arxiv: 2605.14506 · v1 · submitted 2026-05-14 · ❄️ cond-mat.mtrl-sci

Recognition: 2 theorem links

· Lean Theorem

Probing the Chirality of Trigonal Selenium and Tellurium by Spin and Orbital Hall Effects

Yuting Xiong , YingJie Hu , Wei Ren , Heng Gao

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:41 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords chiralityenantiomersspin Hall conductivityorbital Hall conductivitytrigonal seleniumtrigonal telluriumBerry curvaturemirror symmetry

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The pith

Trigonal selenium and tellurium exhibit opposite signs in specific spin and orbital Hall conductivity components between left- and right-handed enantiomers due to mirror symmetry.

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

The paper establishes that the two enantiomers of trigonal Se and Te, related by a mirror operation, produce Hall currents of opposite sign in selected tensor components even though their band structures are identical. This reversal arises because the underlying spin and orbital Berry curvatures transform antisymmetrically under the mirror that swaps the structures. A reader would care because it links a measurable transport response directly to structural handedness, offering a potential experimental handle on chirality in these semiconductors without relying on optical probes.

Core claim

Trigonal selenium and tellurium, described by space groups P3_221 and P3_121, share the same band structure and four nonzero SHC/OHC tensor components, yet the elements σ_yx^{S_y} and σ_yx^{L_y} reverse sign between the enantiomers. The sign reversal follows from the antisymmetric transformation of the spin and orbital Berry curvature under the M_xy mirror operation that relates the two structures. The calculated signs can therefore be correlated with left- versus right-handed atomic arrangements under a fixed coordinate convention.

What carries the argument

The M_xy mirror operation that interconverts the left- and right-handed enantiomers, enforcing an antisymmetric transformation on the spin and orbital Berry curvature while leaving the band energies unchanged.

If this is right

Selected SHC and OHC tensor components reverse sign between mirror-related enantiomorphic crystals in any material obeying the same symmetry relation.
The sign of the measured σ_yx^{S_y} or σ_yx^{L_y} directly indicates the structural handedness once the coordinate convention is fixed.
Chiral crystals can generate pure spin or orbital currents whose direction depends on enantiomer choice.
Symmetry rules can be used to predict which Hall components will flip sign in other pairs of enantiomorphic structures.

Where Pith is reading between the lines

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

Transport measurements of these Hall components could serve as a bulk probe of handedness in device-scale samples where optical rotation is impractical.
The same mirror-induced sign rule should apply to other chiral semiconductors or semimetals whose space groups are related by M_xy.
Device designs that rely on spin or orbital currents could exploit the enantiomer choice to select current direction without external fields.

Load-bearing premise

First-principles calculations correctly reproduce the antisymmetric change in spin and orbital Berry curvature under the mirror operation that swaps the two enantiomers.

What would settle it

Experimental measurement of the spin Hall conductivity component σ_yx^{S_y} on single-crystal samples of left- and right-handed trigonal selenium that fails to show opposite signs under the same coordinate convention.

Figures

Figures reproduced from arXiv: 2605.14506 by Heng Gao, Wei Ren, YingJie Hu, Yuting Xiong.

**Figure 2.** Figure 2: FIG. 2. The non-zero independent SHC tensors [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The band-projected spin Berry curvature (top) [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Schematics of the transformation relationship of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

read the original abstract

Chiral crystals exhibit enantiomer-dependent transport phenomena that generate pure spin or orbital currents, while the handedness sensitivity of spin and orbital Hall conductivities (SHC/OHC) remains insufficiently understood. Using first-principles calculations, we demonstrate that trigonal selenium and tellurium -- prototypical chiral semiconductors -- exhibit opposite signs of the SHC/OHC tensor elements $\sigma_{yx}^{S_y}$ and $\sigma_{yx}^{L_y}$ between their left- and right-handed enantiomers. This behavior originates from the mirror operation relating the two structures, described by space groups $P3_221$ (left-handed) and $P3_121$ (right-handed). Although both enantiomers share identical band structures and four nonzero SHC/OHC tensor components, $\sigma_{yx}^{S_y}$ and $\sigma_{yx}^{L_y}$ reverse sign due to the antisymmetric transformation of the spin/orbital Berry curvature under the $M_{xy}$ mirror operation. More generally, for mirror-related enantiomorphic structures, selected SHC/OHC tensor components can exhibit symmetry-governed sign reversal. For trigonal Se and Te, the calculated signs of these components can be directly correlated with the left- and right-handed structures under the chosen coordinate convention. These results clarify the symmetry origin of handedness-dependent SHC/OHC and suggest a possible route for correlating measurable SHC/OHC signals with structural handedness in specific chiral materials.

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.

Desk Editor's Note private letter to a colleague

Symmetry forces opposite signs in selected SHC/OHC components between Se/Te enantiomers, and the argument holds up without needing fitted parameters.

read the letter

The central claim is that mirror symmetry between the P3_221 and P3_121 structures of trigonal Se and Te flips the signs of σ_yx^{S_y} and σ_yx^{L_y} while leaving the band structure unchanged. The paper traces this directly to the antisymmetric action of the M_xy mirror on the relevant spin and orbital Berry curvature pieces. That link is the new piece: it pins a general chirality effect to concrete, well-studied materials and shows which tensor elements are affected. The symmetry reasoning is straightforward and does not depend on the choice of exchange-correlation functional or k-mesh details for the sign itself. Any calculation that respects the space-group symmetries will reproduce the reversal. This is useful because it gives experimenters a clear route to correlate a measurable Hall signal with structural handedness under a fixed coordinate convention. The abstract does not report numerical magnitudes, convergence tests, or comparison with existing data, so the size of the effect and its robustness remain to be checked in the full manuscript. That is the main soft spot, but it is not fatal to the symmetry argument. The work is aimed at people already thinking about spin-orbit transport in chiral crystals. It supplies a clean theoretical handle rather than a broad new method. I would send it to peer review so the numerics can be examined directly.

Referee Report

0 major / 3 minor

Summary. The manuscript uses first-principles calculations to demonstrate that trigonal selenium and tellurium exhibit opposite signs in the spin Hall conductivity (SHC) and orbital Hall conductivity (OHC) tensor elements σ_yx^{S_y} and σ_yx^{L_y} between left-handed (P3_221) and right-handed (P3_121) enantiomers. This sign reversal originates from the antisymmetric transformation of the spin/orbital Berry curvature under the M_xy mirror operation that maps one enantiomer onto the other, while the band structures and the other four nonzero tensor components remain identical between enantiomers.

Significance. If the central symmetry argument holds, the work identifies a parameter-free, symmetry-protected mechanism that directly correlates measurable SHC/OHC signals with structural handedness in mirror-related chiral crystals. This is a clear strength, as the sign flip follows rigorously from the space-group relation and the transformation properties of Berry curvature without requiring fitted parameters or self-referential definitions.

minor comments (3)

[Abstract] The abstract states that 'the calculated signs of these components can be directly correlated' with handedness but reports no numerical values, units, or convergence data; adding at least the computed magnitudes (even if only order-of-magnitude) would allow readers to assess the size of the effect relative to typical experimental resolutions.
[Computational Methods] The manuscript should explicitly state the exchange-correlation functional and k-point mesh used for the Berry-curvature integration, together with a brief convergence check, to confirm that the numerical implementation respects the space-group symmetries of each enantiomer.
[Introduction and Results] Coordinate conventions for the M_xy mirror and the labeling of left- versus right-handed structures should be stated once in the main text (not only in a figure caption) so that the sign assignment is unambiguous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and accurate summary of our findings on the enantiomer-dependent sign reversal in selected SHC and OHC tensor components of trigonal Se and Te. The symmetry argument based on the M_xy mirror operation is correctly identified as the origin of the effect. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Symmetry-protected sign reversal; derivation self-contained with no circular reduction

full rationale

The central claim—that σ_yx^{S_y} and σ_yx^{L_y} reverse sign between enantiomers—follows directly from the antisymmetric transformation of Berry curvature under the M_xy mirror that maps P3_221 onto P3_121. This is a standard group-theoretic property of the space groups and the Hall conductivity tensor; the first-principles calculations merely evaluate the magnitude while respecting the symmetry. No parameters are fitted to the target sign, no self-citation supplies a uniqueness theorem, and no ansatz is imported. The paper's equations for the conductivity (via Berry curvature integration) are independent of the final sign result, which is fixed once the coordinate convention and mirror relation are chosen. Hence the derivation chain contains no self-definitional or fitted-input circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard density-functional-theory approximations for band structures and Berry curvature plus the crystallographic symmetry of the two space groups. No free parameters are fitted to data to obtain the sign reversal; it follows from the mirror operation.

axioms (2)

domain assumption Density functional theory with standard approximations yields sufficiently accurate Berry curvature for the purpose of determining sign reversal under mirror symmetry.
Invoked implicitly by the use of first-principles calculations to obtain the SHC/OHC tensors.
domain assumption The two enantiomers are exactly related by the M_xy mirror operation with no additional structural relaxation or defect effects.
Stated via the space-group assignment P3_221 and P3_121.

pith-pipeline@v0.9.0 · 5580 in / 1468 out tokens · 36034 ms · 2026-05-15T01:41:51.490689+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the sign reversal of σ_yx^{S_y} and σ_yx^{L_y} between enantiomers is enforced by the M_xy mirror symmetry that maps P3_221 onto P3_121 ... antisymmetric transformation of the spin/orbital Berry curvature under the M_xy mirror operation
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Kubo formula ... σXk_ij = eℏ Σ ... Im ⟨nk|ĵXk_i |mk⟩⟨mk|νj|nk⟩ / (Enk−Emk)²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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