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arxiv: 2510.11342 · v2 · submitted 2025-10-13 · ❄️ cond-mat.mes-hall

One-dimensional topology and topolectrics of nonsymmorphic Kramers degenerate systems

Max Tymczyszyn , Edward McCann This is my paper

Pith reviewed 2026-05-18 07:36 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords nonsymmorphic topologyone-dimensional modelsKramers degeneracytopolectric circuitsZ2 invariantZ4 classificationzero-energy modesdisorder effects
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0 comments X

The pith

Nonsymmorphic four-band models in one dimension support Z2 and Z4 topological phases realized and detected in topolectric circuits.

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

The paper develops nonsymmorphic four-band tight-binding models in one dimension that incorporate Kramers degeneracy and support topological phases with Z2 and Z4 invariants. It extends an open-path winding-number formulation to compute these invariants for the four-band Kramers-degenerate cases. Topolectric circuit implementations are constructed for both the AII Z2 model and the D Z4 model. Impedance measurements in these circuits reproduce the predicted phase boundaries along with the associated zero-energy modes. Disorder analysis for the minimal Z4 model shows that certain domain-wall zero modes remain pinned at zero energy because disorder lacks first-order coupling to the soliton subspace.

Core claim

Nonsymmorphic four-band tight-binding models in one dimension with Kramers degeneracy fall into the AII class with a Z2 invariant and the D class with a Z4 classification. Extending the open-path winding-number formulation allows computation of these invariants. Topolectric circuit realizations reproduce the phase boundaries and zero-energy modes through impedance response. In the minimal nearest-neighbor Z4 model, domain-wall zero modes stay pinned at zero energy under disorder since it has no first-order coupling to the soliton subspace.

What carries the argument

Open-path winding-number formulation extended to Kramers-degenerate four-band systems

If this is right

  • Impedance response in the topolectric circuits matches the predicted topological phase boundaries for the AII and Z4 models.
  • Zero-energy modes appear at the locations expected from the invariants in both circuit realizations.
  • Certain domain-wall zero modes in the minimal nearest-neighbor Z4 model remain pinned at zero energy under disorder.

Where Pith is reading between the lines

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

  • The disorder-pinning mechanism could be tested in fabricated topolectric networks to check robustness beyond the minimal model.
  • The extended winding-number method may apply to related symmetry classes in circuit or lattice systems.
  • Longer-range terms that restore symmetries could be used to tune the protection of the pinned modes.

Load-bearing premise

The open-path winding-number formulation previously used for non-Kramers-degenerate nonsymmorphic Z2 phases extends without modification to compute the invariants of Kramers-degenerate four-band systems.

What would settle it

An impedance measurement in the proposed topolectric circuit that fails to reproduce the predicted phase boundaries and zero-energy modes, or a shift of the domain-wall modes away from zero energy when disorder is added to the minimal Z4 model.

Figures

Figures reproduced from arXiv: 2510.11342 by Edward McCann, Max Tymczyszyn.

Figure 1
Figure 1. Figure 1: FIG. 1. Four distinct topological phases of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The superconducting pairing parameters between [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Distributions of the ratio of consecutive level spac [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Majorana zero modes and winding number for the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Nonsymmorphic AII class with symmorphic time [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) The [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Energy levels and winding number equivalent for the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Probability densities of near zero-energy soliton [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Energy levels in the [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Density of states (DOS) for various systems of the [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Disorder averaged soliton energy for a soliton in the [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Density of states (DOS) for small energy of order [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Schematic of a topolectric circuit realization of the [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Two-point impedance magnitudes across topolectric [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Two-point impedance magnitudes across topolectric [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Topolectric circuit realization of the Kitaev model [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Impedances across the topolectric realization of the [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Examples of topolectric circuit realizations of the [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Circuit diagrams of the phase-control units (PCU) [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Impedances across the topolectric realization of the [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Nonsymmorphic BDI class with symmorphic time [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
read the original abstract

We describe nonsymmorphic four-band tight-binding models in one dimension with Kramers degeneracy, and propose topolectric-circuit realizations of their topological phases. We begin with a representative model in the nonsymmorphic AII class with symmorphic time-reversal symmetry and nonsymmorphic charge-conjugation and chiral symmetries, resulting in a $\mathbb{Z}_2$ invariant. We also provide a Bogoliubov-de Gennes model in the nonsymmorphic D class with symmorphic charge-conjugation symmetry and nonsymmorphic time-reversal and chiral symmetries, which supports a $\mathbb{Z}_4$ classification. To compute these invariants, we extend an open-path winding-number formulation previously used for non-Kramers-degenerate nonsymmorphic $\mathbb{Z}_2$ phases to Kramers-degenerate four-band systems. We propose topolectric circuit implementations of nonsymmorphic systems, beginning with a comparison between the symmorphic Su-Schrieffer-Heeger and nonsymmorphic charge-density-wave models. We then extend this methodology to topolectric realizations of the AII and $\mathbb{Z}_4$ models, finding that the impedance response reproduces the predicted phase boundaries and associated zero energy modes. Finally, we analyze disorder in the $\mathbb{Z}_4$ model and find that, although disorder breaks the nonsymmorphic symmetries, certain domain-wall zero modes in the minimal nearest-neighbor model remain pinned at zero energy because the disorder has no first-order coupling to the soliton subspace; longer-range terms satisfying relevant symmetries lift this emergent property.

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 constructs one-dimensional nonsymmorphic four-band tight-binding models with Kramers degeneracy, one in the AII class (Z2 invariant) with symmorphic time-reversal and nonsymmorphic charge-conjugation/chiral symmetries, and one in the D class (Z4 invariant) with symmorphic charge-conjugation and nonsymmorphic time-reversal/chiral symmetries. It extends a prior open-path winding-number method to evaluate these invariants for the four-band Kramers systems, proposes topolectric-circuit realizations, and reports that the impedance response reproduces the predicted phase boundaries together with associated zero-energy modes. Disorder analysis in the minimal Z4 model shows that certain domain-wall zero modes remain pinned at zero energy because disorder lacks first-order coupling to the soliton subspace.

Significance. If the extension of the open-path winding number is rigorously justified, the work supplies concrete lattice models and circuit blueprints for nonsymmorphic Kramers-degenerate topological phases, together with an impedance-based diagnostic that can be implemented in existing topolectric platforms. The observation that certain zero modes survive disorder through an emergent decoupling mechanism adds a useful robustness result. These elements would be of interest to both the topological-matter and circuit-QED communities.

major comments (2)
  1. [Abstract / computation of invariants paragraph] Abstract (paragraph on computation of invariants): The statement that the open-path winding-number formulation 'extends without modification' to Kramers-degenerate four-band systems is load-bearing for the entire classification and for the subsequent claims that impedance spectra reproduce the Z2 and Z4 phase boundaries. No explicit formula, path choice, or reduction check (e.g., recovery of the two-band SSH limit) is supplied in the abstract or referenced section; if Kramers pairing or the combined symmorphic/nonsymmorphic symmetries introduce an extra phase or force a different open-path definition, the computed invariants and all downstream predictions become unreliable.
  2. [Model definitions] Section on AII and Z4 model definitions: The manuscript states that the models belong to the nonsymmorphic AII and D classes, yet the explicit four-band Hamiltonians, the nonsymmorphic operators, and the symmetry algebra are not displayed in the provided text. Without these, it is impossible to verify that the claimed symmetry classes are realized or that the open-path winding number is being applied to the correct Brillouin-zone path.
minor comments (2)
  1. [Circuit implementations] The abstract mentions 'topolectric-circuit realizations' and 'impedance response' but does not specify the circuit topology (capacitor/inductor values, grounding scheme) or the frequency range used to extract the zero-mode signatures; these details should be added for reproducibility.
  2. [Notation] Notation for the Z4 invariant and the open-path winding number should be introduced with a short equation or diagram early in the text rather than only in the methods section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight opportunities to strengthen the presentation of the invariant computation and model definitions. We address each point below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / computation of invariants paragraph] Abstract (paragraph on computation of invariants): The statement that the open-path winding-number formulation 'extends without modification' to Kramers-degenerate four-band systems is load-bearing for the entire classification and for the subsequent claims that impedance spectra reproduce the Z2 and Z4 phase boundaries. No explicit formula, path choice, or reduction check (e.g., recovery of the two-band SSH limit) is supplied in the abstract or referenced section; if Kramers pairing or the combined symmorphic/nonsymmorphic symmetries introduce an extra phase or force a different open-path definition, the computed invariants and all downstream predictions become unreliable.

    Authors: We agree that additional explicit support is warranted for the extension of the open-path winding number. In the revised manuscript we will augment the abstract with a concise statement of the adapted formula, specify the open path (integrating over half the Brillouin zone with Kramers-pair branch choices), and add a short reduction check demonstrating recovery of the conventional two-band SSH winding number in the appropriate limit. These details will also be expanded in the main text and a new appendix to rigorously justify that the core definition carries over without modification once the path and pairing are accounted for. revision: yes

  2. Referee: [Model definitions] Section on AII and Z4 model definitions: The manuscript states that the models belong to the nonsymmorphic AII and D classes, yet the explicit four-band Hamiltonians, the nonsymmorphic operators, and the symmetry algebra are not displayed in the provided text. Without these, it is impossible to verify that the claimed symmetry classes are realized or that the open-path winding number is being applied to the correct Brillouin-zone path.

    Authors: The full manuscript contains the four-band Hamiltonians, nonsymmorphic operators (glide or screw combined with time-reversal or particle-hole), and the resulting symmetry algebra in the model-definition section. To address the concern directly, we will revise the section to display the Hamiltonians as the first displayed equations, include an explicit table of the symmetry operators and their algebra, and state the precise Brillouin-zone path used for the winding-number integral. This will make verification immediate without altering any results. revision: yes

Circularity Check

1 steps flagged

Classification of AII/Z2 and D/Z4 phases rests on extending prior open-path winding-number without explicit derivation or limit check in this work

specific steps
  1. self citation load bearing [Abstract]
    "To compute these invariants, we extend an open-path winding-number formulation previously used for non-Kramers-degenerate nonsymmorphic Z2 phases to Kramers-degenerate four-band systems."

    The Z2 and Z4 invariants that define the phase boundaries and protected zero modes are computed via this extension. Because the manuscript presents the extension as given without deriving the open-path winding explicitly for the four-band Kramers case or verifying it against a known limit, the classification and all downstream claims about impedance reproduction of boundaries reduce to the assumptions of the cited prior formulation.

full rationale

The paper's topological classification, phase boundaries, and zero-mode predictions are obtained by announcing an extension of a previously published open-path winding-number method to the present Kramers-degenerate four-band case. The abstract states the extension occurs but supplies neither the modified formula nor a reduction check (e.g., to the two-band SSH limit). The subsequent topolectric-circuit impedance results and disorder analysis are independent of this step and provide external grounding, keeping overall circularity moderate rather than forcing the central claims by definition or self-citation chain alone.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard symmetry-based topological classification for the AII and D classes together with the assumption that the prior open-path winding-number method carries over to the four-band Kramers setting; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Nonsymmorphic charge-conjugation and chiral symmetries together with symmorphic time-reversal symmetry place the four-band model in the AII class with a Z2 invariant.
    Invoked when the representative model is introduced in the abstract.
  • domain assumption The open-path winding-number formulation extends directly to Kramers-degenerate four-band systems.
    Stated as the method used to compute the invariants.

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