In-plane Wilson loop for measurement of quantized non-Abelian Berry flux

Tyner, Alexander C.; Sur, Shouvik; Zhou, Qunfei; Puggioni, Danilo; Darancet, Pierre; Rondinelli, James M.; Goswami, Pallab

doi:10.1103/physrevb.109.195149

In-plane Wilson loop for measurement of quantized non-Abelian Berry flux

Journal Article · Wed May 15 00:00:00 EDT 2024 · Physical Review. B

DOI:https://doi.org/10.1103/physrevb.109.195149· OSTI ID:2566125

Tyner, Alexander C. ^[1]; Sur, Shouvik ^[2]; ^[3]; ^[4]; Darancet, Pierre ^[5]; ^[6]; Goswami, Pallab ^[4]

Northwestern Univ., Evanston, IL (United States); Nordic Institute for Theoretical Physics (NORDITA), Stockholm (Sweden); KTH Royal Inst. of Technology, Stockholm (Sweden); Stockholm Univ. (Sweden)
Northwestern Univ., Evanston, IL (United States); Rice Univ., Houston, TX (United States)
Northwestern Univ., Evanston, IL (United States); Argonne National Laboratory (ANL), Argonne, IL (United States). Center for Nanoscale Materials (CNM); Univ. of Kansas, Lawrence, KS (United States)
Northwestern Univ., Evanston, IL (United States)
Argonne National Laboratory (ANL), Argonne, IL (United States). Center for Nanoscale Materials (CNM); Northwestern Argonne Institute for Science and Engineering, Evanston, IL (United States)
Northwestern Univ., Evanston, IL (United States); Northwestern Argonne Institute for Science and Engineering, Evanston, IL (United States)

Band topology of anomalous quantum Hall insulators can be precisely addressed by computing the Chern numbers of constituent nondegenerate bands, describing the presence of quantized, Abelian Berry flux through the two-dimensional Brillouin zone. Can Berry flux be captured for the SU (2) Berry connection of two -fold degenerate bands in spinful materials preserving space -inversion ($$\mathscr{P}$$) and time -reversal ($$\mathscr{T}$$) symmetries without detailed knowledge of underlying basis? We address this question by investigating the correspondence between a non -Abelian generalization of Stokes' theorem and the manifestly gauge -invariant eigenvalues of Wilson loops computed along in -plane contours which preserve the underlying crystalline symmetry. The importance of this correspondence is elucidated by performing natural number resolved classification of ab initio band structures of three-dimensional, Dirac materials. Further, our work underscores how identification of quantized Berry flux, both Abelian and non -Abelian, offers a unified framework for addressing first -order and higher -order topology of insulators and semimetals.

View Accepted Manuscript (DOE)

Research Organization:: Argonne National Laboratory (ANL), Argonne, IL (United States). Center for Nanoscale Materials (CNM)

Sponsoring Organization:: National Science Foundation (NSF); USDOE; USDOE Office of Science (SC), Basic Energy Sciences (BES). Scientific User Facilities (SUF)

Grant/Contract Number:: AC02-06CH11357

OSTI ID:: 2566125

Alternate ID(s):: OSTI ID: 2356892

Journal Information:: Physical Review. B, Journal Name: Physical Review. B Journal Issue: 19 Vol. 109; ISSN 2469-9950

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

References (81)

Non-Abelian Holonomy of BCS and SDW Quasiparticles Demler, Eugene; Zhang, Shou-Cheng Annals of Physics, Vol. 271, Issue 1 https://doi.org/10.1006/aphy.1998.5866	journal	January 1999
Non-Abelian stokes formula Aref'eva, I. Ya. Theoretical and Mathematical Physics, Vol. 43, Issue 1 https://doi.org/10.1007/BF01018469	journal	April 1980
A property of electric and magnetic flux in non-Abelian gauge theories 't Hooft, G. Nuclear Physics B, Vol. 153 https://doi.org/10.1016/0550-3213(79)90595-9	journal	January 1979
WannierTools: An open-source software package for novel topological materials Wu, QuanSheng; Zhang, ShengNan; Song, Hai-Feng Computer Physics Communications, Vol. 224 https://doi.org/10.1016/j.cpc.2017.09.033	journal	March 2018
Topological quantum chemistry Bradlyn, Barry; Elcoro, L.; Cano, Jennifer Nature, Vol. 547, Issue 7663 https://doi.org/10.1038/nature23268	journal	July 2017
Classification of stable three-dimensional Dirac semimetals with nontrivial topology Yang, Bohm-Jung; Nagaosa, Naoto Nature Communications, Vol. 5, Issue 1 https://doi.org/10.1038/ncomms5898	journal	September 2014
Symmetry-based indicators of band topology in the 230 space groups Po, Hoi Chun; Vishwanath, Ashvin; Watanabe, Haruki Nature Communications, Vol. 8, Issue 1 https://doi.org/10.1038/s41467-017-00133-2	journal	June 2017
Strong and fragile topological Dirac semimetals with higher-order Fermi arcs Wieder, Benjamin J.; Wang, Zhijun; Cano, Jennifer Nature Communications, Vol. 11, Issue 1 https://doi.org/10.1038/s41467-020-14443-5	journal	January 2020
Magnetic topological quantum chemistry Elcoro, Luis; Wieder, Benjamin J.; Song, Zhida Nature Communications, Vol. 12, Issue 1 https://doi.org/10.1038/s41467-021-26241-8	journal	October 2021
Spin-resolved topology and partial axion angles in three-dimensional insulators Lin, Kuan-Sen; Palumbo, Giandomenico; Guo, Zhaopeng Nature Communications, Vol. 15, Issue 1 https://doi.org/10.1038/s41467-024-44762-w	journal	January 2024
Efficient topological materials discovery using symmetry indicators Tang, Feng; Po, Hoi Chun; Vishwanath, Ashvin Nature Physics, Vol. 15, Issue 5 https://doi.org/10.1038/s41567-019-0418-7	journal	February 2019
Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe Bouhon, Adrien; Wu, QuanSheng; Slager, Robert-Jan Nature Physics, Vol. 16, Issue 11 https://doi.org/10.1038/s41567-020-0967-9	journal	July 2020
Comprehensive search for topological materials using symmetry indicators Tang, Feng; Po, Hoi Chun; Vishwanath, Ashvin Nature, Vol. 566, Issue 7745 https://doi.org/10.1038/s41586-019-0937-5	journal	February 2019
Catalogue of topological electronic materials Zhang, Tiantian; Jiang, Yi; Song, Zhida Nature, Vol. 566, Issue 7745 https://doi.org/10.1038/s41586-019-0944-6	journal	February 2019
A complete catalogue of high-quality topological materials Vergniory, M. G.; Elcoro, L.; Felser, Claudia Nature, Vol. 566, Issue 7745 https://doi.org/10.1038/s41586-019-0954-4	journal	February 2019
High-throughput calculations of magnetic topological materials Xu, Yuanfeng; Elcoro, Luis; Song, Zhi-Da Nature, Vol. 586, Issue 7831 https://doi.org/10.1038/s41586-020-2837-0	journal	October 2020
Spin-charge separation and quantum spin Hall effect of $$\beta$$-bismuthene Tyner, Alexander C.; Goswami, Pallab Scientific Reports, Vol. 13, Issue 1 https://doi.org/10.1038/s41598-023-38491-1	journal	July 2023
Dirac and Weyl Semimetal in XYBi (X = Ba, Eu; Y = Cu, Ag and Au) Du, Yongping; Wan, Bo; Wang, Di Scientific Reports, Vol. 5, Issue 1 https://doi.org/10.1038/srep14423	journal	September 2015
Bulk crystal growth and electronic characterization of the 3D Dirac semimetal Na ₃ Bi Kushwaha, Satya K.; Krizan, Jason W.; Feldman, Benjamin E. APL Materials, Vol. 3, Issue 4 https://doi.org/10.1063/1.4908158	journal	April 2015
Are the surface Fermi arcs in Dirac semimetals topologically protected? Kargarian, Mehdi; Randeria, Mohit; Lu, Yuan-Ming Proceedings of the National Academy of Sciences, Vol. 113, Issue 31 https://doi.org/10.1073/pnas.1524787113	journal	July 2016
Dirac semimetal in β -CuI without surface Fermi arcs Le, Congcong; Wu, Xianxin; Qin, Shengshan Proceedings of the National Academy of Sciences, Vol. 115, Issue 33 https://doi.org/10.1073/pnas.1803599115	journal	July 2018
Magnetic monopoles in gauge field theories Goddard, P.; Olive, D. I. Reports on Progress in Physics, Vol. 41, Issue 9 https://doi.org/10.1088/0034-4885/41/9/001	journal	September 1978
Highly convergent schemes for the calculation of bulk and surface Green functions Sancho, M. P. Lopez; Sancho, J. M. Lopez; Sancho, J. M. L. Journal of Physics F: Metal Physics, Vol. 15, Issue 4 https://doi.org/10.1088/0305-4608/15/4/009	journal	April 1985
Wannier90 as a community code: new features and applications Pizzi, Giovanni; Vitale, Valerio; Arita, Ryotaro Journal of Physics: Condensed Matter, Vol. 32, Issue 16 https://doi.org/10.1088/1361-648X/ab51ff	journal	January 2020
Electronic structure, Dirac points and Fermi arc surface states in three-dimensional Dirac semimetal Na ₃ Bi from angle-resolved photoemission spectroscopy Liang, Aiji; Chen, Chaoyu; Wang, Zhijun Chinese Physics B, Vol. 25, Issue 7 https://doi.org/10.1088/1674-1056/25/7/077101	journal	July 2016
Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry Bouhon, Adrien; Black-Schaffer, Annica M.; Slager, Robert-Jan Physical Review B, Vol. 100, Issue 19 https://doi.org/10.1103/PhysRevB.100.195135	journal	November 2019
Dirac fermions in antiferromagnetic FeSn kagome lattices with combined space inversion and time-reversal symmetry Lin, Zhiyong; Wang, Chongze; Wang, Pengdong Physical Review B, Vol. 102, Issue 15 https://doi.org/10.1103/PhysRevB.102.155103	journal	October 2020
Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry Ahn, Junyeong; Yang, Bohm-Jung Physical Review B, Vol. 103, Issue 18 https://doi.org/10.1103/PhysRevB.103.184502	journal	May 2021
Subdimensional topologies, indicators, and higher order boundary effects Lange, Gunnar F.; Bouhon, Adrien; Slager, Robert-Jan Physical Review B, Vol. 103, Issue 19 https://doi.org/10.1103/PhysRevB.103.195145	journal	May 2021
Topological correspondence between magnetic space group representations and subdimensions Bouhon, Adrien; Lange, Gunnar F.; Slager, Robert-Jan Physical Review B, Vol. 103, Issue 24 https://doi.org/10.1103/PhysRevB.103.245127	journal	June 2021
Quantized Hall conductance as a topological invariant Niu, Qian; Thouless, D. J.; Wu, Yong-Shi Physical Review B, Vol. 31, Issue 6 https://doi.org/10.1103/PhysRevB.31.3372	journal	March 1985
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set Kresse, G.; Furthmüller, J. Physical Review B, Vol. 54, Issue 16, p. 11169-11186 https://doi.org/10.1103/PhysRevB.54.11169	journal	October 1996
From ultrasoft pseudopotentials to the projector augmented-wave method Kresse, G.; Joubert, D. Physical Review B, Vol. 59, Issue 3, p. 1758-1775 https://doi.org/10.1103/PhysRevB.59.1758	journal	January 1999
Topological invariants of time-reversal-invariant band structures Moore, J. E.; Balents, L. Physical Review B, Vol. 75, Issue 12, Article No. 121306(R) https://doi.org/10.1103/PhysRevB.75.121306	journal	March 2007
Topological insulators with inversion symmetry Fu, Liang; Kane, C. L. Physical Review B, Vol. 76, Issue 4 https://doi.org/10.1103/PhysRevB.76.045302	journal	July 2007
Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1−xSbx Teo, Jeffrey C. Y.; Fu, Liang; Kane, C. L. Physical Review B, Vol. 78, Issue 4, Article No. 045426 https://doi.org/10.1103/PhysRevB.78.045426	journal	July 2008
Robustness of the spin-Chern number Prodan, Emil Physical Review B, Vol. 80, Issue 12 https://doi.org/10.1103/PhysRevB.80.125327	journal	September 2009
Computing topological invariants without inversion symmetry Soluyanov, Alexey A.; Vanderbilt, David Physical Review B, Vol. 83, Issue 23 https://doi.org/10.1103/PhysRevB.83.235401	journal	June 2011
Equivalent expression of Z 2 topological invariant for band insulators using the non-Abelian Berry connection Yu, Rui; Qi, Xiao Liang; Bernevig, Andrei Physical Review B, Vol. 84, Issue 7 https://doi.org/10.1103/PhysRevB.84.075119	journal	August 2011
Dirac semimetal and topological phase transitions in A 3 Bi ( A = Na , K, Rb) Wang, Zhijun; Sun, Yan; Chen, Xing-Qiu Physical Review B, Vol. 85, Issue 19 https://doi.org/10.1103/PhysRevB.85.195320	journal	May 2012
Bulk topological invariants in noninteracting point group symmetric insulators Fang, Chen; Gilbert, Matthew J.; Bernevig, B. Andrei Physical Review B, Vol. 86, Issue 11 https://doi.org/10.1103/PhysRevB.86.115112	journal	September 2012
Three-dimensional Dirac semimetal and quantum transport in Cd 3 As 2 Wang, Zhijun; Weng, Hongming; Wu, Quansheng Physical Review B, Vol. 88, Issue 12 https://doi.org/10.1103/PhysRevB.88.125427	journal	September 2013
Wannier center sheets in topological insulators Taherinejad, Maryam; Garrity, Kevin F.; Vanderbilt, David Physical Review B, Vol. 89, Issue 11 https://doi.org/10.1103/PhysRevB.89.115102	journal	March 2014
Wilson-loop characterization of inversion-symmetric topological insulators Alexandradinata, A.; Dai, Xi; Bernevig, B. Andrei Physical Review B, Vol. 89, Issue 15 https://doi.org/10.1103/PhysRevB.89.155114	journal	April 2014
Dirac semimetals A 3 Bi ( A = Na , K , Rb ) as Z 2 Weyl semimetals Gorbar, E. V.; Miransky, V. A.; Shovkovy, I. A. Physical Review B, Vol. 91, Issue 12 https://doi.org/10.1103/PhysRevB.91.121101	journal	March 2015
Three-dimensional Dirac semimetals: Design principles and predictions of new materials Gibson, Q. D.; Schoop, L. M.; Muechler, L. Physical Review B, Vol. 91, Issue 20 https://doi.org/10.1103/PhysRevB.91.205128	journal	May 2015
Berry-phase description of topological crystalline insulators Alexandradinata, A.; Bernevig, B. Andrei Physical Review B, Vol. 93, Issue 20 https://doi.org/10.1103/PhysRevB.93.205104	journal	May 2016
Type-II Dirac fermions in the PtSe 2 class of transition metal dichalcogenides Huang, Huaqing; Zhou, Shuyun; Duan, Wenhui Physical Review B, Vol. 94, Issue 12 https://doi.org/10.1103/PhysRevB.94.121117	journal	September 2016
Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials Gresch, Dominik; Autès, Gabriel; Yazyev, Oleg V. Physical Review B, Vol. 95, Issue 7 https://doi.org/10.1103/PhysRevB.95.075146	journal	February 2017
Three-dimensional topological critical Dirac semimetal in A MgBi ( A = K, Rb, Cs) Le, Congcong; Qin, Shengshan; Wu, Xianxin Physical Review B, Vol. 96, Issue 11 https://doi.org/10.1103/PhysRevB.96.115121	journal	September 2017
Building blocks of topological quantum chemistry: Elementary band representations Cano, Jennifer; Bradlyn, Barry; Wang, Zhijun Physical Review B, Vol. 97, Issue 3 https://doi.org/10.1103/PhysRevB.97.035139	journal	January 2018
Surface states in Dirac semimetals and topological crystalline insulators Bednik, Grigory Physical Review B, Vol. 98, Issue 4 https://doi.org/10.1103/PhysRevB.98.045140	journal	July 2018
Topological quadrupolar semimetals Lin, Mao; Hughes, Taylor L. Physical Review B, Vol. 98, Issue 24 https://doi.org/10.1103/PhysRevB.98.241103	journal	December 2018
Crystalline topological Dirac semimetal phase in rutile structure β ′ − PtO 2 Kim, Rokyeon; Yang, Bohm-Jung; Kim, Choong H. Physical Review B, Vol. 99, Issue 4 https://doi.org/10.1103/PhysRevB.99.045130	journal	January 2019
Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice Bradlyn, Barry; Wang, Zhijun; Cano, Jennifer Physical Review B, Vol. 99, Issue 4 https://doi.org/10.1103/PhysRevB.99.045140	journal	January 2019
Symmetry representation approach to topological invariants in C 2 z T -symmetric systems Ahn, Junyeong; Yang, Bohm-Jung Physical Review B, Vol. 99, Issue 23 https://doi.org/10.1103/PhysRevB.99.235125	journal	June 2019
Confinement of quarks Wilson, Kenneth G. Physical Review D, Vol. 10, Issue 8 https://doi.org/10.1103/PhysRevD.10.2445	journal	October 1974
Field-strength and dual variable formulations of gauge theory Halpern, M. B. Physical Review D, Vol. 19, Issue 2 https://doi.org/10.1103/PhysRevD.19.517	journal	January 1979
Exact computation of loop averages in two-dimensional Yang-Mills theory Bralić, Ninoslav E. Physical Review D, Vol. 22, Issue 12 https://doi.org/10.1103/PhysRevD.22.3090	journal	December 1980
Model Characterization of Gapless Edge Modes of Topological Insulators Using Intermediate Brillouin-Zone Functions Fidkowski, Lukasz; Jackson, T. S.; Klich, Israel Physical Review Letters, Vol. 107, Issue 3 https://doi.org/10.1103/PhysRevLett.107.036601	journal	July 2011
Topological Node-Line Semimetal and Dirac Semimetal State in Antiperovskite Cu 3 PdN Yu, Rui; Weng, Hongming; Fang, Zhong Physical Review Letters, Vol. 115, Issue 3 https://doi.org/10.1103/PhysRevLett.115.036807	journal	July 2015
Z 2 and Chiral Anomalies in Topological Dirac Semimetals Burkov, Anton A.; Kim, Yong Baek Physical Review Letters, Vol. 117, Issue 13 https://doi.org/10.1103/PhysRevLett.117.136602	journal	September 2016
Type-II Symmetry-Protected Topological Dirac Semimetals Chang, Tay-Rong; Xu, Su-Yang; Sanchez, Daniel S. Physical Review Letters, Vol. 119, Issue 2 https://doi.org/10.1103/PhysRevLett.119.026404	journal	July 2017
Quantized Hall Conductance in a Two-Dimensional Periodic Potential Thouless, D. J.; Kohmoto, M.; Nightingale, M. P. Physical Review Letters, Vol. 49, Issue 6 https://doi.org/10.1103/PhysRevLett.49.405	journal	August 1982
Appearance of Gauge Structure in Simple Dynamical Systems Wilczek, Frank; Zee, A. Physical Review Letters, Vol. 52, Issue 24 https://doi.org/10.1103/PhysRevLett.52.2111	journal	June 1984
Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly" Haldane, F. D. M. Physical Review Letters, Vol. 61, Issue 18 https://doi.org/10.1103/PhysRevLett.61.2015	journal	October 1988
Generalized Gradient Approximation Made Simple Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias Physical Review Letters, Vol. 77, Issue 18, p. 3865-3868 https://doi.org/10.1103/PhysRevLett.77.3865	journal	October 1996
Z2 Topological Order and the Quantum Spin Hall Effect Kane, C. L.; Mele, E. J. Physical Review Letters, Vol. 95, Issue 14, Article No. 146802 https://doi.org/10.1103/PhysRevLett.95.146802	journal	September 2005
Topological Insulators in Three Dimensions Fu, Liang; Kane, C. L.; Mele, E. J. Physical Review Letters, Vol. 98, Issue 10, Article No. 106803 https://doi.org/10.1103/PhysRevLett.98.106803	journal	March 2007
Topology of three-dimensional Dirac semimetals and quantum spin Hall systems without gapless edge modes Tyner, Alexander C.; Sur, Shouvik; Puggioni, Danilo Physical Review Research, Vol. 5, Issue 1 https://doi.org/10.1103/PhysRevResearch.5.L012019	journal	February 2023
Topological Classification of Crystalline Insulators through Band Structure Combinatorics Kruthoff, Jorrit; de Boer, Jan; van Wezel, Jasper Physical Review X, Vol. 7, Issue 4 https://doi.org/10.1103/PhysRevX.7.041069	journal	December 2017
Symmetry Indicators and Anomalous Surface States of Topological Crystalline Insulators Khalaf, Eslam; Po, Hoi Chun; Vishwanath, Ashvin Physical Review X, Vol. 8, Issue 3 https://doi.org/10.1103/PhysRevX.8.031070	journal	September 2018
Weyl and Dirac semimetals in three-dimensional solids Armitage, N. P.; Mele, E. J.; Vishwanath, Ashvin Reviews of Modern Physics, Vol. 90, Issue 1 https://doi.org/10.1103/RevModPhys.90.015001	journal	January 2018
Higher-order topological insulators Schindler, Frank; Cook, Ashley M.; Vergniory, Maia G. Science Advances, Vol. 4, Issue 6 https://doi.org/10.1126/sciadv.aat0346	journal	June 2018
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C. Science, Vol. 314, Issue 5806, p. 1757-1761 https://doi.org/10.1126/science.1133734	journal	December 2006
Discovery of a Three-Dimensional Topological Dirac Semimetal, Na3Bi Liu, Z. K.; Zhou, B.; Zhang, Y. Science, Vol. 343, Issue 6173 https://doi.org/10.1126/science.1245085	journal	January 2014
Observation of Fermi arc surface states in a topological metal Xu, S. -Y.; Liu, C.; Kushwaha, S. K. Science, Vol. 347, Issue 6219 https://doi.org/10.1126/science.1256742	journal	December 2014
Evidence for the chiral anomaly in the Dirac semimetal Na3Bi Xiong, J.; Kushwaha, S. K.; Liang, T. Science, Vol. 350, Issue 6259 https://doi.org/10.1126/science.aac6089	journal	September 2015
Quantized electric multipole insulators Benalcazar, Wladimir A.; Bernevig, B. Andrei; Hughes, Taylor L. Science, Vol. 357, Issue 6346 https://doi.org/10.1126/science.aah6442	journal	July 2017
All topological bands of all nonmagnetic stoichiometric materials Vergniory, Maia G.; Wieder, Benjamin J.; Elcoro, Luis Science, Vol. 376, Issue 6595 https://doi.org/10.1126/science.abg9094	journal	May 2022
Chern Numbers in Discretized Brillouin Zone: Efficient Method of Computing (Spin) Hall Conductances Fukui, Takahiro; Hatsugai, Yasuhiro; Suzuki, Hiroshi Journal of the Physical Society of Japan, Vol. 74, Issue 6 https://doi.org/10.1143/JPSJ.74.1674	journal	June 2005

Similar Records

Berry's connection for non-Abelian, chiral fermions

Journal Article · Sun Jan 14 23:00:00 EST 1990 · Physical Review, D (Particles Fields); (USA) · OSTI ID:6954746

Floquet topological systems with flat bands: Edge modes, Berry curvature, and orbital magnetization

Journal Article · Wed Jun 22 20:00:00 EDT 2022 · Physical Review. B · OSTI ID:1873647

Higher Chern numbers in multilayer Lieb lattices ( N≥2 ): Topological transitions and quadratic band crossing lines

Journal Article · Sun Jun 13 20:00:00 EDT 2021 · Physical Review. B · OSTI ID:1861293

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
Electronic structure of atoms & molecules
First-principles calculations
Topological insulators

In-plane Wilson loop for measurement of quantized non-Abelian Berry flux

Citation Formats

References (81)

Similar Records

Related Subjects