Rashba Spin-Orbit Driven Topological Phase Transitions in Graphene Nanoribbon Heterostructures

Hao-Ru Wu; Hong-Yi Chen; Jhih-Shih You; Yiing-Rei Chen

arxiv: 2602.19568 · v3 · submitted 2026-02-23 · ❄️ cond-mat.mes-hall

Rashba Spin-Orbit Driven Topological Phase Transitions in Graphene Nanoribbon Heterostructures

Hao-Ru Wu , Jhih-Shih You , Yiing-Rei Chen , Hong-Yi Chen This is my paper

Pith reviewed 2026-05-15 20:40 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords graphene nanoribbonsRashba spin-orbit couplingtopological phase transitionsinterface statesheterostructurestopological statesspin-orbit interaction

0 comments

The pith

Increasing Rashba spin-orbit coupling in armchair graphene nanoribbon heterostructures drives a topological phase transition that creates symmetry-protected interface states.

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

The paper examines an armchair graphene nanoribbon with a central segment under Rashba spin-orbit coupling placed between two pristine segments. Raising the Rashba strength produces a gap closing followed by reopening, which marks a topological phase change. This change generates localized states at the two junctions that stay protected by symmetry and survive edge imperfections. The setup shows how spin-orbit interaction alone, without any lattice alteration, can switch the system into a topologically nontrivial regime in finite-width ribbons.

Core claim

The interplay between structural geometry and Rashba spin-orbit coupling generates nontrivial topological phases in honeycomb nanoribbon heterostructures. Increasing the Rashba coupling induces symmetry-protected interface states localized at the junction between topologically distinct regions, which remain robust against edge perturbations. For finite ribbon widths, Rashba spin-orbit coupling drives a gap closing and reopening, signaling a topological phase transition without modifying the lattice structure.

What carries the argument

Rashba spin-orbit coupling applied only to a central embedded region of an armchair nanoribbon heterostructure, which changes the topological character and produces symmetry-protected interface states at the boundaries.

If this is right

Symmetry-protected interface states form at the junctions and remain robust against edge perturbations.
A gap closes and reopens at finite ribbon widths, marking the topological phase transition.
Topological states become tunable through the strength of the spin-orbit interaction alone.
Interfacial geometry cooperates with spin-orbit coupling to engineer the states without lattice modification.

Where Pith is reading between the lines

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

Gate voltages that induce Rashba coupling could provide all-electrical switching of topological features in graphene devices.
The same embedding strategy might produce analogous transitions in other two-dimensional materials with strong spin-orbit effects.
The protected interface states could serve as channels for dissipationless transport in future nanoelectronic circuits.
Transport measurements of conductance plateaus or local density of states at the junctions would test the predicted robustness.

Load-bearing premise

A Rashba spin-orbit coupled region can be cleanly embedded between pristine segments with no extra scattering or lattice distortion at the interfaces.

What would settle it

Direct measurement showing no gap closing and reopening when Rashba strength is increased in finite-width ribbons, or interface states that disappear under small edge disorder, would disprove the topological transition.

Figures

Figures reproduced from arXiv: 2602.19568 by Hao-Ru Wu, Hong-Yi Chen, Jhih-Shih You, Yiing-Rei Chen.

**Figure 1.** Figure 1: The Hamiltonian of the P-R-P heterostructure is 𝐻̂ = ∑𝑡0𝑐̂ 𝑖𝛼 † 𝑐̂𝑗𝛼 〈𝑖,𝑗〉 𝛼 + ∑ 𝑖𝑡𝑅𝑐̂ 𝑖𝛼 † 𝒛 ⋅ (𝝈𝛼𝛽 × 𝜹𝑖𝑗)𝑐𝑗𝛽 〈𝑖,𝑗〉∈RGNR 𝛼𝛽 + h. c. , where 〈𝑖,𝑗〉 includes all configurations of the nearestneighbor sites, 𝛼 and 𝛽 are the spin indices, 𝑐𝑖𝛼 † (𝑐𝑖𝛼) is the creation (annihilation) operator with spin 𝛼 at site 𝑖, 𝑡0 is the nearest-neighbor hopping strength, 𝑡𝑅 is the Rashba SOC strength, 𝝈 are the Pauli matric… view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a)(c) Energy levels for [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. (Color online) (a). The spatial probability distribution [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online) (a) [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (Color online) The winding number phase [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

read the original abstract

We demonstrate that the interplay between structural geometry and Rashba spin-orbit coupling generates nontrivial topological phases in honeycomb nanoribbon heterostructures. We consider an armchair nanoribbon in which a Rashba spin-orbit coupled region is embedded between pristine segments. Increasing the Rashba coupling induces symmetry-protected interface states localized at the junction between topologically distinct regions, which remain robust against edge perturbations. For finite ribbon widths, Rashba spin-orbit coupling drives a gap closing and reopening, signaling a topological phase transition without modifying the lattice structure. Our results reveal a mechanism by which interfacial geometry and spin-orbit interaction cooperatively engineer tunable topological states in graphene-based nanostructures.

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

Rashba coupling closes the gap in these nanoribbon junctions but the topological protection claim rests only on spectral behavior without an invariant check.

read the letter

The paper models an armchair graphene nanoribbon with a central segment under Rashba spin-orbit coupling placed between two pristine segments. As the Rashba strength grows, the bulk gap closes and reopens while localized states form at the two junctions. The authors present this as a geometry-driven topological transition that leaves the lattice unchanged and produces states robust to edge perturbations. The setup is clean and the tight-binding plus Rashba Hamiltonian is standard, so the numerics for different ribbon widths are straightforward to follow and reproduce in principle. That part of the work is solid and incremental in a useful way for people already thinking about spin-orbit effects in carbon nanostructures. The main weakness is that gap closing and reopening by itself does not establish a change in topological class. In a one-dimensional gapped system the two sides must differ by a topological invariant such as the Zak phase or a parity index; without that calculation the interface states could simply be ordinary bound states from the step in the potential. The abstract and the stress-test note give no indication that this invariant was computed, so the claim of symmetry-protected topological states is not yet secured. The model also assumes perfectly clean interfaces with no extra scattering or lattice relaxation, which is fine for theory but narrows how directly the results map to devices. This is aimed at specialists in topological graphene nanoribbons who already know the Rashba literature. It is a modest extension rather than a new framework, but the numerics are concrete enough that a serious referee could usefully ask for the missing invariant and a few robustness checks. I would send it to peer review rather than desk-reject it.

Referee Report

1 major / 1 minor

Summary. The manuscript studies armchair graphene nanoribbon heterostructures consisting of a central Rashba spin-orbit coupled segment embedded between pristine segments. It claims that increasing Rashba coupling strength induces symmetry-protected interface states localized at the junctions between topologically distinct regions; these states remain robust to edge perturbations. For finite ribbon widths, Rashba SOC is asserted to drive a gap closing and reopening that signals a topological phase transition without any modification to the underlying lattice structure.

Significance. If the central claims hold after verification, the results would demonstrate a lattice-preserving route to engineer tunable topological interface states in graphene nanoribbons via Rashba SOC alone. This mechanism could be relevant for spintronic or quantum-transport applications in 2D carbon nanostructures, building on standard tight-binding models with Rashba terms.

major comments (1)

[Abstract] Abstract: the assertion that gap closing and reopening signals a topological phase transition is not supported by any explicit computation of a topological invariant (Zak phase, parity index, or equivalent) that would place the pristine and Rashba-coupled segments in distinct topological classes. In a 1D gapped system, gap closing alone does not establish a change in topological character or guarantee protected interface states.

minor comments (1)

The abstract and introduction should specify the tight-binding parameters (nearest-neighbor hopping, Rashba strength range, ribbon widths considered) and the numerical method used for the band-structure calculations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate the suggested clarification in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that gap closing and reopening signals a topological phase transition is not supported by any explicit computation of a topological invariant (Zak phase, parity index, or equivalent) that would place the pristine and Rashba-coupled segments in distinct topological classes. In a 1D gapped system, gap closing alone does not establish a change in topological character or guarantee protected interface states.

Authors: We agree that an explicit calculation of a topological invariant is required to rigorously confirm the topological character of the two segments. The original manuscript infers the transition from the gap closing/reopening together with the appearance of symmetry-protected interface states, but does not report a direct invariant computation. In the revised manuscript we will add calculations of the Zak phase (or parity index) for both the pristine and Rashba-coupled armchair nanoribbons, explicitly demonstrating that they belong to distinct topological classes. This addition will strengthen the claim that the observed gap closing signals a genuine topological phase transition. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via standard tight-binding spectral analysis

full rationale

The paper constructs a tight-binding Hamiltonian for armchair graphene nanoribbon heterostructures that includes a position-dependent Rashba spin-orbit term between pristine segments. It numerically diagonalizes this model for finite widths, tracks the evolution of the bulk gap with increasing Rashba strength, and notes gap closing/reopening together with the appearance of interface-localized states. None of these steps defines any output quantity in terms of the claimed topological transition itself, nor renames a fitted parameter as a prediction, nor relies on a self-citation chain whose only justification is the present work. The central inference therefore rests on direct computation from the model rather than on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard tight-binding Hamiltonian for graphene plus a Rashba term; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard nearest-neighbor tight-binding model for graphene nanoribbons with added Rashba spin-orbit term
Invoked implicitly to describe the electronic structure and gap closing.

pith-pipeline@v0.9.0 · 5414 in / 1280 out tokens · 42433 ms · 2026-05-15T20:40:45.219492+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

?

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

winding number W = ∫ dk/2πi Tr[h_k^{-1} ∂_k h_k] ... changes from W=2 to W=4
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Rashba SOC drives gap closing and reopening ... without modifying the lattice structure

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

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

Higher Education Sprout Project

By tuning the strength of Rashba SOC 𝑡𝑅 on the embedded region, the winding number can change from 𝑊 = 2 to 𝑊 = 4 . Thus, the difference of winding number between two regions gives rise to the boundary states localized at the interface (shown in Fig. 2) . According to bulk-edge correspondence s, this winding number difference leads to the emergence of fou...

work page
[2]

Ching-Kai Chiu, Jeffrey C. Y. Teo, Andreas P. Schnyder, and Shinsei Ryu , Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016)

work page 2016
[3]

Alexei Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009)

work page 2009
[4]

Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J

Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas W.W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010)

work page 2010
[5]

Poddubny, and Shanhui Fan, Topological Nature of Edge States for One -Dimensional Systems without Symmetry Protection, Phys

Janet Zhong, Heming Wang, Alexander N. Poddubny, and Shanhui Fan, Topological Nature of Edge States for One -Dimensional Systems without Symmetry Protection, Phys. Rev. Lett. 135, 016601 (2025)

work page 2025
[6]

H. C. Wu and L. Jin, Topological zero modes from the interplay of PT symmetry and anti -PT symmetry, Phys. Rev. B 112 , 125146 (2025)

work page 2025
[7]

Yunlin Li, Yufu Liu, Xuezhi Wang, Haoran Zhang, Xunya Jiang, Nontrivial topology in one - and two -dimensional asymmetric systems with chiral boundary states, arXiv:2509.20834

work page arXiv
[8]

Anhua Huang, Shasha Ke, Ji-Huan Guan, Jun Li, and Wen-Kai Lou , Strain-induced topological phase transition in graphene nanoribbons, Phys. Rev. B 109, 045408 (2024)

work page 2024
[9]

Louie, Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains, Phys

Ting Cao, Fangzhou Zhao, and Steven G. Louie, Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains, Phys. Rev. Lett. 119, 076401 (2017)

work page 2017
[10]

Louie, Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons, Nano Lett

Jingwei Jiang and Steven G. Louie, Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons, Nano Lett. 2021, 21, 1, 197–202

work page 2021
[11]

Giessibl, Hiroshi Sakaguchi, Steven G

Shaotang Song, Yu Teng, Weichen Tang, Zhen Xu, Yuanyuan He, Jiawei Ruan, Takahiro Kojima, Wenping Hu, Franz J. Giessibl, Hiroshi Sakaguchi, Steven G. Louie & Jiong Lu, Janus graphene nanoribbons with localized states on a single zigzag edge, Nature 637, 580–586 (2025)

work page 2025
[12]

2018, 18, 11, 7254–7260

Kuan-Sen Lin, and Mei -Yin Chou, Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries , Nano Lett. 2018, 18, 11, 7254–7260

work page 2018
[13]

Louie, Topological Phases in Cove- Edged and Chevron Graphene Nanoribbons: Geometric Structures, Z2 Invariants, and Junction States, Nano Lett

Yea-Lee Lee, Fangzhou Zhao, Ting Cao, Jisoon Ihm, Steven G. Louie, Topological Phases in Cove- Edged and Chevron Graphene Nanoribbons: Geometric Structures, Z2 Invariants, and Junction States, Nano Lett. 2018, 18, 11, 7247–7253

work page 2018
[14]

Jianing Wang, Weiwei Chen, Zhengya Wang, Jie Meng, Ruoting Yin, Miaogen Chen, Shijing Tan, Chuanxu Ma, Qunxiang Li, and Bing Wang, Designer topological flat bands in one-dimensional armchair graphene antidot lattices, Phys. Rev. B 110, 115138 (2024)

work page 2024
[15]

Louie , Topological Phases in Graphene Nanoribbons Tuned by Electric Fields, Phys

Fangzhou Zhao, Ting Cao, and Steven G. Louie , Topological Phases in Graphene Nanoribbons Tuned by Electric Fields, Phys. Rev. Lett. 127, 166401 (2021)

work page 2021
[16]

Phys.: Condens

David M T Kuo, Topological states in finite graphene nanoribbons tuned by electric fields, J. Phys.: Condens. Matter 37, 085304 (2025)

work page 2025
[17]

Yi-Xin Xiao, Guancong Ma, Zhao-Qing Zhang, and C. T. Chan, Topological Subspace -Induced Bound State in the Continuum, Phys. Rev. Lett. 118, 166803 (2017)

work page 2017
[18]

Varykhalov, J

A. Varykhalov, J. Sánchez -Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader, Electronic and Magnetic Properties of Quasifreestanding Graphene on Ni, Phys. Rev. Lett. 101, 157601 (2008)

work page 2008
[19]

Y. A. Bychkov and E. I. Rashba, Properties of a 2D Electron Gas with Lifted Spectral Degeneracy, JETP Lett. 39, 78-83 (1984)

work page 1984
[20]

Bowen Yang, Mark Lohmann, David Barroso, Ingrid Liao, Zhisheng Lin, Yawen Liu, Ludwig Bartels, Kenji Watanabe, Takashi Taniguchi, and Jing Shi, Strong electron -hole symmetric Rashba spin -orbit coupling in graphene/monolayer transition metal dichalcogenide heterostructures, Phys. Rev. B 96, 041409(R) (2017)

work page 2017
[21]

Inglot, V

M. Inglot, V. K. Dugaev, A. Dyrdał, and J. Barnaś, Graphene with Rashba spin -orbit interaction and coupling to a magnetic layer: Electron states localized at the domain wall, Phys. Rev. B 104, 214408 (2021)

work page 2021
[22]

Poszwa, Spin orbit coupling effect on coherent transport properties of graphene nanoscopic rings in external magnetic field , Scientific Reports 14, 31554 (2024)

A. Poszwa, Spin orbit coupling effect on coherent transport properties of graphene nanoscopic rings in external magnetic field , Scientific Reports 14, 31554 (2024)

work page 2024
[23]

Huan Zhang, Zhongshui Ma & Jun-Feng Liu, Equilibrium spin current in graphene with Rashba spin-orbit coupling, Scientific Reports, 4, 6464 (2014)

work page 2014
[24]

C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005). 6

work page 2005
[25]

C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005)

work page 2005
[26]

Topological phases, topological flat bands, and topological excitations in a one -dimensional dimerized lattice with spin -orbit coupling

Zhongbo Yan & Shaolong Wan (2014). Topological phases, topological flat bands, and topological excitations in a one -dimensional dimerized lattice with spin -orbit coupling. Europhysics Letters, 107(4), 47007

work page 2014
[27]

Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice

Masoud Bahari and Mir Vahid Hosseini (2016). Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice. Physical Review B, 94(12), 125119

work page 2016
[28]

Entin -Wohlman, A

Zhi-Hai Liu, O. Entin -Wohlman, A. Aharony, J. Q. You, and H. Q. Xu (2021). Topological states and interplay between spin -orbit and Zeeman interactions in a spinful Su -Schrieffer-Heeger nanowire. Physical Review B, 104(8), 085302

work page 2021
[29]

Kruk, and Yuri Kivshar, Metasurface-Controlled Photonic Rashba Effect for Upconversion Photoluminescence

Aditya Tripathi, Anastasiia Zalogina, Jiayan Liao, Matthias Wurdack, Eliezer Estrecho, Jiajia Zhou, Dayong Jin, Sergey S. Kruk, and Yuri Kivshar, Metasurface-Controlled Photonic Rashba Effect for Upconversion Photoluminescence. Nano Lett. 2023, 23, 6, 2228–2232

work page 2023
[30]

Jingyi Tian, Giorgio Adamo, Hailong Liu, Maciej Klein, Song Han, Hong Liu, and Cesare Soci, Optical Rashba Effect in a Light -Emitting Perovskite Metasurface, Advanced Materials 34, 2109157 (2022)

work page 2022
[31]

Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri, Dekel Veksler, Vladimir Kleiner, and Erez Hasman, Spin -Optical Metamaterial Route to Spin-Controlled Photonics

N. Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri, Dekel Veksler, Vladimir Kleiner, and Erez Hasman, Spin -Optical Metamaterial Route to Spin-Controlled Photonics. Science, 340, 724 -726 (2013)

work page 2013
[32]

Nature Nanotechnology, 15, 927 -933 (2020)

Kexiu Rong, Bo Wang, Avi Reuven, Elhanan Maguid, Bar Cohn, Vladimir Kleiner, Shaul Katznelson, Elad Koren & Erez Hasman, Photonic Rashba effect from quantum emitters mediated by a Berry -phase defective photonic crystal. Nature Nanotechnology, 15, 927 -933 (2020)

work page 2020
[33]

Advanced Optical Materials, 13, 2500152 (2025)

Qi Zhong, Yongsheng Liang, Shiqi Xia, Daohong Song, and Zhiang Chen, Observation of Topological Armchair Edge States in Photonic Biphenylene Network. Advanced Optical Materials, 13, 2500152 (2025)

work page 2025
[34]

Shiqi Xia, Yongsheng Liang, Liqin Tang, Daohong Song, Jingjun Xu, and Zhigang Chen, Photonic Realization of a Generic Type of Graphene Edge States Exhibiting Topological Flat Band. Phy s. Rev. Lett. 131, 013804 (2023)

work page 2023
[35]

Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence

Kyoto Nakada, Mitsutaka Fujita, Gene Dresselhaus, and Mildred S. Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Phys . Rev. B 54, 17954 (1996)

work page 1996
[36]

Brey, and H

L. Brey, and H. A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73, 235411 (2006)

work page 2006
[37]

Katsunori Wakabayashi, Ken-ichi Sasaki, Takeshi Nakanishi and Toshiaki Enoki, Electronic states of graphene nanoribbons and analytical solutions. Sci. Technol. Adv. Mater. 11, 054504 (2010)

work page 2010
[38]

Maria Maffei, Alexandre Dauphin, Filippo Cardano, Maciej Lewenstein and Pietro Massignan, Topological characterization of chiral models through their long -time dynamics. New J. Phys. 20, 013023 (2018)

work page 2018
[39]

Mahdi Zarea and Nancy Sandler, Rashba spin - orbit interaction in graphene and zigzag nanoribbons. Phys. Rev. B 79, 165442 (2009)

work page 2009

[1] [1]

Higher Education Sprout Project

By tuning the strength of Rashba SOC 𝑡𝑅 on the embedded region, the winding number can change from 𝑊 = 2 to 𝑊 = 4 . Thus, the difference of winding number between two regions gives rise to the boundary states localized at the interface (shown in Fig. 2) . According to bulk-edge correspondence s, this winding number difference leads to the emergence of fou...

work page

[2] [2]

Ching-Kai Chiu, Jeffrey C. Y. Teo, Andreas P. Schnyder, and Shinsei Ryu , Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016)

work page 2016

[3] [3]

Alexei Kitaev, Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134, 22 (2009)

work page 2009

[4] [4]

Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J

Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas W.W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12, 065010 (2010)

work page 2010

[5] [5]

Poddubny, and Shanhui Fan, Topological Nature of Edge States for One -Dimensional Systems without Symmetry Protection, Phys

Janet Zhong, Heming Wang, Alexander N. Poddubny, and Shanhui Fan, Topological Nature of Edge States for One -Dimensional Systems without Symmetry Protection, Phys. Rev. Lett. 135, 016601 (2025)

work page 2025

[6] [6]

H. C. Wu and L. Jin, Topological zero modes from the interplay of PT symmetry and anti -PT symmetry, Phys. Rev. B 112 , 125146 (2025)

work page 2025

[7] [7]

Yunlin Li, Yufu Liu, Xuezhi Wang, Haoran Zhang, Xunya Jiang, Nontrivial topology in one - and two -dimensional asymmetric systems with chiral boundary states, arXiv:2509.20834

work page arXiv

[8] [8]

Anhua Huang, Shasha Ke, Ji-Huan Guan, Jun Li, and Wen-Kai Lou , Strain-induced topological phase transition in graphene nanoribbons, Phys. Rev. B 109, 045408 (2024)

work page 2024

[9] [9]

Louie, Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains, Phys

Ting Cao, Fangzhou Zhao, and Steven G. Louie, Topological Phases in Graphene Nanoribbons: Junction States, Spin Centers, and Quantum Spin Chains, Phys. Rev. Lett. 119, 076401 (2017)

work page 2017

[10] [10]

Louie, Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons, Nano Lett

Jingwei Jiang and Steven G. Louie, Topology Classification using Chiral Symmetry and Spin Correlations in Graphene Nanoribbons, Nano Lett. 2021, 21, 1, 197–202

work page 2021

[11] [11]

Giessibl, Hiroshi Sakaguchi, Steven G

Shaotang Song, Yu Teng, Weichen Tang, Zhen Xu, Yuanyuan He, Jiawei Ruan, Takahiro Kojima, Wenping Hu, Franz J. Giessibl, Hiroshi Sakaguchi, Steven G. Louie & Jiong Lu, Janus graphene nanoribbons with localized states on a single zigzag edge, Nature 637, 580–586 (2025)

work page 2025

[12] [12]

2018, 18, 11, 7254–7260

Kuan-Sen Lin, and Mei -Yin Chou, Topological Properties of Gapped Graphene Nanoribbons with Spatial Symmetries , Nano Lett. 2018, 18, 11, 7254–7260

work page 2018

[13] [13]

Louie, Topological Phases in Cove- Edged and Chevron Graphene Nanoribbons: Geometric Structures, Z2 Invariants, and Junction States, Nano Lett

Yea-Lee Lee, Fangzhou Zhao, Ting Cao, Jisoon Ihm, Steven G. Louie, Topological Phases in Cove- Edged and Chevron Graphene Nanoribbons: Geometric Structures, Z2 Invariants, and Junction States, Nano Lett. 2018, 18, 11, 7247–7253

work page 2018

[14] [14]

Jianing Wang, Weiwei Chen, Zhengya Wang, Jie Meng, Ruoting Yin, Miaogen Chen, Shijing Tan, Chuanxu Ma, Qunxiang Li, and Bing Wang, Designer topological flat bands in one-dimensional armchair graphene antidot lattices, Phys. Rev. B 110, 115138 (2024)

work page 2024

[15] [15]

Louie , Topological Phases in Graphene Nanoribbons Tuned by Electric Fields, Phys

Fangzhou Zhao, Ting Cao, and Steven G. Louie , Topological Phases in Graphene Nanoribbons Tuned by Electric Fields, Phys. Rev. Lett. 127, 166401 (2021)

work page 2021

[16] [16]

Phys.: Condens

David M T Kuo, Topological states in finite graphene nanoribbons tuned by electric fields, J. Phys.: Condens. Matter 37, 085304 (2025)

work page 2025

[17] [17]

Yi-Xin Xiao, Guancong Ma, Zhao-Qing Zhang, and C. T. Chan, Topological Subspace -Induced Bound State in the Continuum, Phys. Rev. Lett. 118, 166803 (2017)

work page 2017

[18] [18]

Varykhalov, J

A. Varykhalov, J. Sánchez -Barriga, A. M. Shikin, C. Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O. Rader, Electronic and Magnetic Properties of Quasifreestanding Graphene on Ni, Phys. Rev. Lett. 101, 157601 (2008)

work page 2008

[19] [19]

Y. A. Bychkov and E. I. Rashba, Properties of a 2D Electron Gas with Lifted Spectral Degeneracy, JETP Lett. 39, 78-83 (1984)

work page 1984

[20] [20]

Bowen Yang, Mark Lohmann, David Barroso, Ingrid Liao, Zhisheng Lin, Yawen Liu, Ludwig Bartels, Kenji Watanabe, Takashi Taniguchi, and Jing Shi, Strong electron -hole symmetric Rashba spin -orbit coupling in graphene/monolayer transition metal dichalcogenide heterostructures, Phys. Rev. B 96, 041409(R) (2017)

work page 2017

[21] [21]

Inglot, V

M. Inglot, V. K. Dugaev, A. Dyrdał, and J. Barnaś, Graphene with Rashba spin -orbit interaction and coupling to a magnetic layer: Electron states localized at the domain wall, Phys. Rev. B 104, 214408 (2021)

work page 2021

[22] [22]

Poszwa, Spin orbit coupling effect on coherent transport properties of graphene nanoscopic rings in external magnetic field , Scientific Reports 14, 31554 (2024)

A. Poszwa, Spin orbit coupling effect on coherent transport properties of graphene nanoscopic rings in external magnetic field , Scientific Reports 14, 31554 (2024)

work page 2024

[23] [23]

Huan Zhang, Zhongshui Ma & Jun-Feng Liu, Equilibrium spin current in graphene with Rashba spin-orbit coupling, Scientific Reports, 4, 6464 (2014)

work page 2014

[24] [24]

C. L. Kane and E. J. Mele, Quantum Spin Hall Effect in Graphene, Phys. Rev. Lett. 95, 226801 (2005). 6

work page 2005

[25] [25]

C. L. Kane and E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, 146802 (2005)

work page 2005

[26] [26]

Topological phases, topological flat bands, and topological excitations in a one -dimensional dimerized lattice with spin -orbit coupling

Zhongbo Yan & Shaolong Wan (2014). Topological phases, topological flat bands, and topological excitations in a one -dimensional dimerized lattice with spin -orbit coupling. Europhysics Letters, 107(4), 47007

work page 2014

[27] [27]

Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice

Masoud Bahari and Mir Vahid Hosseini (2016). Zeeman-field-induced nontrivial topological phases in a one-dimensional spin-orbit-coupled dimerized lattice. Physical Review B, 94(12), 125119

work page 2016

[28] [28]

Entin -Wohlman, A

Zhi-Hai Liu, O. Entin -Wohlman, A. Aharony, J. Q. You, and H. Q. Xu (2021). Topological states and interplay between spin -orbit and Zeeman interactions in a spinful Su -Schrieffer-Heeger nanowire. Physical Review B, 104(8), 085302

work page 2021

[29] [29]

Kruk, and Yuri Kivshar, Metasurface-Controlled Photonic Rashba Effect for Upconversion Photoluminescence

Aditya Tripathi, Anastasiia Zalogina, Jiayan Liao, Matthias Wurdack, Eliezer Estrecho, Jiajia Zhou, Dayong Jin, Sergey S. Kruk, and Yuri Kivshar, Metasurface-Controlled Photonic Rashba Effect for Upconversion Photoluminescence. Nano Lett. 2023, 23, 6, 2228–2232

work page 2023

[30] [30]

Jingyi Tian, Giorgio Adamo, Hailong Liu, Maciej Klein, Song Han, Hong Liu, and Cesare Soci, Optical Rashba Effect in a Light -Emitting Perovskite Metasurface, Advanced Materials 34, 2109157 (2022)

work page 2022

[31] [31]

Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri, Dekel Veksler, Vladimir Kleiner, and Erez Hasman, Spin -Optical Metamaterial Route to Spin-Controlled Photonics

N. Shitrit, Igor Yulevich, Elhanan Maguid, Dror Ozeri, Dekel Veksler, Vladimir Kleiner, and Erez Hasman, Spin -Optical Metamaterial Route to Spin-Controlled Photonics. Science, 340, 724 -726 (2013)

work page 2013

[32] [32]

Nature Nanotechnology, 15, 927 -933 (2020)

Kexiu Rong, Bo Wang, Avi Reuven, Elhanan Maguid, Bar Cohn, Vladimir Kleiner, Shaul Katznelson, Elad Koren & Erez Hasman, Photonic Rashba effect from quantum emitters mediated by a Berry -phase defective photonic crystal. Nature Nanotechnology, 15, 927 -933 (2020)

work page 2020

[33] [33]

Advanced Optical Materials, 13, 2500152 (2025)

Qi Zhong, Yongsheng Liang, Shiqi Xia, Daohong Song, and Zhiang Chen, Observation of Topological Armchair Edge States in Photonic Biphenylene Network. Advanced Optical Materials, 13, 2500152 (2025)

work page 2025

[34] [34]

Shiqi Xia, Yongsheng Liang, Liqin Tang, Daohong Song, Jingjun Xu, and Zhigang Chen, Photonic Realization of a Generic Type of Graphene Edge States Exhibiting Topological Flat Band. Phy s. Rev. Lett. 131, 013804 (2023)

work page 2023

[35] [35]

Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence

Kyoto Nakada, Mitsutaka Fujita, Gene Dresselhaus, and Mildred S. Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Phys . Rev. B 54, 17954 (1996)

work page 1996

[36] [36]

Brey, and H

L. Brey, and H. A. Fertig, Electronic states of graphene nanoribbons studied with the Dirac equation, Phys. Rev. B 73, 235411 (2006)

work page 2006

[37] [37]

Katsunori Wakabayashi, Ken-ichi Sasaki, Takeshi Nakanishi and Toshiaki Enoki, Electronic states of graphene nanoribbons and analytical solutions. Sci. Technol. Adv. Mater. 11, 054504 (2010)

work page 2010

[38] [38]

Maria Maffei, Alexandre Dauphin, Filippo Cardano, Maciej Lewenstein and Pietro Massignan, Topological characterization of chiral models through their long -time dynamics. New J. Phys. 20, 013023 (2018)

work page 2018

[39] [39]

Mahdi Zarea and Nancy Sandler, Rashba spin - orbit interaction in graphene and zigzag nanoribbons. Phys. Rev. B 79, 165442 (2009)

work page 2009