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REVIEW 3 major objections 4 minor 72 references

Abundance of $\mathbb{Z}_2$ topological order in exfoliable two-dimensional insulators

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Thirteen exfoliable monolayers are quantum spin Hall insulator candidates, about 1% of the pool.

desk verdict Screening protocol is solid and Pd2HgSe3 is a real find, but the 13-candidate count and ~1% abundance claim are not supported as written. read the letter →

arxiv 1908.08334 v1 pith:D6HMHX5M submitted 2019-08-22 cond-mat.mtrl-sci cond-mat.mes-hall

Antimo Marrazzo , Marco Gibertini , Davide Campi , Nicolas Mounet , Nicola Marzari This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hall
keywords quantumspinHallinsulatorsZ2topologicalordertwo-dimensionalmaterialshigh-throughputscreeningdensityfunctionaltheoryG0W0approximationspin-orbitcouplingexfoliablemonolayers
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

Using first-principles simulations, the paper screens 1,306 exfoliable monolayers drawn from a database of experimentally known layered compounds and asks which of them are quantum spin Hall insulators (QSHIs): two-dimensional insulators whose time-reversal-protected edge states can carry spin-momentum-locked current without dissipation. It identifies 13 candidate monolayers, with the newly synthesized Pd2HgSe3 presented separately as a second Kane-Mele-type example, that are non-magnetic, dynamically stable, and carry a non-trivial $\mathbb{Z}_2$ invariant at the PBE-with-spin-orbit level, sometimes only after a few percent of strain. Standout new materials are AsCuLi2, a previously unknown structural prototype, and Pd2HgSe3, whose G0W0-calculated band inversion is about 41 meV. The screen concludes that the relative abundance of $\mathbb{Z}_2$ topological order among two-dimensional insulators in this exfoliable set is around 1%, a benchmark useful because such materials are rare and promising for low-dissipation nanoelectronics and spintronics.

What carries the argument

The machine that carries the argument is the screening funnel itself. Topological classification uses the $\mathbb{Z}_2$ invariant obtained from hybrid Wannier charge centers, the evolution of Wannier centers across half the Brillouin zone, which the paper computes for every band insulator and, where needed, for direct-gap metals under strain. The robustness measure is the inversion strength, defined as the direct gap at the high-symmetry point where the band inversion occurs: at $K$ for Kane-Mele systems and at $\Gamma$ for Bernevig-Hughes-Zhang systems. The final arbiter for the five most promising materials is the G0W0 approximation with spin-orbit coupling, extrapolated to an infinitely dense $\mathbf{k}$-grid, because two-dimensional dielectric screening makes quasiparticle corrections slow to converge.

What would settle it

Recompute the G0W0-with-spin-orbit $\mathbb{Z}_2$ invariant and band inversion for the PBE-only candidates (TaIrTe4, TaRhTe4, NbIrTe4, ZrBr, ZrCl, Cu2Te3Ti, In2ZnS4); if any of them reverts to a trivial insulator as TiNI does, the candidate list and the 1% abundance claim need downward revision.

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Extended reading notes

Core claim

The paper's central claim is that $\mathbb{Z}_2$ topological order is not exotic within the pool of monolayer crystals that can be peeled from known layered compounds: of the 1,306 monolayers that survive the non-magnetic, structurally relaxed screening, a shortlist of 13 carries a non-trivial $\mathbb{Z}_2$ invariant, and Pd2HgSe3 is added as a second Kane-Mele QSHI, for a relative abundance of roughly 1%. The result is obtained through a funnel: structural relaxation starting from experimental parent crystals, a band-insulator or direct-gap-metal selection (with isotropic strain of 1–3% allowed for the metals), the $\mathbb{Z}_2$ invariant computed from hybrid Wannier charge centers, a magnetic-ground-state filter, and a phonon stability check. For five of the most interesting candidates the band inversion is recomputed at the G0W0 level with spin-orbit coupling; this confirms AsCuLi2 (inversion strength 169 meV) and Pd2HgSe3 (41 meV) as new QSHIs, while exposing TiNI as a PBE false positive that is trivial at G0W0 with a 0.7 eV direct gap. The authors read the 1% figure as the abundance of true two-dimensional bulk insulators whose entire occupied manifold is topologically nontrivial, as opposed to metals with well-defined interband gaps where non-trivial invariants coexist with gapless bulk.

Load-bearing premise

The load-bearing premise is that the PBE-with-spin-orbit $\mathbb{Z}_2$ assignments, which were checked at the G0W0 level for only five of the shortlisted materials, stay correct for the candidates that were not re-examined; the paper itself shows that this premise failed for TiNI.

Editorial extensions

If this is right

  • Monolayer Pd2HgSe3 becomes a concrete experimental target: exfoliable, dynamically stable, and a Kane-Mele QSHI with a G0W0 inversion strength around 41 meV.
  • AsCuLi2 offers a new crystal prototype for spin-Hall physics, with a clean band inversion at $\Gamma$ that strengthens at the G0W0 level (169 meV) compared with PBE (80 meV).
  • Small isotropic strain of 1–3% turns several screened structures into true QSHI insulators, so substrate-induced strain can be used to realize the topological phase in otherwise gapless or weakly gapped monolayers.
  • The 1% abundance estimate quantifies how likely a random exfoliable monolayer is to be a $\mathbb{Z}_2$ topological insulator, giving future computational searches a statistical benchmark.
  • The TiNI case shows that PBE-level $\mathbb{Z}_2$ assignments are provisional: candidates not yet re-examined at the G0W0 level should be treated as predictions pending quasiparticle checks.

Reading between the lines

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

  • The paper leaves the true abundance open: because only five of the shortlisted materials received G0W0 checks and one (TiNI) flipped to trivial, the 1% figure could creep below 1% if the unverified PBE-only candidates behave like TiNI.
  • The finding that every candidate has fewer than 12 atoms per unit cell and a third share the same space group suggests that future screens could pre-filter on structural motifs and cell size, trading a small risk of missed candidates for a large gain in cost.
  • For the strain-driven candidates (ZrBr, ZrCl, WTe2, MoTe2), a concrete testable route is to grow them on lattice-matched substrates that impose 1–3% isotropic strain and to measure the helical edge conductance predicted for the QSHI phase.
  • The 1% abundance refers to exfoliable monolayers from known layered crystals; a screen over broader classes of two-dimensional materials might give a different fraction, so the number is a property of this materials pool, not a universal constant.
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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

3 major / 4 minor

Summary. The paper reports a high-throughput first-principles screening of 1306 exfoliable monolayer materials for quantum spin Hall insulators (QSHIs). Using PBE-SOC band structures, Z2 invariants from Wannier charge centers, phonon stability checks via DFPT, strain engineering for direct-gap metals, and G0W0-with-SOC checks for five leading candidates, the authors identify a set of candidate QSHIs. The most notable new predictions are monolayer Pd2HgSe3, presented as a second Kane-Mele QSHI with a G0W0 inversion strength near 41 meV, and AsCuLi2, a new Bernevig-Hughes-Zhang-type prototype. The manuscript also reports that TiNI, previously proposed as a QSHI, is trivial at the G0W0 level, and it concludes that Z2 topological order has a relative abundance of around 1% among the screened exfoliable two-dimensional insulators.

Significance. If substantiated, this work would provide a valuable, systematically generated set of QSHI candidates and a useful methodological template for database-scale topological screening. Its strengths are the explicit first-principles computation of Z2 invariants (rather than descriptor-based proxies), the inclusion of phonon stability, the use of independent G0W0 quasiparticle corrections that are not fitted to the screening outcome, the honest reporting of a G0W0 false positive (TiNI), and the use of reproducible open-source codes and pseudopotential libraries. The two new material predictions, Pd2HgSe3 and AsCuLi2, are specific and falsifiable and are likely to be of interest to the experimental 2D-materials community. However, the headline candidate count and the associated abundance estimate are not currently supported in their stated form because the candidate table and text are internally inconsistent and because a large fraction of the candidates rest on PBE-level assignments that the paper itself shows can be reversed at G0W0.

major comments (3)
  1. [Results, Table I, and Abstract] The headline count of 13 QSHI candidates is internally inconsistent. The text states 'we find 13 QSHIs candidates... listed in Tab. I', but Table I lists 14 entries (Bi, Pt2HgSe3, Pd2HgSe3, TiNI, AsCuLi2, WTe2, MoTe2, TaIrTe4, TaRhTe4, NbIrTe4, Cu2Te3Ti, In2ZnS4, ZrBr, ZrCl). In addition, the Results paragraph on TiNI reports a G0W0 inversion strength of -705 meV and states explicitly that TiNI is a trivial insulator at G0W0, yet TiNI remains in the candidate table and in the count. This inconsistency directly affects the central claim and the 'around 1%' abundance estimate, and it must be corrected in a revised version.
  2. [Methods (G0W0 subsection) and Results (TiNI paragraph)] Only five candidates receive G0W0 validation, while seven entries in Table I (TaIrTe4, TaRhTe4, NbIrTe4, Cu2Te3Ti, In2ZnS4, ZrBr, ZrCl) have no G0W0 inversion strength. The TiNI case shows that a PBE-SOC inversion strength of 141 meV, well above the 20 meV screening cutoff, can nevertheless be reversed to a trivial assignment at G0W0. The PBE-only Z2 assignments should therefore be presented explicitly as tentative candidates, and the count of '13 QSHI candidates' and the deduced 1% abundance should be restricted to G0W0-validated cases or qualified with the number of PBE-only assignments.
  3. [Conclusions and Results (screening protocol)] The abundance statement 'relative abundance of Z2 topological order in two-dimensional insulators of around 1%' is not directly supported by the reported ratio. The denominator is the 1306 relaxed non-magnetic monolayers, which includes metals and direct-gap metals, not only band insulators. The numerator counts candidates that include strain-induced insulators and at least one G0W0-trivial material. To support the stated claim, the authors should either compute the fraction with respect to the number of band insulators (or an otherwise clearly defined denominator) or rephrase the claim to describe abundance among the full set of non-magnetic exfoliable monolayers.
minor comments (4)
  1. [Throughout] The manuscript contains several typographical errors, including 'unpertubed', 'abudance', 'miminum', 'perfom', and 'hermaphrodite Wannier charge centers' (presumably 'hybrid Wannier charge centers'). These should be corrected.
  2. [Results, TiNI paragraph and Table I] The TiNI paragraph gives a PBE inversion strength of 0.17 eV, while Table I lists 141 meV (0.141 eV) for the same quantity. Please reconcile this discrepancy.
  3. [Methods, Eq. (2) and footnote 20] The 20 meV inversion-strength cutoff and the 0.01 eV direct-gap-metal threshold are introduced without sensitivity analysis or justification. Since the abundance estimate depends on the candidate count, a brief statement on how the results change with these thresholds would strengthen the screening protocol.
  4. [Methods, Eq. (1)] The G0W0 inversion strengths obtained from the extrapolation in Eq. (1) are reported without uncertainty estimates; providing the fitted parameters or an error estimate would make the quoted values, especially the 41 meV result for Pd2HgSe3, more informative.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the screening's Z2 labels and G0W0 checks are independent first-principles computations, not fits or renamings of the input database.

full rationale

The paper's central claims — 13 candidate quantum spin Hall insulators and a ~1% abundance among exfoliable monolayers — are derived by an explicit high-throughput funnel: PBE-SOC band structures, hybrid Wannier charge-center Z2 invariants computed with Z2Pack, phonon stability via DFPT, and G0W0-with-SOC checks for the five most interesting candidates. None of these steps fits a parameter to the target result and then re-predicts it; the Z2 invariant is computed independently from the Wannier evolution, and the G0W0 inversion strengths are extrapolated from dense k-point grids using Eq. (1), which is a convergence extrapolation, not a fit to the PBE answer. The paper does rely on its own prior work: the exfoliable-monolayer database of Ref. [16] and the jacutingaite prediction of Ref. [33] are authored by overlapping groups, and the SSSP pseudopotential library (Ref. [57]) is also self-cited. However, these are input infrastructure and prior external predictions, not assumptions that already contain the 13-candidate list or the 1% abundance; the screening recomputes all properties from first principles rather than reading off the cited results. The TiNI case (PBE topological, G0W0 trivial) is presented as a cautionary result from the paper's own independent G0W0 calculation, and retaining TiNI in the candidate count is a correctness/consistency issue, not a circular-derivation issue. No equation in the paper reduces to its own input, and no fitted parameter is renamed as a prediction. The overall circularity score is therefore very low, 1 out of 10, reflecting only the presence of self-cited infrastructure that is not load-bearing for the central derivation.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, dimensions, or conserved quantities are introduced. The free parameters are screening thresholds and a numerical extrapolation fit; the axioms are standard DFT/GW domain assumptions plus the inherited-structure assumption that underlies the exfoliable database.

free parameters (4)
  • Inversion-strength cutoff = 20 meV
    Compounds with DFT-PBE band inversions below 20 meV are excluded from QSHI screening even if insulating (Footnote 20), so this threshold sets the candidate list and the 1% abundance.
  • Direct-gap-metal threshold = 0.01 eV
    A metallic system is treated as a direct-gap metal if the minimum direct gap exceeds 0.01 eV (Eq. 2); this determines which metals enter the strain branch of the funnel.
  • Strain steps = +/-1, 2, 3%
    Hydrostatic strains applied to direct-gap metals to look for strain-induced insulators; the minimum strain values in Table I depend on this discrete grid.
  • G0W0 inversion-strength extrapolation parameters = a, b, c in IS(Nk) = a/Nk + b/sqrt(Nk) + c
    Eq. (1) fits three parameters to finite-k-grid G0W0 calculations to extrapolate direct gaps; the Pd2HgSe3 inversion of 41 meV inherits the uncertainty of this fit.
assumptions (5)
  • domain assumption PBE+SOC band structures are accurate enough for Z2 classification of the candidates not rechecked with G0W0.
    Only five of thirteen candidates receive G0W0 treatment; TiNI demonstrates that PBE can assign nontrivial Z2 to a material that is trivial at G0W0.
  • domain assumption Monolayers inherit parent bulk crystal symmetries and do not undergo structural transitions on exfoliation.
    Stated in the screening protocol paragraph; if false, the relaxed monolayer structures and Z2 results could be wrong.
  • domain assumption The non-magnetic ground-state assumption is valid for the screened candidates.
    Magnetic screening uses collinear DFT configurations only; non-collinear magnetism or missing magnetic order would break time-reversal symmetry and the Z2 classification.
  • domain assumption Exfoliability is captured by the vdW-DFT database of Ref. 16.
    The starting set of 1825 monolayers defines the search space and the abundance denominator; the database was built by the same group with van der Waals DFT, not by experiment.
  • standard math Z2 invariant from hybrid Wannier charge centers is correct for gapped time-reversal invariant systems.
    Uses the published Z2Pack implementation; this is a standard mathematical result for time-reversal invariant insulators.

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Cite this review

Pith. "Pith review of Abundance of $\mathbb{Z}_2$ topological order in exfoliable two-dimensional insulators." pith.science (2026). https://pith.science/paper/D6HMHX5M

@misc{pith2026190808334,
  author       = {Pith},
  title        = {Pith review of: Abundance of $\mathbbZ_2$ topological order in exfoliable two-dimensional insulators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D6HMHX5M}},
  note         = {Machine review of arXiv:1908.08334}
}
abstract

Quantum spin Hall insulators are a class of two-dimensional materials with a finite electronic band gap in the bulk and gapless helical edge states. In the presence of time-reversal symmetry, $\mathbb{Z}_2$ topological order distinguishes the topological phase from the ordinary insulating one. Some of the phenomena that can be hosted in these materials, from one-dimensional low-dissipation electronic transport to spin filtering, could be very promising for many technological applications in the fields of electronics, spintronics and topological quantum computing. Nevertheless, the rarity of two-dimensional materials that can exhibit non-trivial $\mathbb{Z}_2$ topological order at room temperature hinders development. Here, we screen a comprehensive database we recently created of 1825 monolayers that can be exfoliated from experimentally known compounds, to search for novel quantum spin Hall insulators. Using density-functional and many-body perturbation theory simulations, we identify 13 monolayers that are candidates for quantum spin Hall insulators, including high-performing materials such as AsCuLi$_2$ and jacutingaite (Pt$_2$HgSe$_3$). We also identify monolayer Pd$_2$HgSe$_3$ as a novel Kane-Mele quantum spin Hall insulator, and compare it with jacutingaite. Such a handful of promising materials are mechanically stable and exhibit $\mathbb{Z}_2$ topological order, either unpertubed or driven by a small amount of strain. Such screening highlights a relative abundance of $\mathbb{Z}_2$ topological order of around 1%, and provides an optimal set of candidates for experimental efforts.

Figures

Figures reproduced from arXiv: 1908.08334 by the authors.

Figure 1
Figure 1. Computational protocol for the screening [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Band gap at the DFT-PBE level with SOC versus average atomic number for exfoliable 2D materials identified in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Left panel: DFT-PBE band structure with (green) and without (orange) SOC for monolayer Pd [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Left panel: Crystal structure of AsCuLi2 (top and lateral views), a potentially exfoliable material identified in Ref. [16]. Right panel: DFT-PBE band structure with (green) and without (orange) SOC for monolayer AsCuLi2, where SOC drives a band inversion at Γ. small b…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.