Measuring the non-Abelian Quantum Phase with the Algorithm of Quantum Phase Estimation

Chia-Hsiu Hsieh; Seng Ghee Tan; Son-Hsien Chen; Yen-Fu Chen; Yen-Lin Chen; Ying-Cheng Yang

REVIEW

Measuring the non-Abelian Quantum Phase with the Algorithm of Quantum Phase Estimation

Seng Ghee Tan, Son-Hsien Chen, Ying-Cheng Yang, Yen-Fu Chen, Yen-Lin Chen, Chia-Hsiu Hsieh

Quantum Physics · Submitted 9 Sep 2025 · 22 references

Pith's verdict: A plausible two-module QPE scheme for extracting a spin-orbit phase from measured amplitudes, with internally consistent algebra and matching Qiskit simulations, but the physical electron-to-qubit transfer is an unaddressed gap and the 'direct phase readout' claim is overstated.

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arXiv ID2509.07716 (version 2)

Pith IDAB64TTCJ, Pith's permanent record of this paper and its review

CitationChia-Hsiu Hsieh, Seng Ghee Tan, Son-Hsien Chen, Yen-Fu Chen, Yen-Lin Chen, Ying-Cheng Yang. "Measuring the non-Abelian Quantum Phase with the Algorithm of Quantum Phase Estimation." arXiv:2509.07716 (2025). https://pith.science/paper/AB64TTCJ

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Editor's note

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Dear colleague,

Quick take: this paper proposes a two-module QPE scheme (QPEV and QPEH) for extracting the non-Abelian phase theta of an electron traversing an L-shaped spin-orbit path without interferometry. The specific reconstruction formulas (Eqs. 16-17) are new relative to the group's earlier discrete-path papers, and the internal algebra is consistent. The Qiskit histograms match the analytic amplitudes for the chosen test angles, so the paper does what it says at the level of ideal qubit simulation.

What's genuinely good: the idea of using QPE to measure amplitudes in a projected basis, then combining those measurements to infer a phase, is reasonable and potentially useful. The authors are honest that the QPE readouts give squared amplitudes and that the phase is then computed, not directly read. They say this explicitly in the QPEV discussion. So the paper is not deceptive about that.

Where it gets shaky: the load-bearing assumption is that the electron state after the L-shaped path is exactly Ry(delta)Rx(-eta)|0> and that this state can be coherently loaded into the QPE qubit register. Neither is demonstrated. A real spin-orbit nanostructure will include corners, spin relaxation, stray dynamic phases, and a non-unitary loading/unloading step. If the state at the collector differs from the ideal spinor, the measured amplitudes don't correspond to C, S, |A| as defined, and theta from Eq. (16) is not the physical electron's phase. The paper doesn't address this gap beyond a schematic figure.

Also, the demonstration is self-consistent rather than independent: the Qiskit simulation uses the same unitary that defines the amplitudes, so it confirms the algebra, not the physical model. Missing code, error bars, and real-device numbers would be acceptable in a proposal paper if the limitations were stated, but the conclusion claims 'accuracy confirmed' and a 'direct interface' with quantum processors, which overreaches.

Bottom line: it's a plausible, clearly written protocol paper with a modest but real theoretical contribution. It is not a demonstration of measuring a physical non-Abelian phase. I'd send it to peer review because the protocol idea deserves scrutiny and could be useful to the spintronics/quantum metrology community, but the authors need to dial back the claims and explicitly flag the interface problem.

Referee report

Major objections (4)

[Quantum Phase Generator and Figures 2 and 5] The central claim that the electron's non-Abelian phase is measured requires a coherent transfer of the electron state from the collector quantum dot into the QPE qubit register, but no such interface is described. The Qiskit simulations apply the ideal gates Rx(−η), Ry(δ), Ry(δa), and Rx(ηa) directly to |0⟩, so they validate the algebra of the amplitudes but not the physical measurement chain, including loading, transfer, and readout. Please either specify a concrete transfer mechanism and its assumptions, or explicitly restate the contribution as an idealized protocol whose experimental realization is left open.
[QPE Measurements, Eqs. (16) and (17)] The QPE modules return squared amplitudes |C|², |S|², and |A|² only. Eq. (17) requires the signed product SC, and Eq. (16) requires the signed difference S−C, so the phase θ is not uniquely determined unless the signs of C and S are known or separately measured. For example, a path angle that changes the sign of C while preserving |C| changes the ratio (S−C)/(S+C) and hence changes θ, even though the QPE histograms are identical. Please add a procedure to fix the signs of C and S, or state and justify a parameter range in which C and S are guaranteed positive.
[Hamiltonian of Eq. (1) and the gate model of Eq. (3)] The Hamiltonian in Eq. (1) includes the kinetic term ℏ²k²/2m, which contributes a dynamical phase factor to the electron wavefunction. The gate model U = Ry(δ)Rx(−η) used in Eqs. (3)–(16) omits this factor, so the quantity θ extracted by the proposed procedure is not identified as the non-Abelian phase unless the kinetic contribution is explicitly canceled or is shown to be common to both paths and eliminated by the measurement protocol. Please address this omission.
[Conclusion and Figures 4 and 6] The conclusion states that accuracy is confirmed by Qiskit simulation, but the simulations implement the same unitary that defines the model amplitudes. They therefore demonstrate internal consistency of the algebra rather than agreement with a physical spin-orbit device. Please soften this claim or provide an independent check, such as comparing against a separately derived Hamiltonian evolution or against experimental data from a spin-orbit nanostructure.

Minor objections (5)

[QPE Measurements, Figure 4 caption] The text says the QPE circuits use 10 Hadamard gates, but the histogram labels are 5-bit strings such as 0.00010 and 0.11110; please clarify the number of ancilla qubits and the binary representation used in the figures.
[Eq. (2)] The definition arg(x + iy) ≡ tan⁻¹(y/x) should specify the quadrant, since tan⁻¹ alone is ambiguous for phases outside the range (−π/2, π/2).
[Abstract and Introduction] The phrase 'phase is measured by reading off of measurable qubit states' is imprecise; the QPE readout gives probabilities of ancilla eigenstates, from which amplitudes and then the inferred phase are obtained.
[QPE Measurements, Eq. (13)] The phases γ₁ and γ₂ are described as 'ready phase factors transparent to the QPE algorithm'; this should be clarified because QPE measures eigenvalues of the controlled unitary, not these state-dependent phase factors, and the distinction is central to why only amplitudes are extracted.
[General presentation] There are numerous typographical and stylistic errors, including the heading 'ABSTRACTS', 'ph as', 'as though', and 'First off'; a careful proofreading pass is needed.

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The paper

The paper claims

Two quantum phase estimation modules can extract the non-Abelian phase of an electron on an L-shaped spin-orbit path, eliminating the interferometer.

The reading

The paper claims that the non-Abelian quantum phase of an electron sent through an L-shaped spin-orbit path can be extracted from the output statistics of two quantum phase estimation (QPE) modules, with no interferometric apparatus. In the proposed protocol, one module measures the two path-1 amplitude coefficients $C$ and $S$, the other measures the overall magnitude $|A|$ after both legs, and the phase angle $\theta$ is then obtained from a closed trigonometric formula. Because all operations are standard quantum gates plus an inverse quantum Fourier transform, the measurement becomes a qubit readout problem compatible with gate-based quantum processors. The paper validates the amplitude-reading step with circuit-level simulations whose results match the analytic estimates.

What carries the argument

The machinery is a pair of QPE modules used as amplitude meters in a rotated measurement basis. QPEV applies controlled $R_y(\delta_a)$ gates to the path-1 state and an inverse quantum Fourier transform, so that the IQFT histogram records $C^2=\cos^2(\pi/4-\eta/2)$ and $S^2=\sin^2(\pi/4-\eta/2)$ at the eigenvalues $\pm\delta_a/2$; here the auxiliary angle $\delta_a$ is only a readout location, not part of the physical evolution. QPEH applies controlled $R_x(\eta_a)$ gates to the two-leg state and reads $\tfrac12|A|^2$ and $\tfrac12|B|^2$. Ready phase factors of the $|\pm_x\rangle$ basis are transparent to QPE, which is why the projected-basis amplitude, rather than the eigenphase, carries the physical information. The final identities $\sin\delta=(|A|^2-1)/(2SC)$ and $\tan\theta = \frac{S-C}{S+C}\tan(\delta/2)$ complete the extraction.

Extended reading notes

Core claim

The paper's core claim is that the non-Abelian phase of an electron traversing a discrete spin-orbit path can be decoded from probability amplitudes rather than from interference fringes. For an L-shaped generator with leg rotations $R_x(-\eta)$ and $R_y(\delta)$, the state at the collector is $|\psi_2\rangle = \frac{1}{\sqrt{2}}(A|+_x\rangle + B|-_x\rangle)$, and the total non-cyclic phase appears in $\langle 0|\psi_2\rangle = \sqrt{\tfrac12 + SC\cos\delta}\,e^{i\theta}$, where $C=\cos(\pi/4-\eta/2)$ and $S=\sin(\pi/4-\eta/2)$. The paper shows that QPEV reads $C$ and $S$ by projecting onto the $y$ basis with an auxiliary $R_y(\delta_a)$ rotation, and QPEH reads $|A|$ by projecting onto the $x$ basis with an auxiliary $R_x(\eta_a)$ rotation. From those readings, $\delta$ is inferred via $\sin\delta=(|A|^2-1)/(2SC)$ and the phase via $\tan\theta = \frac{S-C}{S+C}\tan(\delta/2)$. The protocol thus turns a non-Abelian phase measurement into the statistics of two QPE readouts.

Load-bearing premise

The load-bearing premise is that the electron's passage through the L-shaped spin-orbit structure is exactly the unitary $U=R_y(\delta)R_x(-\eta)$ acting on $|0\rangle$, with no scattering, spin relaxation, or extra phases, and that the electron state collected at the dot can be loaded into the QPE qubits without corruption.

Editorial extensions

If this is right

The non-Abelian phase is recovered from qubit readout statistics, so an interferometric apparatus with beam splitters and precise alignment is no longer needed for this measurement.
For a fixed $\eta$, one QPEV run calibrates $C$ and $S$; each QPEH run then yields $\delta$ and $\theta$ on that calibration, so the protocol reduces ongoing phase measurements to a single type of readout.
All required operations are standard gates (Hadamard, controlled rotations, inverse quantum Fourier transform), so the protocol can be implemented on gate-based quantum processors without new measurement hardware.
The same two-module amplitude strategy applies to other non-Abelian platforms such as Weyl or topological surfaces, where the relevant path unitary replaces $R_y(\delta)R_x(-\eta)$.

Reading between the lines

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

The scheme suggests a practical calibration use: because $\delta$ encodes both path length and effective spin-orbit field strength, a fixed device could use the QPE readouts to map spin-orbit coupling strength across a chip.
Repeating the two-module readout at different traversal speeds should separate geometric from dynamic phase, since only the dynamic part scales with dwell time; the paper does not carry out this decomposition.
Adding a third non-commuting segment would likely require a third QPE module, so the protocol naturally extends to longer discrete paths at the cost of more readout qubits and a longer inverse Fourier transform.
The strongest test on real hardware would be a direct comparison against a conventional interferometer on the same sample; the paper's simulations stop short of that.

Original abstract

We propose an approach to measure the quantum phase of an electron in a non-Abelian system using the algorithm of Quantum Phase Estimation (QPE). The discrete-path systems were previously studied in the context of square or rectangular rings. Present focus is on measuring the quantum phases. The merit of the algorithm approach is two-fold. First off, it eliminates the need for an interferometric set up. Quantum phase is measured by reading off of measurable qubit states of the QPE modules. Secondly, the QPE works by subjecting the quantum state to a sequence of quantum computing operations that eventually map the phase information into measurable qubit states. All the operations are realizable by standard quantum computer gates and algorithms, placing the new effort within the reach of standard quantum computational framework.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,018 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We propose an approach to measure the quantum phase of an electron in a non-Abelian system using the algorithm of Quantum Phase Estimation (QPE). The discrete-path systems were previously studied in the context of square or rectangular rings. Present focus is on measuring the quantum phases. The merit of the algorithm approach is two-fold. First off, it eliminates the need for an interferometric set up. Quantum phase is measured by reading off of measurable qubit states of the QPE modules. Secondly, the QPE works by subjecting the quantum state to a sequence of quantum computing operations that eventually map the phase information into measurable qubit states. All the operations are realizable by standard quantum computer gates and algorithms, placing the new effort within the reach of standard quantum computational framework.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose an approach to measure the quantum phase of an electron in a non-Abelian system using the algorithm of Quantum Phase Estimation (QPE).	contribution 1.00	0.64	0.36	asserted 0.72
s1	The discrete-path systems were previously studied in the context of square or rectangular rings.	background 1.00	0.04	0.02	asserted 1.00
s2	Present focus is on measuring the quantum phases.	contribution 0.54	0.33	0.05	asserted 0.88
s3	The merit of the algorithm approach is two-fold.	contribution 0.83	0.44	0.05	asserted 0.98
s4	First off, it eliminates the need for an interferometric set up.	contribution 0.85	0.46	0.12	asserted 1.00
s5	Quantum phase is measured by reading off of measurable qubit states of the QPE modules.	method 0.93	0.43	0.19	asserted 1.00
s6	Secondly, the QPE works by subjecting the quantum state to a sequence of quantum computing operations that eventually map the phase information into measurable qubit states.	method 0.96	0.29	0.10	asserted 1.00
s7	All the operations are realizable by standard quantum computer gates and algorithms, placing the new effort within the reach of standard quantum computational framework.	interpretation 0.59	0.45	0.15	asserted 0.93

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from argument 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** An L-shaped structure (spin-orbit material) comprising of path-1 and path-2 act as a quantum phase generator. Electron is launched into path-1 via the emitter quantum dot and is collected at the collector quantum dot for measurement. +𝜎𝑦 −𝜎𝑥 𝑥 𝑦 𝑝𝑎𝑡ℎ 1 𝑝𝑎𝑡ℎ 2 𝑒𝑚𝑖𝑡𝑡𝑒𝑟 𝑄𝐷 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑜𝑟 𝑄𝐷

**Figure 6.** FIG.6. Qiskit simulation of the QPEH to measure the non

Showing 3 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2509.07716 by the authors.

References

Works cited (22)

[1]

S. Pancharatnam, Proceedings of the Indian Academy of Sciences A (New Delhi: Springer India) Vol. 44, No. 5, 247-262 (1956)

View on Pith 1956
[2]

M.V. Berry, Proceedings of the Royal Society of London A (Mathematical and Physical Sciences) 392 (1802), 45-57 (1984)

View on Pith 1984
[3]

T. Fujita, M.B.A. Jalil, S.G. Tan and S. Murakami, Journal of Applied Physics (Applied Physics Review) 110, 121301 (2011)

View on Pith 2011
[4]

Seng Ghee Tan and M.B.A. Jalil, Mod. Phys. Lett. A 31, 1630003 (2016)

View on Pith 2016
[5]

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, Nature Reviews Physics 1, 437 (2019). 12

View on Pith 2019
[6]

F. Nagasawa, D. Frustaglia, H. Saarikoski, K. Richter, J. Nitta, Nature Communications 4, 2526 (2013)

View on Pith 2013
[7]

Seng Ghee Tan, Son-Hsien Chen, Cong Son Ho, Che-Chun Huang, Mansoor B.A. Jalil, Ching-Ray Chang, and Shuichi Murakami, Physics Reports 882 1–36 (2020)

View on Pith 2020
[8]

Kenji Kondo, A derivation of Aharonov –Casher phase and another adiabatic approximation for pure gauge under general rashba effects, SPIN 6 (2), 1640006 (2016)

View on Pith 2016
[9]

Che-Chun Huang, Seng Ghee Tan, Ching-Ray Chang, New Journal of Physics 25, 063026 (2023)

View on Pith 2023
[10]

Ying-Cheng Yang, Seng Ghee Tan , Ching-Ray Chang, Chinese Journal of Physics 96 , 53–62 (2025)

View on Pith 2025
[11]

Marcin Markiewicz, Mahasweta Pandit & Wiesław Laskowski, Scientific Rep orts 11, 15669 (2021)

View on Pith 2021
[12]

Luis Garza -Soto, Nathan Hagen, Dorilian Lopez -Mago, and Yukitoshi Otani, Differences between the geometric phase and propagation phase: clarifying the boundedness problem, Applied Optics 63, 645-653 (2024)

View on Pith 2024
[13]

A.A. Voitiv, M.T. Lusk, M.E. Siemens, Experimental measurement of the geometric phase of non-geodesic circles. Optics Letters, 48, 2680-2683 (2023)

View on Pith 2023
[14]

Clément Godfrin, Rafik Ballou, Edgar Bonet, Mario Ruben, Svetlana Klyatskaya, Wolfgang Wernsdorfer & Franck Balestro, npj Quantum Information volume 4, 53 (2018)

View on Pith 2018
[15]

Z. Zhang, T. Wang, L. Xiang, J. Yao, J. Wu, & Y. Yin, Measuring the Berry phase in a superconducting phase qubit by a shortcut to adiabaticity, Physical Review A, 95, 042345 (2017)

View on Pith 2017
[16]

B. Kolok, & A. Pályi, Protocols to measure the non-Abelian Berry phase by pumping a spin qubit through a quantum-dot loop. Physical Review B, 109, 045438 (2024)

View on Pith 2024
[17]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010

View on Pith 2010
[18]

Seng Ghee Tan, Ching Hua Lee, Mansoor B.A. Jalil, Quantum physics and modern applications: problems and solutions , World Scientific (2023), ISBN 978 -981-127-101-4 (Soft); ISBN 978 -981- 127- 039-0 (Hard)

View on Pith 2023
[19]

T. Fujita, M.B.A. Jalil, S.G. Tan, Applied Physics Express 4(9), 094201 (2011)

View on Pith 2011
[20]

Shengyuan A. Yang, Dirac and Weyl materials: Fundamental aspects and some spintronics applications, SPIN 6, 1640003 (2016)

View on Pith 2016
[21]

X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Topological semimetal and Fermi -arc surface states in the electronic structue of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011)

View on Pith 2011
[22]

S.-Y. Xu et al., Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide, Nature Physics 11, 748 (2015). Acknowledgement We would like to thank the National Science and Technology Council of Taiwan for supporting this work under Grant. No.: NSTC 113-2112-M-034-002 and NSTC 114-2112-M-034-001. Data Availability Statement The data that support...

View on Pith 2015

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

S. Pancharatnam, Proceedings of the Indian Academy of Sciences A (New Delhi: Springer India) Vol. 44, No. 5, 247-262 (1956)

View on Pith 1956

[2] [2]

M.V. Berry, Proceedings of the Royal Society of London A (Mathematical and Physical Sciences) 392 (1802), 45-57 (1984)

View on Pith 1984

[3] [3]

T. Fujita, M.B.A. Jalil, S.G. Tan and S. Murakami, Journal of Applied Physics (Applied Physics Review) 110, 121301 (2011)

View on Pith 2011

[4] [4]

Seng Ghee Tan and M.B.A. Jalil, Mod. Phys. Lett. A 31, 1630003 (2016)

View on Pith 2016

[5] [5]

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, Nature Reviews Physics 1, 437 (2019). 12

View on Pith 2019

[6] [6]

F. Nagasawa, D. Frustaglia, H. Saarikoski, K. Richter, J. Nitta, Nature Communications 4, 2526 (2013)

View on Pith 2013

[7] [7]

Seng Ghee Tan, Son-Hsien Chen, Cong Son Ho, Che-Chun Huang, Mansoor B.A. Jalil, Ching-Ray Chang, and Shuichi Murakami, Physics Reports 882 1–36 (2020)

View on Pith 2020

[8] [8]

Kenji Kondo, A derivation of Aharonov –Casher phase and another adiabatic approximation for pure gauge under general rashba effects, SPIN 6 (2), 1640006 (2016)

View on Pith 2016

[9] [9]

Che-Chun Huang, Seng Ghee Tan, Ching-Ray Chang, New Journal of Physics 25, 063026 (2023)

View on Pith 2023

[10] [10]

Ying-Cheng Yang, Seng Ghee Tan , Ching-Ray Chang, Chinese Journal of Physics 96 , 53–62 (2025)

View on Pith 2025

[11] [11]

Marcin Markiewicz, Mahasweta Pandit & Wiesław Laskowski, Scientific Rep orts 11, 15669 (2021)

View on Pith 2021

[12] [12]

Luis Garza -Soto, Nathan Hagen, Dorilian Lopez -Mago, and Yukitoshi Otani, Differences between the geometric phase and propagation phase: clarifying the boundedness problem, Applied Optics 63, 645-653 (2024)

View on Pith 2024

[13] [13]

A.A. Voitiv, M.T. Lusk, M.E. Siemens, Experimental measurement of the geometric phase of non-geodesic circles. Optics Letters, 48, 2680-2683 (2023)

View on Pith 2023

[14] [14]

Clément Godfrin, Rafik Ballou, Edgar Bonet, Mario Ruben, Svetlana Klyatskaya, Wolfgang Wernsdorfer & Franck Balestro, npj Quantum Information volume 4, 53 (2018)

View on Pith 2018

[15] [15]

Z. Zhang, T. Wang, L. Xiang, J. Yao, J. Wu, & Y. Yin, Measuring the Berry phase in a superconducting phase qubit by a shortcut to adiabaticity, Physical Review A, 95, 042345 (2017)

View on Pith 2017

[16] [16]

B. Kolok, & A. Pályi, Protocols to measure the non-Abelian Berry phase by pumping a spin qubit through a quantum-dot loop. Physical Review B, 109, 045438 (2024)

View on Pith 2024

[17] [17]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010

View on Pith 2010

[18] [18]

Seng Ghee Tan, Ching Hua Lee, Mansoor B.A. Jalil, Quantum physics and modern applications: problems and solutions , World Scientific (2023), ISBN 978 -981-127-101-4 (Soft); ISBN 978 -981- 127- 039-0 (Hard)

View on Pith 2023

[19] [19]

T. Fujita, M.B.A. Jalil, S.G. Tan, Applied Physics Express 4(9), 094201 (2011)

View on Pith 2011

[20] [20]

Shengyuan A. Yang, Dirac and Weyl materials: Fundamental aspects and some spintronics applications, SPIN 6, 1640003 (2016)

View on Pith 2016

[21] [21]

X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Topological semimetal and Fermi -arc surface states in the electronic structue of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011)

View on Pith 2011

[22] [22]

S.-Y. Xu et al., Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide, Nature Physics 11, 748 (2015). Acknowledgement We would like to thank the National Science and Technology Council of Taiwan for supporting this work under Grant. No.: NSTC 113-2112-M-034-002 and NSTC 114-2112-M-034-001. Data Availability Statement The data that support...

View on Pith 2015