One-dimensional Dirac modes in the core of a pentagonal topological crystalline insulator nanowire

Marta A. Chabowska; Rafa{\l} Rechci\'nski; Ryszard Buczko; Saeed Samadi

REVIEW 3 major objections 4 minor 66 references

One-dimensional Dirac modes in the core of a pentagonal topological crystalline insulator nanowire

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Pentagonal SnTe-class nanowires with cationic twin planes are predicted to host two spatially separated one-dimensional helical Dirac crossings—one at the wire core and one at the outer surface—whenever the bulk band structure is inverted.

desk verdict A solid idealized-model prediction of two spatially separated helical Dirac channels in pentagonal TCI nanowires, with the main open question being how the ~2% closure strain affects the narrow alloy gap. read the letter →

arxiv 2608.05908 v1 pith:SMWCVY6T submitted 2026-08-06 cond-mat.mes-hall

Saeed Samadi , Rafa{l} Rechciński , Marta A. Chabowska , Ryszard Buczko This is my paper · ORCID

classification cond-mat.mes-hall

keywords topologicalcrystallineinsulatorSnTehelicalDiracmodepentagonalnanowiretwinplaneone-dimensionaltight-bindingmodel

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

This paper predicts that pentagonal nanowires of SnTe-class topological crystalline insulators—wires whose five rock-salt wedges meet at five twin planes along the axis—host two one-dimensional helical Dirac crossings when the bulk bands are inverted and the twin planes are cationic: one bound to the nanowire core and one to the outer surface. The paper argues the crossings arise because the five helical channels living on the twin-plane edges hybridize; since five is odd, one Kramers pair is forced to remain gapless at each boundary. For anionic twin planes the spectrum stays gapped, and for topologically trivial PbTe no crossings appear. Using a material-specific model of Pb0.4Sn0.6Te, the paper finds both modes well developed at wire thicknesses of about 50 nm and above, within the range of experimentally grown pentagonal nanowires. If correct, this makes the fabricated nanowires an accessible platform for spatially separated one-dimensional helical channels.

What carries the argument

The load-bearing object is the odd set of five helical channels localized at the edges of the five cationic twin planes. Under the fivefold rotational symmetry $C_5$ of the ideal nanowire, the Hamiltonian block diagonalizes into sectors labeled by eigenvalues $\lambda_\nu = e^{-i\pi(2\nu+5)/5}$; the self-conjugate $\nu=0$ sector has eigenvalue $-1$ and, by time-reversal and mirror constraints, its mass term $\Delta_0$ must vanish, forcing a linear crossing. The paper proves that this $C_5=-1$ block is equivalent, up to the small strain used to close the pentagon, to a cylindrical SnTe shell containing a single twin plane with a twisted seam hopping, so the two crossings of the shell are the core and surface crossings of the nanowire. A complementary selection rule from the axial atomic column (spin plus orbital plus site phase must equal $5/2$ modulo 5) shows that $s$ and $p$ orbitals on the central column cannot contribute to the core mode, which is carried by surrounding atomic rings.

What would settle it

Tunneling or angle-resolved photoemission maps on a 50-nm-thick pentagonal Pb0.4Sn0.6Te nanowire whose twin-plane sublattice is identified by atomically resolved imaging: the predicted signature is two linearly dispersing crossings at the zone center, one localized near the wire axis and one at the surface hinges, present for cationic twin planes and absent for anionic twin planes. Finding a gap in the cationic wire, or crossings in the anionic wire, would rule out the mechanism as stated. A simpler numerical check is a calculation that includes the deformation-potential corrections the paper omits: if the inverted gap closes, the modes vanish.

Watch

Extended reading notes

Core claim

The central claim is that a pentagonal SnTe-class nanowire with cationic {111} twin planes and an inverted bulk band structure has two spatially separated helical Dirac crossings near the zone center $\Gamma$: one localized at the wire core and one at the outer surface, each consisting of a single time-reversal-protected Kramers pair. The crossings are not features of the specific core termination: they survive for occupied and hollow cores, and the core crossing has zero weight on the axial atomic column in the p-orbital model. The anionic-twin-plane wire shows no such crossings, and the same geometry in topologically trivial PbTe does not. The paper interprets the two modes as the inner and outer boundaries of an effective quantum-spin-Hall-like ribbon, with the five helical twin-plane-edge channels reorganizing under fivefold rotation into two gapped conjugate sectors and one self-conjugate sector that is forced to be gapless.

Load-bearing premise

The prediction assumes the roughly 2% uniaxial strain that closes the five wedges into a regular pentagon does not close or reorder the inverted bulk gap—strain-induced gap changes are estimated at 10-20% and dropped from the Hamiltonian—and that wires with cationic twin planes can actually be grown, although the paper's own DFT calculations find anionic twin planes energetically preferred in SnTe.

Editorial extensions

If this is right

Cationic-twin-plane pentagonal SnTe-class nanowires provide an experimentally accessible realization of two spatially separated helical one-dimensional channels, one along the axial defect and one along the surface.
The core helical mode persists across the microscopic core realizations studied (occupied and hollow), so its existence does not depend on the central atomic column or on dangling-bond states.
In Pb0.4Sn0.6Te the two modes decouple exponentially with wire thickness, reaching well-developed massless dispersions at about 50 nm, within the experimentally grown thickness range.
Nonmagnetic disorder and moderate structural distortions cannot gap a single helical Kramers pair by themselves, so the crossings are expected to persist as long as the protecting bulk and twin-plane gaps stay open.
The Z-sector spectrum remains gapped for both twin-plane types, so the gapless physics is specific to the $\Gamma$ sector and the core/surface pair.

Reading between the lines

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

The odd-channel counting suggests a general rule: any multiply twinned nanowire whose $n$-fold rotation axis meets an odd number of twin planes should leave one gapless helical pair in the self-conjugate rotational sector, whereas an even-numbered analogue would be fully gapped at the center by the same logic.
If the core and surface modes can be selectively coupled to a superconductor and to magnets, the geometry is a natural testbed for Majorana bound states, since the two channels are spatially separated and could be gated or proximity-coupled independently—an extension the paper only sketches.
The energetic preference for anionic twin planes found in the paper's DFT calculations is the main experimental hurdle; alloy composition or growth conditions might flip this preference, which would be a concrete target for growth studies.
The exponential anticrossing fit suggests a simple design rule: the required wire thickness scales with the inverse localization length of the boundary states, so larger-gap alloys will need thicker wires.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper finds something real. In pentagonal SnTe-class nanowires with cationic twin planes and an inverted bulk gap, the spectrum contains two spatially separated helical Dirac crossings, one at the core and one at the outer surface. The mechanism is clean: five helical channels from the twin-plane edges, an odd number, so one Kramers pair survives at each boundary after hybridization. The core and surface crossings appear in two independent tight-binding models and in the C5-decomposed spectrum, and the massless nu=0 sector follows directly from the symmetry constraint Delta_nu = -Delta_-nu. The single-TP shell mapping in Appendix B is a nice piece of analysis and it checks out numerically. The authors also disclose weak spots honestly: the geometric arguments in Appendix C are marked as conceptual, the anionic-TP control spectrum involves a hand-picked 40 meV shift, and their own DFT calculation finds anionic TPs energetically preferred for SnTe.

Main soft spot is strain. The pentagonal geometry needs a ~2% uniaxial deformation per wedge, and the TB Hamiltonian includes only bond-orientation changes. Deformation potentials, Poisson strain, and bond-length rescaling are estimated to shift the bulk gap by less than 10% for SnTe and 20% for the alloy, then dropped. The alloy gap is about 104 meV, so a 20% shift is 20 meV—not obviously fatal, but the estimate is not a calculation, and uniaxial strain can split the projected L valleys. The PbTe control shows bulk inversion is essential, so the material-specific prediction rests on an assumption that is plausible but unquantified. The 50 nm claim itself is an extrapolation from a four-point exponential fit; fine as a scaling estimate, but it should be labeled as such.

The anionic-stability problem is real but properly flagged. The paper's own conclusion says control of the TP sublattice type is a central experimental challenge. That tempers the 'experimentally accessible' framing but does not invalidate the idealized-model prediction.

Overall, the central physics claim is well supported within the idealized model. For a serious referee, I would want the strain treatment and the anionic control spectrum resolved. This paper deserves peer review rather than a desk reject, and I would cite it for the odd-channel mechanism and the core-surface Dirac separation.

Referee Report

3 major / 4 minor

Summary. The paper studies pentagonal IV–VI nanowires built from five rocksalt wedges separated by radial {111} twin planes meeting at the wire axis. Tight-binding calculations with a minimal p3 model and a material-specific sp3d5 model show that, when the bulk band structure is inverted and the twin planes are cationic, the nanowire spectrum contains two helical Dirac crossings at the Gamma point: one localized near the core and one at the outer surface. For anionic twin planes the corresponding spectra remain gapped. The authors attribute the crossings to five helical twin-plane-edge channels whose C5 decomposition leaves a gapless self-conjugate sector, with the vanishing mass Delta_0 = 0 following from symmetry in Eq. (7). The mechanism is corroborated by C5-resolved spectra, a single-TP shell calculation that embeds as the C5 = -1 sector, and a PbTe control calculation. Material-specific calculations for Pb0.4Sn0.6Te predict well-developed modes at thicknesses of about 50 nm, and DFT calculations are used to compare structural stability of core and twin-plane configurations.

Significance. The central existence result is well supported: the two crossings are directly visible in the TB spectra of Fig. 3(a), they belong to the predicted C5 = -1 sector in Fig. 9, and the low-energy argument in Eqs. (4)-(7) is a clean symmetry derivation. The paper also includes strong controls: anionic twin planes, topologically trivial PbTe, occupied versus hollow cores, and DFT comparison with the earlier core-chain band of Ref. [30]. If the prediction survives quantitative strain and stability checks, it identifies a realistic nanowire geometry with spatially separated helical channels, which is an interesting analogue of a quantum spin Hall ribbon and potentially relevant for Majorana proposals. The main limitations are also acknowledged in the manuscript: the idealized strain treatment and the DFT result that anionic twin planes are energetically favored for SnTe. These caveats do not undermine the symmetry-based existence proof, but they do affect the strength of the experimental-accessibility claim.

major comments (3)

[Section III and Section IV D] The central prediction for Pb0.4Sn0.6Te depends on the approximately 104 meV inverted bulk gap surviving the roughly 2% closure strain. However, the TB Hamiltonian includes only bond-orientation changes; deformation potentials, Poisson strain, and Slater-Koster bond-length rescaling of hoppings are estimated to change the gap by less than 10-20% and are then dropped. This estimate is not a calculation, and uniaxial strain can split the projected L valleys, so the statement in Section III that the corrections "do not alter the band ordering" is an assertion. Because Fig. 7(e) shows that removing bulk inversion removes both crossings, this is a load-bearing assumption. Please compute the p3 and sp3d5 gaps under the applied strain with deformation potentials and distance-rescaled hoppings, or provide a worst-case bound that demonstrates the band ordering remains inverted.
[Abstract, Conclusions, and Appendix D] The abstract and conclusions describe these modes as establishing an "experimentally accessible realization" of spatially separated helical channels, but the DFT calculations in Fig. 18 find that for the investigated SnTe structures the anionic-TP configuration is energetically favored, and the entire predicted mode structure requires cationic twin planes. The paper acknowledges this as a challenge, but it does not provide evidence that cationic twin planes can be stabilized in Pb0.4Sn0.6Te or under realistic growth conditions. Please either give a quantitative argument for how the TP sublattice preference depends on alloy composition or growth conditions, or explicitly reframe the central claim as a conditional prediction for cationic-TP wires. As written, the experimental-accessibility claim goes beyond the evidence presented.
[Eq. (8) and Fig. 7(c)] The statement that the modes are "well developed" at approximately 50 nm and above is based on an exponential fit to four thicknesses with a single decay constant and no reported uncertainty. Four points over a range whose largest value is the quoted 50 nm is a thin basis for the extrapolation "and above," especially because the sp3d5 alloy model itself inherits the approximate strain treatment discussed above. Please report the fit residuals, include additional thicknesses if possible, and state more cautiously that the 50 nm estimate is specific to the virtual-crystal sp3d5 model with the adopted strain approximations.

minor comments (4)

[Fig. 3(b) and Section IV A] The anionic-TP spectrum in Fig. 3(b) is shown only after applying a 40 meV onsite shift to core orbitals, and the unshifted spectrum is not shown. Because this is the main control for the gapped anionic case in the simplified model, please include the unshifted spectrum in an appendix or state explicitly why the shift cannot remove a protected crossing.
[Appendix B, Eq. (B24)] The approximation arccos(1/3) approximately equals 2 pi/5 is central to identifying the single-TP shell with the C5 = -1 sector. Please state explicitly that the residual 1.47 degree difference per wedge is the same closure strain already imposed in the pentagonal geometry, so that the correspondence is quantitative rather than approximate in a separate sense.
[Fig. 12] The labels H and F appear in the spectra of Fig. 12 but are not defined in the caption; please define them in the caption or point to the subsection where they are introduced.
[General] The paper would benefit from a short statement in Section III about the number of k-points, convergence criteria, and the exact procedure used to extract the anticrossing Delta_ac in Fig. 7(c), since the main quantitative claim depends on that extraction.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Dirac-mode claim is derived from explicit TB spectra and a symmetry constraint, not from a fitted parameter or self-citation chain.

full rationale

The central claim—two spatially separated helical Dirac crossings for cationic-TP pentagonal nanowires—is demonstrated numerically inside the paper: TB spectra in Fig. 3, C5-resolved spectra in Fig. 9, the single-TP shell in Fig. 4, and the material-specific sp3d5 model in Fig. 7, with anionic-TP and PbTe controls in Figs. 7(d,e). The low-energy model's key step is not fitted: Eq. (7) imposes Delta_nu = -Delta_{-nu} by time-reversal and C5 symmetry, so the self-conjugate sector nu=0 has Delta_0=0 and therefore a linear crossing (Eq. (6)). The equivalence between the C5=-1 block of the full nanowire and the single-TP shell is derived in Appendix B (Eqs. (B20)-(B27)) rather than assumed. The exponential fit Eq. (8) quantifies the core-surface anticrossing versus thickness; the 50 nm result is a direct calculation, not a fitted quantity renamed as a prediction. The only self-citation of note is Ref. [33] (by two of the present authors), used for the p3 parameter set and for the earlier cationic/anionic TP classification; the paper re-derives the TP-sublattice distinction in its own single-TP and full-nanowire spectra, so the citation is corroborative and not load-bearing. The approximate treatment of closure strain (deformation potentials estimated at 10-20% and dropped) is a modeling assumption and a robustness risk, but it is not a circular step: the prediction would fail if the gap reordered, but the existence of the modes is computed from the stated Hamiltonian, not imported as the conclusion. Overall, no step reduces by construction to its input.

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

The existence argument for the Dirac crossings is essentially symmetry-only: given five helical channels and the stated symmetry representations, Delta_0 = 0 forces a crossing in the nu = 0 sector. The claim therefore does not rest on fitted constants. What the claim does rest on are inherited inputs: the inverted bulk band structure, the twin-plane topological classification of Ref [33], the idealized wire geometry with strain corrections dropped, and the single-particle TB description. The free parameters listed are either fitted for the finite-size scaling statement or chosen by hand for the perturbed control calculations.

free parameters (4)

Anticrossing decay rate gamma = 0.1065 nm^{-1}
Fit of Delta_ac proportional to exp(-gamma d) to TB results at 14, 21, 36, and 50 nm (Eq. 8); used to argue that core and surface modes decouple at about 50 nm.
Core-region onsite shift (anionic-TP NW) = 40 meV within 1 nm of the axis
Applied in Fig. 3(b) to move a trivial core subband away from the small surface gap; the unshifted spectrum is not shown.
Surface perturbation magnitudes (Figs. 5 and 12) = not specified (weak electrostatic potentials of different magnitudes)
Hand-chosen potentials used to expose the five TP-edge channels and the ten Z-sector channels; values are not stated, so exact reproduction is not possible.
Tight-binding parameter sets (p3 and sp3d5) = bulk values; virtual-crystal average for the alloy
Inherited from Refs [33] and [38]; the central claim depends on these band-structure inputs, but they are not fitted in this paper.

assumptions (5)

domain assumption SnTe and Pb0.4Sn0.6Te have inverted bulk band gaps (TCI regime) with the band ordering used in the tight-binding models.
Taken from prior literature (Refs [5,7,8,38]); the entire calculation assumes this inverted ordering, and the PbTe control (Fig. 7e) is used to show the modes require it.
domain assumption Cationic and anionic (111) twin planes have the distinct topological characters assigned in Ref [33]: cationic twin planes bind protected side-surface crossings near the Gamma analogue, anionic ones do not.
The paper re-derives the single-twin-plane physics in its own shell calculations (Fig. 4), partially self-contained, but the classification language and the expectation that this carries over to the nanowire originate in the authors' prior paper.
ad hoc to paper The roughly 2% wedge deformation used to close the pentagonal cross section, with no deformation-potential, Poisson, or bond-length corrections, preserves the inverted band ordering.
Sec. II estimates valley-dependent gap changes below 10% for SnTe and 20% for the alloy and drops the corrections without explicit calculation; Sec. III states all parameters are retained at bulk values.
domain assumption The five twin planes meet at a single uncompensated cationic (or anionic) atomic column, and the electronic structure is described by a single-particle tight-binding Hamiltonian.
Sec. II adopts the idealized occupied-core structure based on an approximately C5-symmetric TEM image in Ref [30]; Appendix A and D test variants (hollow core, relaxed stoichiometric core) but only within the same idealized framework.
standard math Spinful rotation acts with U(2pi) = -I, and the C5 block decomposition of Appendix B is valid for the nanowire Hamiltonian.
Standard representation theory for spin-1/2 systems; the decomposition is numerically verified in the spectra of Fig. 9.

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Pith. "Pith review of One-dimensional Dirac modes in the core of a pentagonal topological crystalline insulator nanowire." pith.science (2026). https://pith.science/paper/SMWCVY6T

@misc{pith2026260805908,
  author       = {Pith},
  title        = {Pith review of: One-dimensional Dirac modes in the core of a pentagonal topological crystalline insulator nanowire},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SMWCVY6T}},
  note         = {Machine review of arXiv:2608.05908}
}

abstract

We investigate the electronic band topology of recently fabricated pentagonal IV-VI semiconductor nanowires, which contain five radial $\{111\}$ twin planes meeting at the nanowire axis. Tight-binding calculations show that, when the bulk band structure is inverted and the twin planes in the nanowire are cationic, the spectrum contains two spatially separated helical Dirac crossings near $\overline{\Gamma}$: one localized at the core and the other at the outer surface. When the twin planes are anionic, the corresponding spectra remain gapped. The crossings originate from the hybridization of five helical channels associated with the twin-plane edges, whose odd number leaves one Kramers pair near the nanowire axis and the other at the outer boundary. Realistic multiorbital calculations for $\mathrm{Pb}_{0.4}\mathrm{Sn}_{0.6}\mathrm{Te}$ predict well-developed core and surface modes at nanowire thicknesses of approximately 50~nm and above. These results establish pentagonal SnTe-class nanowires as an experimentally accessible realization of spatially separated helical channels bound to the axial defect and the outer boundary.

Figures

Figures reproduced from arXiv: 2608.05908 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Band structures and wave-function localization in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: FIG. 4. Band structures of cylindrical SnTe shells containing [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Band structures of pentagonal SnTe NWs after apply [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Low-energy spectrum and spin structure of a NW with cationic TPs. (a) Representative spectrum of the symmetry [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Band structures and finite-size scaling obtained with the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Decomposition of the pentagonal NW, shown in cross [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. Rotationally resolved band structures of pentagonal [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: FIG. 10. Closed-surface origin of the gap near [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

**Figure 11.** Figure 11: FIG. 11. Mirror-related helical channels near [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: FIG. 12. Band structures of pentagonal SnTe NWs after ap [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

**Figure 13.** Figure 13: FIG. 13. Mirror-resolved spectra of the doubled low-energy [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]

**Figure 14.** Figure 14: FIG. 14. Geometric interpretation of the mirror-resolved chiralities near [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

**Figure 15.** Figure 15: FIG. 15. Band structures near [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]

**Figure 16.** Figure 16: FIG. 16. Thickness dependence of the binding energy of [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]

**Figure 17.** Figure 17: FIG. 17. Atomic structures and electronic band structures of four-ring pentagonal SnTe NWs with cationic TPs. Panels (a), [PITH_FULL_IMAGE:figures/full_fig_p022_17.png]

**Figure 18.** Figure 18: FIG. 18. Binding energies of non-stoichiometric pentagonal [PITH_FULL_IMAGE:figures/full_fig_p023_18.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [30]

W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole mo- ment pumping, and chiral hinge states in crys- talline insulators, Phys. Rev. B96, 245115 (2017), arXiv:1708.04230

arXiv 2017

[2] [1]

We label these lattice sites byn= 0,

Off-axis orbital content of theC 5 =−1sector Every off-axis atomic site belongs to an orbit of five sites related by successiveC 5 rotations. We label these lattice sites byn= 0, . . . ,4 in counterclockwise order when the cross section is viewed from the positive end of the NW axis toward the origin, so that an active coun- terclockwise rotation mapsnont...

work page

[3] [2]

,4 and ordered counterclockwise as shown in Fig

Block diagonalization of the pentagonal NW Hamiltonian We divide the pentagonal NW into five symmetry- related wedges, labeled by the indexn= 0, . . . ,4 and ordered counterclockwise as shown in Fig. 8. For clarity, we first present the block diagonalization for the case in which each wedge couples only to its two nearest neigh- boring wedges. The inclusi...

work page

[4] [3]

(B19) becomes H {0} NW =H 0 −T 01U5 −U † 5 T † 01.(B20) λ0 = −1 other λν C5 sectors NW with cat

Relation of theC 5 =−1sector to a single-TP shell Forν= 0, one hasλ 0 =−1, and Eq. (B19) becomes H {0} NW =H 0 −T 01U5 −U † 5 T † 01.(B20) λ0 = −1 other λν C5 sectors NW with cat. TPs NW with an. TPs (b) (a) FIG. 9. Rotationally resolved band structures of pentagonal SnTe NWs with (a) cationic and (b) anionic TPs. The spectra correspond to those shown in ...

work page

[5] [4]

The basis consists of five helical Kramers pairs ordered counterclockwise around the NW, with a two-component spinor attached to each channel

Projected symmetries of the five-channel model We finally derive the symmetry matrices used in the k·ptheory near Γ. The basis consists of five helical Kramers pairs ordered counterclockwise around the NW, with a two-component spinor attached to each channel. The labeling is chosen so thatC 5 maps channelnonto n+ 1 and channeln= 0 lies in the mirror plane...

work page

[6] [5]

Closed-surface quantization We begin with a defect-free analogue of the single-TP shell, obtained by wrapping a SnTe (100) slab around the

work page

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Ando and L

Y. Ando and L. Fu, Topological crystalline insulators and topological superconductors: From concepts to materi- als, Annu. Rev. Condens. Matter Phys.6, 361 (2015), arXiv:1501.00531

arXiv 2015

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Mirror-related TP-edge and face-midline channels The same gapped spectrum admits a complementary real-space interpretation. The cylindrical shell contain- ing one TP has two mirror planes relevant near Z: the mirror plane coinciding with the TP and a regular (01 ¯1) mirror plane on the opposite side of the shell. Consider a time-reversal-preserving surfac...

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