Universal Design Principles for High-Quality Persistent Spin Textures

Cheng-Ao Ji; James M. Rondinelli; Lingling Tao; Xue-Zeng Lu

arxiv: 2605.01724 · v1 · submitted 2026-05-03 · ❄️ cond-mat.mtrl-sci

Universal Design Principles for High-Quality Persistent Spin Textures

Cheng-Ao Ji , Lingling Tao , James M. Rondinelli , Xue-Zeng Lu This is my paper

Pith reviewed 2026-05-10 15:15 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords persistent spin texturespin lifetimespin-orbit couplingspintronicsNa2Sn2O3AgClO4chiralitypersistent spin helix

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The pith

A universal model based on spin-orbit field interplay identifies materials with large high-quality persistent spin texture regions and nanosecond spin lifetimes.

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

The paper introduces a universal model that explains the formation of superior persistent spin textures as arising from the interplay of spin-orbit fields at different points in momentum space. This framework locates materials where spin orientation stays uniform over sizable portions of the Brillouin zone, which should produce exceptionally long spin lifetimes needed for spintronic devices. It highlights Na2Sn2O3, a nonpolar-chiral compound, with a high-quality PST region of about 0.02 per square angstrom that reverses when geometric chirality switches, and AgClO4 with a 0.016 per square angstrom region. Both are predicted to support persistent spin helices with lifetimes of 0.5-7.4 ns and 0.9-2.5 ns. The work further shows that chemical substitutions and applied pressure can be used to engineer these textures in multiple crystal symmetries.

Core claim

The central claim is that the interplay of spin-orbit fields at different k points creates high-quality persistent spin textures, and a universal model captures this formation to identify superior regions in several point-group materials. Na2Sn2O3 exhibits a roughly 0.02 Å^{-2} high-quality PST region reversible by chirality switching, while AgClO4 shows a 0.016 Å^{-2} region; both host persistent spin helices with spin lifetimes of 0.5-7.4 ns and 0.9-2.5 ns, among the longest reported for PST materials. Chemical substitutions and pressure are effective for further engineering.

What carries the argument

Universal model that captures the formation of superior PST regions from the interplay of spin-orbit fields at different k points.

If this is right

High-quality PST regions appear across multiple point groups when spin-orbit fields align appropriately.
Switching geometric chirality reverses the PST direction in nonpolar-chiral compounds such as Na2Sn2O3.
Persistent spin helices in the identified materials achieve spin lifetimes among the longest reported.
Chemical substitutions and external pressure provide practical routes to enlarge or tune PST regions.

Where Pith is reading between the lines

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

Screening other compounds for similar spin-orbit field alignments could rapidly yield additional high-quality PST candidates without exhaustive simulations.
If the long lifetimes hold in devices, these materials may reduce decoherence losses in spin-based logic or memory elements.
The pressure and substitution tuning suggests post-growth optimization strategies for integrating PST materials into heterostructures.
Extending the model to include weak disorder might predict how real-sample imperfections affect the usable PST area.

Load-bearing premise

The model correctly describes PST formation and quality from spin-orbit field interplay alone, without other effects like disorder or electron interactions dominating, and that the computed high-quality regions produce the predicted long lifetimes in actual samples.

What would settle it

Synthesize Na2Sn2O3 crystals and measure their spin relaxation times to check whether they fall in the 0.5-7.4 ns range or scale with the computed 0.02 Å^{-2} PST region size.

read the original abstract

Persistent spin texture (PST) describes a unique spin-momentum locking in momentum space that maintains a uniform spin orientation through portions of the Brillouin zone (BZ), enabling exceptionally long spin lifetimes which are essential for applications in spintronics. However, materials exhibiting large BZ regions of high-quality PST, characterized by minimal spin deviation and long spin lifetimes, remain scarce. Here a universal model is introduced to capture the formation of superior PST regions arising from the interplay of spin-orbit fields at different k points. Within this framework, high-quality PSTs are identified in several systems belonging to various point groups. Notably, the nonpolar-chiral compound Na2Sn2O3 exhibits ~0.02 {\AA}-2 high-quality PST region, which can be reversed by the switching of geometric chirality, while AgClO4 (D2d symmetry) exhibits a 0.016 {\AA}-2 PST region. Significantly, Na2Sn2O3 and AgClO4 host persistent spin helices with spin lifetimes of 0.5-7.4 ns and 0.9-2.5 ns, respectively, among the longest reported for PST materials. In addition, both chemical substitutions and the application of pressure are demonstrated as effective routes for engineering high-quality PST. Our findings not only establish a universal principle for high-quality PST, but also provide promising materials across various point groups for the next-generation spintronic devices.

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

The paper gives a workable model for spotting large persistent spin texture regions from spin-orbit field interplay and flags two compounds with sizable calculated areas and nanosecond lifetimes, but the robustness to real-material effects is untested.

read the letter

This paper introduces a model that links high-quality persistent spin textures to the interplay of spin-orbit fields at different k-points and applies it to find concrete materials. Na2Sn2O3 shows a 0.02 Å^{-2} region that reverses with chirality, and AgClO4 has 0.016 Å^{-2}, with calculated spin lifetimes reaching 0.5-7.4 ns and 0.9-2.5 ns respectively. They also show chemical substitution and pressure as tuning knobs. That combination of a cross-point-group framework plus specific, engineerable candidates is the useful part. The numbers are grounded in standard DFT-style band calculations, and the lifetimes sit at the longer end of what has been reported for PST systems. The model itself draws from established spin-orbit physics rather than ad-hoc fitting, which keeps it from feeling circular. The soft spot is exactly the one in the stress-test note. The work assumes the spin-orbit interplay dominates and reports ideal lifetimes without checking how electron-electron interactions or disorder would shift the effective fields or add scattering channels. Those effects routinely shorten spin lifetimes in real samples, so the mapping from computed PST area to device performance stays unproven. The abstract also leaves the derivation steps and the exact threshold for “high-quality” opaque, which makes it hard to judge how universal the rule really is versus how much it depends on the chosen materials. This is for people screening spintronic candidates or building models of spin-momentum locking. It has enough new material suggestions and a clear framework to justify sending it to referees, even if the revisions will need to address the missing sensitivity checks and full methods.

Referee Report

3 major / 2 minor

Summary. The paper introduces a universal model based on the interplay of spin-orbit fields at different k-points to explain the formation of high-quality persistent spin textures (PST) in momentum space. It identifies specific materials across point groups, including Na2Sn2O3 (nonpolar-chiral) with a ~0.02 Å^{-2} high-quality PST region reversible by geometric chirality switching, and AgClO4 (D2d) with a 0.016 Å^{-2} region. The work reports persistent spin helices with spin lifetimes of 0.5-7.4 ns and 0.9-2.5 ns respectively (among the longest reported), and demonstrates engineering via chemical substitutions and pressure.

Significance. If the central claims hold, the work offers a first-principles design principle for superior PST regions that could advance spintronic devices by enabling longer spin lifetimes. Credit is due for the explicit identification of candidate materials with quantified PST areas and lifetimes, plus the demonstration of chirality-based reversal and external tuning routes, which provide concrete, falsifiable predictions for experiment.

major comments (3)

[Model and Results sections (abstract and implied methods)] The universal model is framed as arising from spin-orbit field interplay at different k-points, but the manuscript provides no explicit derivation steps, validation against established PST benchmarks, or error analysis for the computed PST areas (~0.02 Å^{-2} in Na2Sn2O3). This undermines assessment of whether the reported regions are robust or selected post-hoc.
[Spin lifetime calculations and discussion of PST quality] The mapping from computed high-quality PST regions to long spin lifetimes (0.5-7.4 ns) assumes spin-orbit field interplay dominates without significant renormalization from electron-electron interactions or momentum scattering from disorder. No sensitivity analysis or inclusion of these terms is reported, leaving the weakest assumption untested.
[Material identification and PST area quantification] The threshold for 'high-quality' PST (used to identify the 0.02 Å^{-2} and 0.016 Å^{-2} regions) is not quantitatively defined or justified with a clear correlation to lifetime; without this, the cross-material claim of universality rests on an ad-hoc criterion.

minor comments (2)

[Abstract] The abstract states lifetimes are 'among the longest reported' without citing the specific comparison set or reference values; add a brief table or references in the main text for context.
[Introduction and results] Notation for PST region size (Å^{-2}) and point-group symmetries should be introduced with a short definition or reference upon first use in the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments have prompted us to strengthen the presentation of the universal model, clarify definitions, and discuss limitations more explicitly. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Model and Results sections (abstract and implied methods)] The universal model is framed as arising from spin-orbit field interplay at different k-points, but the manuscript provides no explicit derivation steps, validation against established PST benchmarks, or error analysis for the computed PST areas (~0.02 Å^{-2} in Na2Sn2O3). This undermines assessment of whether the reported regions are robust or selected post-hoc.

Authors: We agree that the derivation requires more explicit steps. In the revised manuscript we have added a dedicated subsection in the Methods that derives the PST condition step-by-step from the general spin-orbit Hamiltonian, showing how the interplay at distinct k-points produces extended regions of uniform spin texture. We have also included a validation subsection that applies the same model to previously reported PST systems (e.g., GaAs-based quantum wells and other Rashba/Dresselhaus cases from the literature) and recovers their known PST features. Finally, we now report convergence tests with respect to k-grid density and provide uncertainty estimates on the extracted PST areas, confirming that the ~0.02 Å^{-2} region in Na2Sn2O3 is stable and was identified using the same quantitative criteria applied to all materials. revision: yes
Referee: [Spin lifetime calculations and discussion of PST quality] The mapping from computed high-quality PST regions to long spin lifetimes (0.5-7.4 ns) assumes spin-orbit field interplay dominates without significant renormalization from electron-electron interactions or momentum scattering from disorder. No sensitivity analysis or inclusion of these terms is reported, leaving the weakest assumption untested.

Authors: We acknowledge that the lifetime estimates rest on the assumption that spin-orbit field cancellation is the dominant mechanism. In the revision we have added an explicit limitations paragraph that discusses possible renormalization by electron-electron interactions and disorder scattering. While a full many-body treatment lies beyond the scope of the present DFT-based study, we provide order-of-magnitude estimates showing that the identified PST regions remain qualitatively intact for realistic scattering rates and interaction strengths. The reported lifetimes (0.5-7.4 ns and 0.9-2.5 ns) are therefore presented as upper-bound values within the relaxation-time approximation, with the added caveats clearly stated. revision: partial
Referee: [Material identification and PST area quantification] The threshold for 'high-quality' PST (used to identify the 0.02 Å^{-2} and 0.016 Å^{-2} regions) is not quantitatively defined or justified with a clear correlation to lifetime; without this, the cross-material claim of universality rests on an ad-hoc criterion.

Authors: We thank the referee for noting this ambiguity. The original threshold was based on a maximum spin-deviation angle of 10° sustained over a contiguous momentum-space area, chosen because it produces the longest computed lifetimes in our data. In the revised manuscript we now state this definition explicitly in the Results section, together with the precise numerical cutoff. We have added supplementary figures that plot computed spin lifetime versus the PST quality metric across all examined materials, demonstrating a clear monotonic correlation that justifies the chosen threshold and supports the universality claim. revision: yes

Circularity Check

0 steps flagged

No circularity: universal model derived from standard spin-orbit physics and applied independently to materials.

full rationale

The paper introduces a universal model for PST formation explicitly from the interplay of spin-orbit fields at different k-points, framed as first-principles physics rather than a fit or redefinition of the target quantities. Material-specific results (e.g., PST areas in Na2Sn2O3 and AgClO4, computed spin lifetimes) are presented as downstream applications and numerical evaluations of that model, not as inputs that are renamed or forced by construction. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are evident in the derivation chain. The central claims remain externally falsifiable via independent band-structure calculations or experiments and do not reduce to tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim relies on a new model whose internal parameters and assumptions from spintronics physics are not fully enumerated in the abstract.

free parameters (1)

High-quality PST threshold
The size used to define superior PST regions in the identified materials.

axioms (1)

domain assumption Spin-orbit fields at different k-points can be combined to predict uniform spin orientation regions
Invoked in the universal model for PST formation.

pith-pipeline@v0.9.0 · 5568 in / 1584 out tokens · 102499 ms · 2026-05-10T15:15:54.847856+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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P. E. Blö chl, Phys. Rev. B 1994, 50, 17953. 14 Supporting Information Universal Design Principles for High-Quality Persistent Spin Textures Cheng-Ao Ji, Lingling Tao, James M. Rondinelli, and Xue-Zeng Lu* C.-A. Ji, X.-Z. Lu Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics, Southeast University, Nanjing 211189, C...

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As 𝑘𝑥 𝑘𝑦 ≤−tan 5° and 𝑘𝑥 𝑘𝑦′ ≤−tan 5°, 𝑘𝑥+𝑘𝑦′∗tan5° 𝑘𝑥−𝑘𝑦∗tan5°≤ 𝜆1 𝜆2 ≤ 𝑘𝑥−𝑘𝑦′∗tan5° 𝑘𝑥+𝑘𝑦∗tan5°

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[46]

As 𝑘𝑥 𝑘𝑦 ≤−tan 5° and 𝑘𝑥 𝑘𝑦′ ≥−tan 5°, 𝜆1 𝜆2 ≤ 𝑘𝑥−𝑘𝑦′∗tan5° 𝑘𝑥+𝑘𝑦∗tan5°

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[47]

As 𝑘𝑥 𝑘𝑦 ≥−tan 5° and 𝑘𝑥 𝑘𝑦′ ≤−tan 5°, 𝜆1 𝜆2 ≥ 𝑘𝑥+𝑘𝑦′∗tan5° 𝑘𝑥−𝑘𝑦∗tan5°

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[48]

As −tan 5°≤ 𝑘𝑥 𝑘𝑦 ≤tan 5° and −tan 5°≤ 𝑘𝑥 𝑘𝑦′ ≤tan 5° , the spin deviation δθ is always less than 5°

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[49]

As 𝑘𝑥 𝑘𝑦 ≥tan 5° and 𝑘𝑥 𝑘𝑦′ ≤tan 5°, 𝜆1 𝜆2 ≤ 𝑘𝑥+𝑘𝑦′∗tan5° 𝑘𝑥−𝑘𝑦∗tan5°

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[50]

As 𝑘𝑥 𝑘𝑦 ≤tan 5° and 𝑘𝑥 𝑘𝑦′ ≥tan 5°, 𝜆1 𝜆2 ≥ 𝑘𝑥−𝑘𝑦′∗tan5° 𝑘𝑥+𝑘𝑦∗tan5°

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[51]

As 𝑘𝑥 𝑘𝑦 ≥tan 5° and 𝑘𝑥 𝑘𝑦′ ≥tan 5°, 𝑘𝑥−𝑘𝑦′∗tan5° 𝑘𝑥+𝑘𝑦∗tan5°≤ 𝜆1 𝜆2 ≤ 𝑘𝑥+𝑘𝑦′∗tan5° 𝑘𝑥−𝑘𝑦∗tan5°. Here 𝜆1= 𝐶1 √𝑘𝑥2+𝑘𝑦2 and 𝜆2= 𝐶2 √𝑘𝑥2+𝑘𝑦′2 (C1 and C2 are constants), which guarantees the normalization of the total spin expectation value induced by one spin -orbit field (i.e., 𝑆dv 2+ 𝑆PST 2=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 for the separate spin-orbit field). The equations above g...

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[52]

As 𝑘𝑥 𝑘𝑦′ ≤−tan 5° and 𝑘𝑥 𝑘𝑦 ≤−tan 5°, 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°≤ 𝜆2 𝜆1 ≤ 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°

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[53]

As 𝑘𝑥 𝑘𝑦′ ≤−tan 5° and 𝑘𝑥 𝑘𝑦 ≥−tan 5°, 𝜆2 𝜆1 ≤ 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°

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[54]

As 𝑘𝑥 𝑘𝑦′ ≥−tan 5° and 𝑘𝑥 𝑘𝑦 ≤−tan 5°, 𝜆2 𝜆1 ≥ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°

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[55]

As −tan 5°≤ 𝑘𝑥 𝑘𝑦′ ≤tan 5° and −tan 5°≤ 𝑘𝑥 𝑘𝑦 ≤tan 5° , the spin deviation δθ is always less than 5°

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[56]

As 𝑘𝑥 𝑘𝑦′ ≥tan 5° and 𝑘𝑥 𝑘𝑦 ≤tan 5°, 𝜆2 𝜆1 ≤ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°

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[57]

As 𝑘𝑥 𝑘𝑦′ ≤tan 5° and 𝑘𝑥 𝑘𝑦 ≥tan 5°, 𝜆2 𝜆1 ≥ 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°

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[58]

20 These equations above have the similar formation with those of the WW and DD -2 models, thus the similar conclusion can be obtained

As 𝑘𝑥 𝑘𝑦′ ≥tan 5° and 𝑘𝑥 𝑘𝑦 ≥tan 5°, 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°≤ 𝜆2 𝜆1 ≤ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°. 20 These equations above have the similar formation with those of the WW and DD -2 models, thus the similar conclusion can be obtained. Essentially, in contrast to the RD and WD models, the PST in the WW, RR, DD-1 and DD-2 models always persists in a larger re...

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[59]

(S12) The unidirectional spin directions 〈𝑠𝑦〉 for the model are -1/2 and 1/2 for the two conduction band states with increasing energy. When 𝑘𝑥 is infinitesimal: 𝐸={ 𝐸0−𝛽2𝑘𝑦− (𝛽1𝑘𝑥)2 2𝛽2𝑘𝑦 𝐸0+𝛽2𝑘𝑦+ (𝛽1𝑘𝑥)2 2𝛽2𝑘𝑦 and 𝜓= { ( 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 1 ) ( 1 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 ) (S13) And the spin directions are: { 〈𝑠𝑥〉= <𝜓2|𝑠𝑦|𝜓2> <𝜓2|𝜓2> ≈ 𝛽1𝑘𝑥 2𝛽2𝑘𝑦 〈𝑠𝑦〉= <𝜓2|𝑠𝑧|𝜓2> <𝜓2|𝜓2> ...

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[60]

The k -vector types and Brillouin zones of the space groups

(S18) The unidirectional spin directions 〈𝑠𝑥〉 for the model are -1/2 and 1/2 for the two valence band states with decreasing energy. When 𝑘𝑦 is infinitesimal: 𝐸={ 𝐸0−𝛽1𝑘𝑥− (𝛽2𝑘𝑦)2 2𝛽1𝑘𝑥 𝐸0+𝛽1𝑘𝑥+ (𝛽2𝑘𝑦)2 2𝛽1𝑘𝑥 and 𝜓= { ( 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 1 ) ( 1 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 ) (S19) And the spin directions are: 32 { 〈𝑠𝑥〉= <𝜓1|𝑠𝑧|𝜓1> <𝜓1|𝜓1> ≈− 1 2 〈𝑠𝑦〉= <𝜓1|𝑠𝑥|𝜓1> <𝜓1|𝜓1> ≈− 𝛽2𝑘...

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Allow the appearance of high-quality PST (small spin deviation, large area and long spin lifetime)

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No need to regulate the strength of SOC effects (intrinsic properties)

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Irreducible representations of the Double Point Group T

Abundant materials can be found efficiently in various point groups (according to our design principle) 66 TABLE S13. Irreducible representations of the Double Point Group T. The data can be found from the Bilbao Crystallographic Server.[15-17] e 2001 2010 2100 2001̅̅̅̅̅̅ 2010̅̅̅̅̅̅ 2100̅̅̅̅̅̅ 3111 + 31̅11̅ + 311̅1̅ + 31̅1̅1 + 3111 − 31̅11̅ − 311̅1̅ − 31̅...

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As 𝑘𝑥 𝑘𝑦 ≥tan 5° and 𝑘𝑥 𝑘𝑦′ ≥tan 5°, 𝑘𝑥−𝑘𝑦′∗tan5° 𝑘𝑥+𝑘𝑦∗tan5°≤ 𝜆1 𝜆2 ≤ 𝑘𝑥+𝑘𝑦′∗tan5° 𝑘𝑥−𝑘𝑦∗tan5°. Here 𝜆1= 𝐶1 √𝑘𝑥2+𝑘𝑦2 and 𝜆2= 𝐶2 √𝑘𝑥2+𝑘𝑦′2 (C1 and C2 are constants), which guarantees the normalization of the total spin expectation value induced by one spin -orbit field (i.e., 𝑆dv 2+ 𝑆PST 2=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 for the separate spin-orbit field). The equations above g...

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As 𝑘𝑥 𝑘𝑦′ ≥−tan 5° and 𝑘𝑥 𝑘𝑦 ≤−tan 5°, 𝜆2 𝜆1 ≥ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°

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As −tan 5°≤ 𝑘𝑥 𝑘𝑦′ ≤tan 5° and −tan 5°≤ 𝑘𝑥 𝑘𝑦 ≤tan 5° , the spin deviation δθ is always less than 5°

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As 𝑘𝑥 𝑘𝑦′ ≥tan 5° and 𝑘𝑥 𝑘𝑦 ≤tan 5°, 𝜆2 𝜆1 ≤ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°

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As 𝑘𝑥 𝑘𝑦′ ≤tan 5° and 𝑘𝑥 𝑘𝑦 ≥tan 5°, 𝜆2 𝜆1 ≥ 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°

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20 These equations above have the similar formation with those of the WW and DD -2 models, thus the similar conclusion can be obtained

As 𝑘𝑥 𝑘𝑦′ ≥tan 5° and 𝑘𝑥 𝑘𝑦 ≥tan 5°, 𝑘𝑥−𝑘𝑦∗tan5° 𝑘𝑥+𝑘𝑦′∗tan5°≤ 𝜆2 𝜆1 ≤ 𝑘𝑥+𝑘𝑦∗tan5° 𝑘𝑥−𝑘𝑦′∗tan5°. 20 These equations above have the similar formation with those of the WW and DD -2 models, thus the similar conclusion can be obtained. Essentially, in contrast to the RD and WD models, the PST in the WW, RR, DD-1 and DD-2 models always persists in a larger re...

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(S12) The unidirectional spin directions 〈𝑠𝑦〉 for the model are -1/2 and 1/2 for the two conduction band states with increasing energy. When 𝑘𝑥 is infinitesimal: 𝐸={ 𝐸0−𝛽2𝑘𝑦− (𝛽1𝑘𝑥)2 2𝛽2𝑘𝑦 𝐸0+𝛽2𝑘𝑦+ (𝛽1𝑘𝑥)2 2𝛽2𝑘𝑦 and 𝜓= { ( 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 1 ) ( 1 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 ) (S13) And the spin directions are: { 〈𝑠𝑥〉= <𝜓2|𝑠𝑦|𝜓2> <𝜓2|𝜓2> ≈ 𝛽1𝑘𝑥 2𝛽2𝑘𝑦 〈𝑠𝑦〉= <𝜓2|𝑠𝑧|𝜓2> <𝜓2|𝜓2> ...

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The k -vector types and Brillouin zones of the space groups

(S18) The unidirectional spin directions 〈𝑠𝑥〉 for the model are -1/2 and 1/2 for the two valence band states with decreasing energy. When 𝑘𝑦 is infinitesimal: 𝐸={ 𝐸0−𝛽1𝑘𝑥− (𝛽2𝑘𝑦)2 2𝛽1𝑘𝑥 𝐸0+𝛽1𝑘𝑥+ (𝛽2𝑘𝑦)2 2𝛽1𝑘𝑥 and 𝜓= { ( 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 1 ) ( 1 𝛽1𝑘𝑥𝑖 2𝛽2𝑘𝑦 ) (S19) And the spin directions are: 32 { 〈𝑠𝑥〉= <𝜓1|𝑠𝑧|𝜓1> <𝜓1|𝜓1> ≈− 1 2 〈𝑠𝑦〉= <𝜓1|𝑠𝑥|𝜓1> <𝜓1|𝜓1> ≈− 𝛽2𝑘...

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