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arxiv: 2311.15814 · v3 · pith:YJYBTKKSnew · submitted 2023-11-27 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci· cond-mat.supr-con

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

Seishiro Ono , Ken Shiozaki This is my paper

Pith reviewed 2026-05-24 06:21 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-scicond-mat.supr-con
keywords topological invariantsAtiyah-Hirzebruch spectral sequencetopological superconductorsspace groupsK-theorysymmetry indicatorstime-reversal symmetry
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The pith

Atiyah-Hirzebruch spectral sequence in momentum space yields topological invariants that fully characterize K-groups for time-reversal symmetric spinful superconductors in 159 space groups.

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

The paper develops a systematic construction of topological invariants for symmetry classes where existing symmetry-indicator methods are incomplete. It applies the Atiyah-Hirzebruch spectral sequence directly in momentum space to time-reversal symmetric spinful superconductors with conventional pairing. The resulting invariants are shown to completely classify the relevant K-groups across 159 space groups after validation on random symmetric Hamiltonians. For normal conducting phases the invariants are initially defined under gauge conditions, with procedures given to extract gauge-independent forms. This framework supplies a general route to invariants rather than case-by-case discovery.

Core claim

Using the Atiyah-Hirzebruch spectral sequence in momentum space, the authors construct topological invariants for all space groups in the class of time-reversal symmetric spinful superconductors with conventional pairing. These invariants completely characterize the K-groups in 159 space groups. The same method applies to a large part of other symmetry classes, and gauge-fixed invariants for normal conducting phases can be converted to gauge-independent versions.

What carries the argument

Atiyah-Hirzebruch spectral sequence in momentum space, which filters successive approximations to produce explicit topological invariants for given symmetry classes.

If this is right

  • The invariants diagnose topological phases in classes where symmetry indicators return trivial results.
  • Validation across random Hamiltonians confirms the quantities function as topological invariants.
  • The construction supplies a uniform method for a large fraction of symmetry classes instead of isolated examples.
  • Gauge-independent forms derived from the gauge-fixed invariants enable direct numerical evaluation for normal conducting phases.

Where Pith is reading between the lines

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

  • Pairing the invariants with first-principles band calculations could systematically enlarge the catalog of candidate topological materials.
  • The spectral-sequence procedure may extend without major change to other pairing symmetries or to insulators in the same symmetry classes.
  • Direct comparison of the new invariants against known classifications in low-dimensional or high-symmetry space groups offers an immediate consistency check.

Load-bearing premise

The spectral sequence applied in momentum space together with the chosen gauge conditions produces quantities unchanged under symmetry-preserving continuous deformations of the Hamiltonian.

What would settle it

Compute the constructed invariants on a family of random symmetric Hamiltonians known to realize a nontrivial K-group element in one of the 159 space groups and verify that the invariant takes the expected nonzero value while remaining zero for topologically trivial cases.

Figures

Figures reproduced from arXiv: 2311.15814 by Ken Shiozaki, Seishiro Ono.

Figure 1
Figure 1. Figure 1: FIG. 1. Characterization of stable topological phases. All stable [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Cell decomposition for layer groups [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of the cell decomposition of the fundamental [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Real-space pictures of topological phases in layer groups [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of 1-cell [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the cell decomposition of the fundamental do [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a, b, c, d) Illustrations of topological crystalline insulators [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a, b) Illustrations of topological crystalline insulators intro [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Fundamental domain in the momentum space of mag [PITH_FULL_IMAGE:figures/full_fig_p037_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a, b, c) Real-space pictures of topological crystalline su [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Cell decomposition of magnetic space group [PITH_FULL_IMAGE:figures/full_fig_p044_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The representation matrix [PITH_FULL_IMAGE:figures/full_fig_p049_12.png] view at source ↗
read the original abstract

The past decade has witnessed significant progress in topological materials investigation. Symmetry-indicator theory and topological quantum chemistry provide an efficient scheme to diagnose topological phases from only partial information of wave functions without full knowledge of topological invariants, which has resulted in a recent comprehensive materials search. However, not all topological phases can be captured by this framework, and topological invariants are needed for a more refined diagnosis of topological phases. In this study, we present a systematic framework to construct topological invariants for a large part of symmetry classes, which should be contrasted with the existing invariants discovered through one-by-one approaches. Our method is based on the recently developed Atiyah-Hirzebruch spectral sequence in momentum space. As a demonstration, we construct topological invariants for time-reversal symmetric spinful superconductors with conventional pairing symmetries of all space groups, for which symmetry indicators are silent. We also validate that the obtained quantities work as topological invariants by computing them for randomly generated symmetric Hamiltonians. Remarkably, the constructed topological invariants completely characterize $K$-groups in 159 space groups. Our topological invariants for normal conducting phases are defined under some gauge conditions. To facilitate efficient numerical simulations, we discuss how to derive gauge-independent topological invariants from the gauge-fixed topological invariants through some examples. Combined with first-principles calculations, our results will help us discover topological materials that could be used in next-generation devices and pave the way for a more comprehensive topological materials database.

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 / 3 minor

Summary. The manuscript develops a systematic method to construct topological invariants for a large class of symmetry-protected topological phases using the Atiyah-Hirzebruch spectral sequence applied in momentum space. As a key demonstration, it constructs such invariants for time-reversal symmetric spinful superconductors with conventional pairing symmetries across all space groups, where symmetry indicators do not apply. The invariants are validated through numerical computations on randomly generated symmetric Hamiltonians, and the authors claim that these invariants fully characterize the K-groups for 159 space groups. For normal conducting phases, gauge-fixed definitions are provided along with a method to obtain gauge-independent versions.

Significance. If the constructions hold, this provides a powerful general tool for diagnosing topological phases beyond current symmetry-indicator approaches, potentially enabling more complete searches for topological materials in databases and first-principles calculations. The explicit construction for superconductors in all space groups is a notable strength, as is the numerical validation approach.

major comments (2)
  1. [Abstract] The central claim that the constructed topological invariants completely characterize the K-groups in 159 space groups (abstract) is load-bearing; the manuscript should include an explicit comparison, such as a table matching the number and type of constructed invariants to the known rank and structure of each K-group, to substantiate completeness rather than asserting it from the spectral sequence application alone.
  2. [Demonstration and validation] The validation section states that the quantities work as invariants on random Hamiltonians, but lacks details on the ensemble (number of samples, parameter ranges, and exact procedure for enforcing symmetries in Hamiltonian generation), which is necessary to evaluate whether the numerical evidence robustly supports the claim of invariance and quantization for the full set of constructed invariants.
minor comments (3)
  1. The abstract refers to 'a large part of symmetry classes' without enumerating them beyond the TR-symmetric spinful superconductor case; a brief list or reference to the relevant Altland-Zirnbauer classes would improve clarity.
  2. The discussion of deriving gauge-independent invariants from gauge-fixed ones is illustrated through examples; a general algorithmic outline or pseudocode would aid readers in applying the method to other cases.
  3. Notation for the spectral sequence pages and the resulting invariants should be made consistent throughout to avoid ambiguity in the construction steps.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity and substantiation of our claims.

read point-by-point responses

  1. Referee: [Abstract] The central claim that the constructed topological invariants completely characterize the K-groups in 159 space groups (abstract) is load-bearing; the manuscript should include an explicit comparison, such as a table matching the number and type of constructed invariants to the known rank and structure of each K-group, to substantiate completeness rather than asserting it from the spectral sequence application alone.

    Authors: We agree that an explicit comparison would strengthen the presentation and allow readers to directly verify the completeness. While the Atiyah-Hirzebruch spectral sequence guarantees that our construction captures the full structure of the K-groups (including all generators and relations), we will add a supplementary table in the revised manuscript that lists, for each of the 159 space groups, the number and type of invariants we construct alongside the known rank and torsion structure of the corresponding K-group. This will make the completeness explicit rather than relying solely on the spectral sequence argument. revision: yes

  2. Referee: [Demonstration and validation] The validation section states that the quantities work as invariants on random Hamiltonians, but lacks details on the ensemble (number of samples, parameter ranges, and exact procedure for enforcing symmetries in Hamiltonian generation), which is necessary to evaluate whether the numerical evidence robustly supports the claim of invariance and quantization for the full set of constructed invariants.

    Authors: We acknowledge that the current description of the numerical ensemble is insufficient for full reproducibility and evaluation. In the revised manuscript we will expand the validation section to specify the total number of random Hamiltonians generated, the ranges and distributions of the random parameters, and the exact algorithmic procedure used to enforce the space-group symmetries (including how the pairing terms and time-reversal symmetry are imposed). These additions will allow readers to assess the robustness of the numerical evidence for invariance and quantization across all constructed invariants. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external spectral sequence

full rationale

The manuscript constructs topological invariants by direct application of the Atiyah-Hirzebruch spectral sequence in momentum space to the relevant symmetry classes and space groups. The central claim of completeness for K-groups in 159 space groups follows from the filtration and convergence properties of that spectral sequence, which is an established mathematical tool independent of the present paper. Validation consists of explicit construction plus numerical checks on randomly generated symmetric Hamiltonians; these checks corroborate invariance and quantization but are not used to define or fit the invariants themselves. No self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation chain appears in the derivation. The gauge-fixed definitions for normal phases are converted to gauge-independent forms by explicit procedures, again without circularity. The result is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Atiyah-Hirzebruch spectral sequence to momentum-space Hamiltonians of the stated symmetry classes and on the validity of the chosen gauge fixing for normal phases; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Atiyah-Hirzebruch spectral sequence applies in momentum space to classify topological phases for the symmetry classes considered
    The method is explicitly based on this spectral sequence (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. Crystalline topological invariants in quantum many-body systems

    cond-mat.str-el 2026-04 unverdicted novelty 1.0

    Reviews characterization, classification, and detection of crystalline symmetry-protected topological invariants in 2D integer and fractional Chern insulators, focusing on translation, rotation, and charge conservation.

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