G-valued 1-cocycle, and M is the ...

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[4] [4]

Consider a lattice L with vertices v ∈ V

G-SPT phases in 2 + 1D In this subsection, we construct the 2 + 1D G-SPT Hamiltonian and ground state wavefuncion. Consider a lattice L with vertices v ∈ V. We assign one qudit per vertex. The qudit state is given by |g⟩, g ∈ G. The trivial G-symmetric ground state can be written as, |Ψ⟩ =   1 |G| X g∈G |g⟩   ⊗N . (A35) The global symmetry operator is...

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[5] [5]

Consider we have a 1 + 1D S3 × S3 state, and the Z2 × Z2 subgroup breaks in the the diagonal group Z2

1 + 1D (Z3 × Z3) ⋊ Z2 SPT phases As an illustrative example, and also for the benefit of later discussion, let us consider constructing the lattice model for the 1 + 1D (Z3 × Z3) ⋊ Z2 SPT. Consider we have a 1 + 1D S3 × S3 state, and the Z2 × Z2 subgroup breaks in the the diagonal group Z2. The corresponding nontrivial SPTs are classified by the nontrivia...

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[6] [6]

Hd (G × G′, M) = dM k=0 Hk G, Hd−k (G′, M)

K¨ unneth formula and the classification of 1d domain walls When M is Abelian, finitely-generated, and a trivial G × G′ module, we have the following K¨ unneth formula [81]. Hd (G × G′, M) = dM k=0 Hk G, Hd−k (G′, M) . (A53) In 1+1d, the K¨ unneth formula can help us understand the domain wall classifications. Let us look at one example. When d = 2, M = U...

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[7] [37]

[72], with some adjustments, any action on the Pauli operators can given by the conjugation of the following canonical unitary operator U = ΩΠΩ′

Proof of Lemma 2 According to the theorem 1 in Ref. [72], with some adjustments, any action on the Pauli operators can given by the conjugation of the following canonical unitary operator U = ΩΠΩ′. The three unitary operators are Ω = nY i,j=1 CZ Γi,j i,j , Ω′ = nY i,j=1 CZ Γ′ i,j i,j , Π =( Y i̸=j CX ∆i,j i,j )S( nY i=1 H hi i )( Y i̸=j CX ∆′ i,j i,j ), (...

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[8] [38]

diagonal

Proof of Theorem 1 In this section, we prove that G × G × G SPT-sewing gives rise to all the invertible domain walls in a 2d G quantum double for any Abelian group G. An Abelian group is always isomorphic to a product of cyclic group, G = Zn1 × Zn2 × · · ·Znq . Thus the most general form of topological action that characterizes 1d G × G × G SPT is given b...

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