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arxiv: 2310.19266 · v2 · submitted 2023-10-30 · ❄️ cond-mat.str-el · math-ph· math.MP

Correspondence of boundary theories between internal and crystalline symmetry protected topological phases

Jian-Hao Zhang , Shang-Qiang Ning This is my paper

Pith reviewed 2026-05-24 06:25 UTC · model grok-4.3

classification ❄️ cond-mat.str-el math-phmath.MP
keywords symmetry protected topological phasesanomaly indicatorsboundary theoriestime-reversal symmetrymirror symmetrysymmetry enriched topological ordersconformal field theoryhigher-order topological phases
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The pith

Boundary anomalies of internal and crystalline SPT phases correspond via mapped indicators.

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

The paper shows that anomalous boundary states of symmetry-protected topological phases with internal symmetries correspond directly to those with crystalline symmetries in three or higher dimensions. It demonstrates this correspondence first for surface symmetry-enriched topological orders by mapping time-reversal anomaly indicators onto mirror anomaly indicators, proving that a given 2D topological order carries one anomaly exactly when it carries the other. The same logic extends to continuous symmetries by generating new anomaly indicators from crystalline counterparts. For near-critical boundaries the paper relates 1+1D conformal field theory edge theories of internal fermionic SPTs to 0+1D corner modes of higher-order crystalline fermionic SPTs, treating the corner modes as perturbations of the CFT.

Core claim

The central claim is a one-to-one correspondence between the anomalous boundary theories of internal and crystalline SPT phases. For SET surfaces this takes the explicit form of a direct mapping between time-reversal and mirror anomaly indicators, so that any 2D topological order realizing one anomaly realizes the other. For critical boundaries the correspondence appears as an edge-corner relation in which the 1+1D CFT describing the edge of an internal fermionic SPT is identified with the 0+1D corner modes of the crystalline counterpart, with the perturbed-CFT viewpoint also yielding boundary theories for intrinsically interacting fermionic SPTs.

What carries the argument

The direct mapping of time-reversal anomaly indicators to mirror anomaly indicators for SET boundaries, together with the edge-corner correspondence that treats corner modes as perturbed 1+1D CFT for critical boundaries.

If this is right

  • A 2D topological order carries the time-reversal anomaly if and only if it carries the mirror anomaly.
  • New anomaly indicators for continuous symmetries can be read off from their crystalline counterparts.
  • Quantum anomalies of one symmetry type can be explored using the crystalline equivalent.
  • Boundary theories of some intrinsically interacting fermionic SPTs become accessible by viewing corner modes as perturbed CFT.

Where Pith is reading between the lines

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

  • The same indicator-mapping technique may apply to other pairs of internal and crystalline symmetries beyond time-reversal and mirror.
  • Methods already developed to detect mirror anomalies could be repurposed to detect time-reversal anomalies in the same orders.
  • The correspondence suggests that classification results obtained in one symmetry setting can be translated into physical boundary predictions in the other setting.

Load-bearing premise

The known abstract classification correspondence between internal and crystalline SPT phases extends without further assumptions to a concrete, indicator-level correspondence on their anomalous boundary theories.

What would settle it

Existence of a specific 2D topological order whose computed time-reversal anomaly indicator is nonzero while the mapped mirror anomaly indicator is zero (or vice versa) would falsify the claimed indicator mapping.

Figures

Figures reproduced from arXiv: 2310.19266 by Jian-Hao Zhang, Shang-Qiang Ning.

Figure 1
Figure 1. Figure 1: FIG. 1: The anomalous boundary correspondence. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Mirror plane decoration for 3D mirror SPT. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The n-Sphere angles traced by two [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The cell decomposition of the 2-fold [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

Symmetry-protected topological phases protected by crystalline symmetries and internal symmetries are shown to enjoy a fascinating one-to-one correspondence in classification. Here we investigate the physics content behind the abstract correspondence in three or higher-dimensional systems. We show correspondence between anomalous boundary states, which provides a new way to explore the quantum anomaly of symmetry from its crystalline equivalent counterpart. We show such a correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical boundary states. (1) First of all, for the surface SET correspondence, we demonstrate it by considering examples involving time-reversal symmetry and mirror symmetry. We show that one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa, by directly establishing the mapping of the time reversal anomaly indicators and mirror anomaly indicators. Besides, we also consider other cases involving continuous symmetry, which leads us to introduce some new anomaly indicators for symmetry from its counterpart. (2) Furthermore, we also build up direct correspondence for (near) critical boundaries. In this perspective, we first consider the edge-corner correspondence between edge theory as 1+1D conformal field theory of internal fermionic SPT and the 0+1D corner modes of (higher-order) crystalline fermionic SPT. By viewing the corner modes on 1D boundary as perturbed CFT is crucial insight for the correspondence, but also help to discover the boundary theory of some intrinsically interacting fermionic SPT, which are challenging.

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 / 2 minor

Summary. The manuscript claims a one-to-one correspondence between anomalous boundary theories of internal and crystalline SPT phases in d≥3. It demonstrates this via explicit mappings of time-reversal and mirror anomaly indicators on 2D SET boundaries (showing that a given topological order can host the TR anomaly iff it can host the mirror anomaly) and via an edge-corner correspondence that relates 1+1D CFT edge modes of internal fermionic SPTs to 0+1D corner modes of higher-order crystalline fermionic SPTs, including new anomaly indicators for continuous symmetries.

Significance. If the indicator mappings hold beyond the presented cases, the work supplies a concrete dictionary that lets anomalies of one symmetry type be read off from its crystalline counterpart, which is useful for classification and for discovering boundary theories of interacting fermionic SPTs. The explicit example computations and the CFT-perturbation insight for corner modes are concrete strengths.

major comments (2)
  1. [surface SET correspondence section] The central claim that 'one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa' is supported only by explicit computation on a handful of Abelian SET examples (toric-code variants and similar). No general argument or exhaustive check is given that the indicator map is bijective for arbitrary (including non-Abelian) 2D topological orders or for inequivalent symmetry realizations (projective vs linear).
  2. [anomaly indicator mapping discussion] The abstract states that the correspondence is 'directly established' by mapping the indicators, yet the text supplies no closed-form expression or proof that every possible nonzero indicator value for one symmetry is matched by a unique value for the other; the mapping therefore remains example-dependent rather than derived from the bulk-boundary correspondence.
minor comments (2)
  1. [continuous symmetry subsection] Notation for the new continuous-symmetry anomaly indicators is introduced without a dedicated table comparing them to the discrete-symmetry cases.
  2. [figures in SET examples] Several figure captions for the SET lattice models do not list the explicit symmetry action on the anyons, making it harder to reproduce the indicator calculations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive criticism. The comments correctly identify that our demonstrations rely on explicit example computations rather than general proofs. We address each point below and will revise the manuscript accordingly to clarify the scope of the results.

read point-by-point responses

  1. Referee: [surface SET correspondence section] The central claim that 'one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa' is supported only by explicit computation on a handful of Abelian SET examples (toric-code variants and similar). No general argument or exhaustive check is given that the indicator map is bijective for arbitrary (including non-Abelian) 2D topological orders or for inequivalent symmetry realizations (projective vs linear).

    Authors: We agree that the central claim is supported only by explicit computations on a limited set of Abelian SET examples. The manuscript does not contain a general argument establishing bijectivity of the indicator map for arbitrary topological orders (including non-Abelian ones) or for inequivalent symmetry realizations. This is a genuine limitation of the present work. We will revise the relevant section to explicitly state that the correspondence is verified through these examples and to note the absence of a general proof as an open question. revision: yes

  2. Referee: [anomaly indicator mapping discussion] The abstract states that the correspondence is 'directly established' by mapping the indicators, yet the text supplies no closed-form expression or proof that every possible nonzero indicator value for one symmetry is matched by a unique value for the other; the mapping therefore remains example-dependent rather than derived from the bulk-boundary correspondence.

    Authors: We acknowledge that the abstract phrasing 'directly established' may suggest a more general derivation than what is provided. The mappings are constructed case-by-case for the examples considered, without a closed-form expression or a derivation from bulk-boundary correspondence that would guarantee matching for every possible indicator value. We will revise the abstract and the anomaly-indicator discussion to accurately describe the example-dependent nature of the mappings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; correspondence shown via explicit example computations.

full rationale

The paper's central claim is a direct mapping between time-reversal and mirror anomaly indicators for 2D SETs, established by explicit computation on specific Abelian examples (toric code variants). The abstract states this is done 'by directly establishing the mapping,' with no quoted equations or steps that reduce the result to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation relies on case-by-case verification rather than any ansatz or uniqueness theorem imported from prior self-work that would force the outcome by construction. This is the normal non-circular outcome for an example-driven correspondence paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5803 in / 1051 out tokens · 16945 ms · 2026-05-24T06:25:02.346762+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Unraveling the Bott spiral

    math-ph 2026-05 unverdicted novelty 8.0

    A new homotopy model for the Bott spiral of fermionic SPTs is built via twisted ABS orientation and IFT spiral maps, showing IFTs need more symmetry data than K-theory and relying on an extraspecial group isomorphism ...

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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    So the anomaly specified at the UV limit will always be present in the low energy limit (IR limit)

    Exact anomaly The quantum anomaly of symme- try is invariant under renormalization flow. So the anomaly specified at the UV limit will always be present in the low energy limit (IR limit). There might be an emergent anomaly of symmetry 94 and 4 even anomaly of emergent symmetry (in contrast to the exact symmetry that is defined in UV limit), but we do not...

    work page

  2. [2]

    That means two boundary phases with the same quantum anomaly could transfer into each other under by tuning some parameter of the surface Hamiltonian

    Symmetry-preserving operation In the UV limit, the quantum anomaly of symmetry is invariant under symmetry preserving operation (such as tuning pa- rameters of surface Hamiltonian) even though phase transition happens in the ground state 95. That means two boundary phases with the same quantum anomaly could transfer into each other under by tuning some pa...

    work page

  3. [3]

    For orientation changing symme- try, there is alternative way to detect the anomaly that is determined by the bulk topology

    T opological class of anomaly For orientation pre- serving symmetry, one way to see the quantum anomaly of boundary theories is to use the anomaly in- flow approach which is cancelled by the presence of the topological bulk 96. For orientation changing symme- try, there is alternative way to detect the anomaly that is determined by the bulk topology. In o...

    work page

  4. [4]

    The anomaly-preserving operation is defined to the trans- formation that can not change the topological class of the quantum anomaly of symmetry

    Anomaly-preserving operation The boundary cor- respondence is assumed to be in the sense of quan- tum anomaly, namely the boundary correspondence is defined up to anomaly-preserving operation. The anomaly-preserving operation is defined to the trans- formation that can not change the topological class of the quantum anomaly of symmetry. In fact, the two s...

    work page

  5. [5]

    One-to-one correspondence Together with the above mentioned points that any two theories on the SPT can be transformed into each other by symmetry- preserving operation, then the one-to-one correspon- dence can be made if we can make a connection (up to some proper anomaly-preserving operation) between arbitrary one boundary theory of the internal SPT and...

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    Application One important application of the boundary correspondence is that we can explore the quantum anomaly of symmetry by studying its crys- talline partner. Recalling that for a phase or the- ory (such the underlyding topological order of SET), whether it can carry quantum anomaly of certain in- ternal symmetry is equivalent to say whether it can be...

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    Two of the roots are protected by only time reversal or mirror symmetry, which can be detected by the above mentioned two anomaly indicators η1 and ηT or ηM

    Topological insulator and topological crystalline insulator protected by mirror symmetry For 3D bosonic topological insulators with U(1) × Z T 2 and their crystalline counterparts—topological crys- talline insulator with U(1) × Z M 2 , their classification are both (Z2)4. Two of the roots are protected by only time reversal or mirror symmetry, which can b...

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    symmetry enforced gaplessness

    Topological phase with SU (2) × Z T 2 and topological crystalline phase with SU (2) × Z M 2 The classification for 3D bosonic topological phase pro- tected by SU (2)×Z T 2 and SU (2)×Z M 2 is (Z2)3. The first two roots do not need the protection of SU (2) and have already been discussed above. So we focus on the third root here. As the phase is still prot...

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    Two roots of them are protected only by time reversal

    Topological phase with SO(N) × Z T 2 and topological crystalline phase with SO(N) × Z M 2 The classification for 3D topological phases protected by SO(3) × Z T (M) 2 is classified by ( Z2)4. Two roots of them are protected only by time reversal. And the other two ones are one-to-one correspondence to the ones by breaking SO(3) down to U(1)z = {eiα ˆSz|α∈[...

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    So in turn, we can conjecture the two anomaly indicators (13) and (14) are also the same for SO(N) × Z M 2 which might be derived using the folding trick in Ref. 105. Therefore, the topological order C can or cannot be put on the surface of 3D topological phases protected by SO(N) × Z T 2 and SO(N) × Z M 2

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    This builds up the surface SET correspondence of them. IV. SURF ACE CRITICALITY CORRESPONDENCE OF 3D TOPOLOGICAL PHASES: MIRROR SYSTEM In this section, we discuss the (near) critical sur- face of 3D mirror symmetry-protected topological phase and also its counterpartner—3D time-reversal topologi- cal phase—that is classified by group cohomology. A. The ne...

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    The extension of ⃗ n(u, τ) is a choice of the convention as one can use the south pole instead of the north one at the position of u = 0

    Review on Haldane’s derivation To begin with, we first review that effective descrip- tion of 0+1d spin one-half system that is the 0+1d O(3) nonlinear sigma model with level one WZW term S0 = Z dτ 1 g (∂τ ⃗ n)2 + 2πkΓ[⃗ n(u, τ)] (25) We have level k = 1 and Γ[⃗ n(u, τ)] is the 0+1D WZW Γ[⃗ n] = 1 4π Z dudτ⃗ n· (∂u⃗ n× ∂τ ⃗ n) (26) where we have extended ...

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    narrow ribbon

    Therefore, the second term of summed S0[ni] gives the O(3) topological theta term with Θ = π. Now we give a more intuitive derivation of Eq.(31). Re- call that the physical meaning of Γ[⃗ n′ i] is the surface angle on the sphere surrounded by the trajectory of physical vector ⃗ n′ i(τ) from τ = 0 to τ = β. Then Γ[ ⃗ n′ 2i] − Γ[⃗ n′ 2i−1] is just the diffe...

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    narrow ribbon

    Generalization to (n + 1)D Let us begin with the nD O (n + 2) nonlinear sigma model with level one WZW terms Sn = Z dxn 1 g (∂µ⃗ n)2 + 2πkΓn+1[⃗ n] (33) where ⃗ nis the n + 2 component unit vector k = 1 is the level and Γ n[⃗ n] is the WZW term Γn+1[⃗ n] = 1 Ωn+1 Z dudxnϵabc...dna∂unb∂xnc...∂τ nd (34) where Ω n+1 is the volume of n + 1 sphere. We have ext...

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    Generalize to three and higher fermionic dimension

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    Study the surface criticality correspondence for SPT beyond group cohomology

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    Local unitary trans- formation, long-range quantum entanglement, wave func- tion renormalization, and topological order,

    Generalize to SPT phases with average symmetry114–118. Note − While preparing this manuscript, we notice that the mapping between the time reversal and reflection data in 2D SET are also discussed in Ref. 119. Acknowledgements – Stimulating discussions with Zhen Bi, Ruochen Ma, Liujun Zou, Chong Wang, and Chenjie Wang are acknowledged. JHZ is supported 14...

    work page doi:10.1038/ncomms4507 2010