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arxiv: 2304.01827 · v2 · pith:Y5OGPNBTnew · submitted 2023-04-04 · ❄️ cond-mat.mes-hall · cond-mat.supr-con

Atiyah-Hirzebruch spectral sequence for topological insulators and superconductors: E₂ pages for 1651 magnetic space groups

Ken Shiozaki , Seishiro Ono This is my paper

Pith reviewed 2026-05-24 09:09 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.supr-con
keywords Atiyah-Hirzebruch spectral sequencetopological crystalline insulatorstopological superconductorsmagnetic space groupsK-theorysymmetry-protected topological phasesthree-dimensional classificationcrystalline symmetries
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The pith

The Atiyah-Hirzebruch spectral sequence determines the topological K-groups for about 59 percent of three-dimensional magnetic space group settings.

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

This paper computes the E2 pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequence for topological crystalline insulators and superconductors in up to three dimensions. The computation applies to all 1651 magnetic space groups and uses a cell decomposition in which any group element that fixes a cell setwise also fixes it pointwise. Under a physically reasonable assumption the authors enumerate the K-groups consistent with those E2 pages in both versions of the sequence, thereby fixing the classification for roughly 59 percent of the three-dimensional symmetry settings.

Core claim

We compute the E2 pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequence for topological crystalline insulators and superconductors up to three spatial dimensions, considering the cell decomposition in which if a group action fixes a cell setwise then its group action fixes the same cell pointwise. Under a physically reasonable assumption, we enumerate all possible K-groups that are compatible with the E2 pages for both momentum and real-space AHSS. As a result, we determine the K-groups for approximately 59% of symmetry settings in three spatial dimensions.

What carries the argument

The Atiyah-Hirzebruch spectral sequence applied to a cell decomposition of momentum or real space, where the group action fixes fixed cells pointwise.

If this is right

  • Explicit lists of allowed topological invariants now exist for 59 percent of three-dimensional magnetic space groups.
  • Both momentum-space and real-space E2 pages are supplied so that classifications can be cross-checked.
  • The tabulated results cover every one of the 1651 magnetic space groups considered.
  • The same cell-decomposition procedure yields E2 pages in lower dimensions for the same groups.

Where Pith is reading between the lines

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

  • Screening of candidate materials can now use the enumerated K-groups to predict which invariants are possible under given symmetries.
  • Cases outside the 59 percent may require computation of higher pages of the spectral sequence or additional physical constraints.
  • The dual momentum and real-space computations could reveal when extra consistency conditions beyond the E2 pages are needed.

Load-bearing premise

The actual K-groups are precisely the ones compatible with the computed E2 pages of both the momentum-space and real-space versions.

What would settle it

A numerical or experimental determination of a topological invariant for one of the enumerated symmetry settings that lies outside the listed possible K-groups.

Figures

Figures reproduced from arXiv: 2304.01827 by Ken Shiozaki, Seishiro Ono.

Figure 1
Figure 1. Figure 1: FIG. 1. A cell decomposition of BZ for MSG [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A cell decomposition of real space for MSG [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The relationship between the boundary [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. [a] The higher differentials [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

We compute the $E_2$ pages of the momentum-space and real-space Atiyah-Hirzebruch spectral sequence (AHSS) for topological crystalline insulators and superconductors up to three spatial dimensions, considering the cell decomposition in which if a group action fixes a cell setwise then its group action fixes the same cell pointwise. We provide a detailed description of the implementation for computing the $E_2$ pages of AHSS. Under a physically reasonable assumption, we enumerate all possible $K$-groups that are compatible with the $E_2$ pages for both momentum and real-space AHSS. As a result, we determine the $K$-groups for approximately 59\% of symmetry settings in three spatial dimensions. All the results can be found at this http \href{https://www2.yukawa.kyoto-u.ac.jp/~ken.shiozaki/ahss/e2.html}{URL}.

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

1 major / 2 minor

Summary. The manuscript computes the E₂ pages of both the momentum-space and real-space Atiyah-Hirzebruch spectral sequences (AHSS) for topological crystalline insulators and superconductors up to three spatial dimensions. It employs a cell decomposition in which a group action that fixes a cell setwise also fixes it pointwise, provides a detailed implementation description, and, under one physically reasonable assumption, enumerates all K-groups compatible with the computed E₂ pages from both AHSS constructions. This yields unique K-group determinations for approximately 59% of the 1651 magnetic space groups in three dimensions, with all tabulated results posted online.

Significance. If the enumerated K-groups hold, the work supplies a large, directly computed reference database for topological classifications in crystalline systems, covering a majority fraction of 3D magnetic space groups without fitted parameters or circular reductions. The explicit online tabulation and implementation details constitute a reproducible resource that can accelerate material searches and further spectral-sequence studies in the field.

major comments (1)
  1. [Enumeration procedure (near abstract and § on K-group compatibility)] The 59% coverage claim (abstract and enumeration procedure) rests on a single 'physically reasonable assumption' whose precise statement, domain of applicability, and justification are not reproduced in the abstract and must be located and verified in the main text. Because this assumption directly gates the uniqueness of the K-group determinations, its explicit formulation and any counter-examples should be highlighted in a dedicated subsection.
minor comments (2)
  1. [Implementation description] The cell-decomposition convention (group action fixes cell pointwise when fixing setwise) is central to the E₂-page computation; a short dedicated paragraph or figure illustrating the distinction from other common decompositions would improve readability for readers outside the immediate subfield.
  2. [Abstract] The online URL in the abstract appears as a raw hyperlink; ensure the published version contains a stable, citable link or DOI to the tabulated E₂ pages and K-group lists.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation and recommendation of minor revision. We address the major comment below and will incorporate the suggested changes to improve clarity.

read point-by-point responses

  1. Referee: [Enumeration procedure (near abstract and § on K-group compatibility)] The 59% coverage claim (abstract and enumeration procedure) rests on a single 'physically reasonable assumption' whose precise statement, domain of applicability, and justification are not reproduced in the abstract and must be located and verified in the main text. Because this assumption directly gates the uniqueness of the K-group determinations, its explicit formulation and any counter-examples should be highlighted in a dedicated subsection.

    Authors: We agree that the assumption merits more prominent and self-contained presentation. The assumption is formulated in the main text (specifically in the section on K-group compatibility), but we acknowledge that its placement and the abstract's brevity make verification less immediate. In the revised manuscript we will insert a dedicated subsection immediately preceding the enumeration results. This subsection will: (i) state the assumption verbatim, (ii) delineate its domain of applicability (including the symmetry settings and dimensions to which it applies), (iii) supply the physical justification, and (iv) discuss any known or potential counter-examples together with their implications for the 59 % figure. We will also revise the abstract to include a concise parenthetical description of the assumption so that the coverage claim is transparent on first reading. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct computation of spectral-sequence pages

full rationale

The paper performs explicit computation of E2 pages for the Atiyah-Hirzebruch spectral sequence under a fixed cell decomposition (group action fixes cells pointwise when fixing them setwise). It then enumerates K-groups compatible with both momentum- and real-space E2 pages under one explicitly stated physical assumption. No equation reduces an output to a fitted input by construction, no parameter is renamed as a prediction, and no load-bearing step relies on a self-citation whose content is itself unverified or tautological. The derivation chain consists of group-cohomology calculations and compatibility checks that remain independent of the final enumerated K-groups.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to the single domain assumption stated in the abstract.

axioms (1)
  • domain assumption Physically reasonable assumption allowing enumeration of all K-groups compatible with the computed E2 pages
    Invoked to convert E2 pages into a list of possible K-groups; details not supplied in abstract.

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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. Towards complete characterization of topological insulators and superconductors: A systematic construction of topological invariants based on Atiyah-Hirzebruch spectral sequence

    cond-mat.mes-hall 2023-11 unverdicted novelty 8.0

    A systematic construction via Atiyah-Hirzebruch spectral sequence yields topological invariants that completely characterize K-groups for TRS spinful superconductors in 159 space groups.

Reference graph

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    ch shφh φhch sh ch shφh φhch sh TABLE I

    Z classification For each irrep of the little group for each orbit, we con- struct a vector ⃗ ap,−n∈ Ep 0 consisting of the matrix di- mensions of the generating Dirac Hamiltonians with the 11 Wα T Wα C Wα Γ n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 A 0 0 0 (Z, 1) 0 ( Z, 2) 0 ( Z, 4) 0 ( Z, 8) 0 AIII 0 0 1 0 ( Z, 2) 0 ( Z, 4) 0 ( Z, 8) 0 ( Z, 16) AI ...

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    Z2 classification No sign difference exists in the Z2 number. For each irrep of each orbit, we construct a vector ⃗bp,−n ∈ Ep 0 consisting of matrix sizes of generator Dirac Hamilto- nians as follows. Pick a representative p-cell Dp i of an orbit of equivalent p-cells inCp. For an irrep αr(kp i ), if the classification of degree n is Z2, we definebp,−n ir by...

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    Z classification For each irrep of each orbit, we construct a vector ⃗ ap,−n∈ E0 p. Pick a representative p-cell Dp i of an or- bit of equivalent p-cells inCp. For an irrep αr(xp i ), if the classification of degree −n is Z, we set ap,−n ir to be the matrix size of generator Dirac Hamiltonian listed in Table II. The components for other equivalent irreps h[...

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    Z2 classification For an irrep αr(xp i ), if the classification of degree −n is Z2, set bp,−n ir = dim (H), the matrix dimension listed in Table II. For other equivalent irreps h[α(xp i )] at h(x)∈ X, set bp,−n h(i)h(r) = bp,−n ir . Constructing the vectors⃗bp,−n 1 ,⃗bp,−n 2 ,..., for all inequivalent irreps and inequivalent orbits, we have E1Z2 p,−n :=⟨⃗bp...

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