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arxiv: 1906.11847 · v1 · pith:X6SNLSREnew · submitted 2019-06-27 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall· cond-mat.str-el

Robust topological invariants of topological crystalline phases in the presence of impurities

Ian Mondragon-Shem , Taylor L. Hughes This is my paper

Pith reviewed 2026-05-25 13:35 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hallcond-mat.str-el
keywords topological crystalline phasesprojected symmetry operatorsreal-space topological markersimpurity robustnessspatial symmetriesinversion symmetrymirror symmetryboundary invariants
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The pith

Projected symmetry operators yield basis-independent invariants for topological crystalline phases that stay stable under impurities away from symmetry centers.

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

Topological crystalline phases rely on spatial symmetries such as inversion or mirror symmetry for their protection. Standard invariants are formulated at high-symmetry momenta in the Brillouin zone and therefore assume perfect translational order. The paper constructs invariants directly from symmetry operators projected onto the occupied subspace. These invariants produce real-space markers that decay exponentially away from the symmetry fixed points. Because the markers sit only at those fixed points, impurities placed elsewhere leave the invariants intact and allow a mesh of local markers to distinguish the phase throughout the crystal.

Core claim

Robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators. The real-space topological markers of these invariants are exponentially localized to the fixed points of the spatial symmetry. As a result, this real-space structure protects them against the presence of impurities that are located away from the fixed points. By considering all possible symmetry centers a mesh of markers is obtained that distinguishes the phase locally, and the same construction applies to boundary modes such as the gapless Majorana edges of mirror-symmetric topological superconductors.

What carries the argument

Projected symmetry operators onto the occupied subspace, whose eigenvalues define the invariants and whose real-space matrix elements furnish exponentially localized markers at symmetry fixed points.

If this is right

  • A complete mesh of markers, one per possible symmetry center, supplies a local topological diagnostic throughout the sample.
  • The same projected-operator construction produces quantized invariants on the boundaries of these phases, including an integer edge invariant for mirror-symmetric topological superconductors that equals a quantized effective edge polarization.
  • The approach covers one- and two-dimensional examples protected by inversion, rotation, and mirror symmetries.
  • Because the markers are exponentially localized, the invariants remain well-defined for any impurity configuration that spares the immediate vicinity of the fixed points.

Where Pith is reading between the lines

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

  • Local spectroscopic probes placed near symmetry centers could read out the topology without requiring knowledge of the entire sample or momentum-space periodicity.
  • The localization property suggests that controlled disorder engineered far from fixed points might be used to test the separation between bulk and boundary invariants in experiment.
  • Extending the projection technique to interacting or Floquet systems could produce analogous real-space markers for those settings.

Load-bearing premise

Spatial symmetries remain well-defined and their operators can still be projected onto the occupied subspace even when impurities sit away from the symmetry fixed points.

What would settle it

Observation that the topological markers lose their quantization or spread significantly when impurities are introduced near the symmetry fixed points would falsify the claimed robustness.

Figures

Figures reproduced from arXiv: 1906.11847 by Ian Mondragon-Shem, Taylor L. Hughes.

Figure 1
Figure 1. Figure 1: FIG. 1. Inversion-symmetric 1D insulator. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Rotationally-symmetric 2D superconductors. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Mirror-symmetric DIII 2D superconductor. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

Topological crystalline phases (TCPs) are topological states protected by spatial symmetries. A broad range of TCPs have been conventionally studied by formulating topological invariants (symmetry indicators) at invariant momenta in the Brillouin zone, which leaves open the question of their stability in the absence of translational invariance. In this work, we show that robust basis-independent topological invariants can be generically constructed for TCPs using projected symmetry operators. Remarkably we show that the real-space topological markers of these invariants are exponentially localized to the fixed points of the spatial symmetry. As a result, this real-space structure protects them against the presence of impurities that are located away from the fixed points. By considering all possible symmetry centers in a crystalline system we can generate a mesh of real-space topological markers that can provide a local topological distinction for TCPs. We illustrate the robustness of this mesh of invariants with 1D and 2D TCPs protected by inversion, rotational and mirror symmetries. Finally, we find that the boundary modes of these TCPs can also exhibit robust topological invariants with localized markers on the edges. We illustrate this with the gapless Majorana boundary modes of mirror-symmetric topological superconductors, and relate their integer topological edge invariant with a quantized effective edge polarization.

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 proposes constructing robust, basis-independent topological invariants for topological crystalline phases (TCPs) via projected symmetry operators. It claims that the associated real-space topological markers localize exponentially at spatial-symmetry fixed points, thereby remaining insensitive to impurities placed elsewhere (provided the symmetry is preserved). The construction is illustrated for 1D and 2D TCPs protected by inversion, rotational, and mirror symmetries; the approach is further applied to gapless Majorana boundary modes of mirror-symmetric topological superconductors, where an integer edge invariant is related to quantized effective edge polarization. A mesh of such markers is proposed to furnish local topological distinction.

Significance. If the localization property and the associated robustness are rigorously established, the work supplies a real-space route to topological classification of TCPs that does not rely on translational invariance. This could be useful for systems with impurities or weak disorder that preserve the relevant spatial symmetries. The explicit connection between the projected-operator invariants and boundary-mode polarization is a concrete strength.

major comments (2)
  1. [Abstract / projected symmetry operators section] Abstract and § on projected symmetry operators: the claim that the projection onto the occupied subspace remains well-defined when impurities are present away from fixed points is stated but not accompanied by an explicit construction or error bound; the manuscript should demonstrate that the occupied projector commutes with the symmetry operator to the required accuracy even in the presence of such impurities.
  2. [real-space markers section] Section on real-space markers: the exponential localization is asserted as the key protection mechanism, yet no decay length, analytic bound, or systematic numerical check (e.g., versus impurity strength or distance) is referenced; without such evidence the robustness statement remains qualitative.
minor comments (2)
  1. Figure captions should explicitly label the symmetry fixed points and the spatial decay of the markers so that the localization claim can be read off visually.
  2. Notation for the projected operators (e.g., P S P) should be introduced once and used consistently; a short appendix collecting all definitions would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / projected symmetry operators section] Abstract and § on projected symmetry operators: the claim that the projection onto the occupied subspace remains well-defined when impurities are present away from fixed points is stated but not accompanied by an explicit construction or error bound; the manuscript should demonstrate that the occupied projector commutes with the symmetry operator to the required accuracy even in the presence of such impurities.

    Authors: When impurities preserve the spatial symmetry, the full Hamiltonian (including the impurity potential) commutes exactly with the symmetry operator S. The occupied projector P is the spectral projector onto states below the gap and is uniquely determined by the Hamiltonian; it therefore inherits the exact commutation relation [P, S] = 0 with no approximation or error bound required. This holds regardless of the location of the impurities provided the symmetry is unbroken. We will insert a short clarifying paragraph in the projected-symmetry-operator section making this commutation explicit. revision: yes

  2. Referee: [real-space markers section] Section on real-space markers: the exponential localization is asserted as the key protection mechanism, yet no decay length, analytic bound, or systematic numerical check (e.g., versus impurity strength or distance) is referenced; without such evidence the robustness statement remains qualitative.

    Authors: The exponential decay of the real-space markers follows directly from the exponential decay of the single-particle density matrix in a gapped system (a standard result for insulators with a finite gap). Because the marker is built from the projected symmetry operator, its support is confined to the fixed-point region on the same length scale. While we did not include explicit numerical scans versus impurity distance or strength, the analytic argument based on the gapped density-matrix decay already supplies the protection mechanism. We will add a brief reference to this standard decay property and one illustrative numerical panel in the revised manuscript. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained construction

full rationale

The paper constructs topological invariants directly from projected symmetry operators applied to the occupied subspace. The claimed exponential localization of real-space markers at symmetry fixed points follows from the algebraic properties of these operators and the definition of the fixed points, without any parameter fitting, renaming of known results, or load-bearing self-citation chains. The robustness to distant impurities is a direct geometric consequence of that localization rather than an input redefined as output. No step reduces the target invariants to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no new free parameters or invented entities. It relies on the standard assumption that spatial symmetries are present.

axioms (1)
  • domain assumption The crystalline system possesses well-defined spatial symmetries (inversion, rotation, mirror) whose fixed points can be identified.
    The entire construction of projected operators and localized markers presupposes the existence of these symmetries.

pith-pipeline@v0.9.0 · 5759 in / 1191 out tokens · 27164 ms · 2026-05-25T13:35:14.031530+00:00 · methodology

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Reference graph

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