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  • Letter
  • Open Access

Higher-order Van Hove singularity in magic-angle twisted trilayer graphene

Daniele Guerci1,2,3, Pascal Simon2, and Christophe Mora1,4

  • 1Université de Paris, Laboratoire Matériaux et Phénomènes Quantiques, CNRS, F-75013 Paris, France
  • 2Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
  • 3Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
  • 4Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

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Phys. Rev. Research 4, L012013 – Published 9 February, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L012013

Abstract

Twisted trilayer graphene (TTG) has recently emerged experimentally as a fascinating playground to study correlated and exotic superconducting phases. We have found that TTG hosts a zero-energy higher-order Van Hove singularity with an exponent −1/3 that is stronger than the one predicted in twisted bilayer graphene. This singularity is protected by a threefold rotation symmetry and a combined mirror-particle-hole symmetry and can be tuned with only the twist angle and a perpendicular electric field. It arises from the combined merging of Van Hove singularities and Dirac cones at zero energy, a scheme that goes beyond the recent classifications of Van Hove singularities in single-band models. This structure gives a topological Lifshitz transition, with anomalous exponent −2/5, which can be achieved in TTG by varying a third control parameter such as the atomic corrugation. The interplay between the nonstandard class of higher-order Van Hove singularities and interaction effects offers an unprecedented platform for studying correlation and superconductivity.

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References (97)

  1. L. Van Hove, The occurrence of singularities in the elastic frequency distribution of a crystal, Phys. Rev. 89, 1189 (1953).
  2. I. M. Lifshitz, Anomalies of electron characteristics of a metal in the high pressure region, Sov. Phys. JETP 11, 1130 (1960).
  3. A. Shtyk, G. Goldstein, and C. Chamon, Electrons at the monkey saddle: A multicritical Lifshitz point, Phys. Rev. B 95, 035137 (2017).
  4. N. F. Q. Yuan, H. Isobe, and L. Fu, Magic of high-order Van Hove singularity, Nat. Commun. 10, 5769 (2019).
  5. N. F. Q. Yuan and L. Fu, Classification of critical points in energy bands based on topology, scaling, and symmetry, Phys. Rev. B 101, 125120 (2020).
  6. A. Chandrasekaran, A. Shtyk, J. J. Betouras, and C. Chamon, Catastrophe theory classification of fermi surface topological transitions in two dimensions, Phys. Rev. Research 2, 013355 (2020).
  7. Y. Sherkunov and J. J. Betouras, Electronic phases in twisted bilayer graphene at magic angles as a result of Van Hove singularities and interactions, Phys. Rev. B 98, 205151 (2018).
  8. H. Isobe, N. F. Q. Yuan, and L. Fu, Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene, Phys. Rev. X 8, 041041 (2018).
  9. Y.-P. Lin and R. M. Nandkishore, Parquet renormalization group analysis of weak-coupling instabilities with multiple high-order Van Hove points inside the Brillouin zone, Phys. Rev. B 102, 245122 (2020).
  10. L. Classen, A. V. Chubukov, C. Honerkamp, and M. M. Scherer, Competing orders at higher-order Van Hove points, Phys. Rev. B 102, 125141 (2020).
  11. D. V. Chichinadze, L. Classen, and A. V. Chubukov, Valley magnetism, nematicity, and density wave orders in twisted bilayer graphene, Phys. Rev. B 102, 125120 (2020).
  12. Z. Bi and L. Fu, Excitonic density wave and spin-valley superfluid in bilayer transition metal dichalcogenide, arXiv:1911.04493.
  13. D. V. Efremov, A. Shtyk, A. W. Rost, C. Chamon, A. P. Mackenzie, and J. J. Betouras, Multicritical Fermi Surface Topological Transitions, Phys. Rev. Lett. 123, 207202 (2019).
  14. C. H. Mousatov, E. Berg, and S. A. Hartnoll, Theory of the strange metal Sr3Ru2O7, Proc. Natl. Acad. Sci. USA 117, 2852 (2020).
  15. A. Ramires, P. Coleman, A. H. Nevidomskyy, and A. M. Tsvelik, β−YBALB4: A Critical Nodal Metal, Phys. Rev. Lett. 109, 176404 (2012).
  16. K. Gofron, J. C. Campuzano, A. A. Abrikosov, M. Lindroos, A. Bansil, H. Ding, D. Koelling, and B. Dabrowski, Observation of an “Extended” Van Hove Singularity in YBa2Cu4o8 by Ultrahigh Energy Resolution Angle-Resolved Photoemission, Phys. Rev. Lett. 73, 3302 (1994).
  17. H. Isobe and L. Fu, Supermetal, Phys. Rev. Research 1, 033206 (2019).
  18. J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Graphene Bilayer with a Twist: Electronic Structure, Phys. Rev. Lett. 99, 256802 (2007).
  19. R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. USA 108, 12233 (2011).
  20. G. Trambly de Laissardière, D. Mayou, and L. Magaud, Localization of dirac electrons in rotated graphene bilayers, Nano Lett. 10, 804 (2010).
  21. Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature (London) 556, 80 (2018).
  22. Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature (London) 556, 43 (2018).
  23. E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020).
  24. Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule, J. Mao, and E. Y. Andrei, Charge order and broken rotational symmetry in magic-angle twisted bilayer graphene, Nature (London) 573, 91 (2019).
  25. U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern et al., Cascade of phase transitions and Dirac revivals in magic-angle graphene, Nature (London) 582, 203 (2020).
  26. Y. Saito, J. Ge, L. Rademaker, K. Watanabe, T. Taniguchi, D. A. Abanin, and A. F. Young, Hofstadter subband ferromagnetism and symmetry-broken Chern insulators in twisted bilayer graphene, Nat. Phys. 17, 478 (2021).
  27. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Flavour Hund's coupling, Chern gaps and charge diffusivity in moiré graphene, Nature (London) 592, 43 (2021).
  28. D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Cascade of electronic transitions in magic-angle twisted bilayer graphene, Nature (London) 582, 198 (2020).
  29. Y. Xie, B. Lian, B. Jäck, X. Liu, C.-L. Chiu, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Spectroscopic signatures of many-body correlations in magic-angle twisted bilayer graphene, Nature (London) 572, 101 (2019).
  30. Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All Magic Angles in Twisted Bilayer Graphene are Topological, Phys. Rev. Lett. 123, 036401 (2019).
  31. H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99, 195455 (2019).
  32. J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum Valley Hall Effect, Orbital Magnetism, and Anomalous Hall Effect in Twisted Multilayer Graphene Systems, Phys. Rev. X 9, 031021 (2019).
  33. S.-Y. Li, Y. Zhang, Y.-N. Ren, J. Liu, X. Dai, and L. He, Experimental evidence for orbital magnetic moments generated by moiré-scale current loops in twisted bilayer graphene, Phys. Rev. B 102, 121406(R) (2020).
  34. X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature (London) 574, 653 (2019).
  35. C. L. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia, J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. E. Huber, and A. F. Young, Imaging orbital ferromagnetism in a moiré Chern insulator, arXiv:2006.08053.
  36. D. Guerci, P. Simon, and C. Mora, Moiré lattice effects on the orbital magnetic response of twisted bilayer graphene and Condon instability, Phys. Rev. B 103, 224436 (2021).
  37. Z. Wu, Z. Zhan, and S. Yuan, Lattice relaxation, mirror symmetry and magnetic field effects on ultraflat bands in twisted trilayer graphene, Sci. China Phys. Mech. Astron. 64, 267811 (2021).
  38. Y. Choi, H. Kim, Y. Peng, A. Thomson, C. Lewandowski, R. Polski, Y. Zhang, H. S. Arora, K. Watanabe, T. Taniguchi, J. Alicea, and S. Nadj-Perge, Tracing out correlated Chern insulators in magic angle twisted bilayer graphene, arXiv:2008.11746.
  39. K. P. Nuckolls, M. Oh, D. Wong, B. Lian, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Strongly correlated Chern insulators in magic-angle twisted bilayer graphene, Nature (London) 588, 610 (2020).
  40. P. Stepanov, M. Xie, T. Taniguchi, K. Watanabe, X. Lu, A. H. MacDonald, B. A. Bernevig, and D. K. Efetov, Competing Zero-Field Chern Insulators in Superconducting Twisted Bilayer Graphene, Phys. Rev. Lett. 127, 197701 (2021).
  41. A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science 365, 605 (2019).
  42. M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrinsic quantized anomalous Hall effect in a moiré heterostructure, Science 367, 900 (2020).
  43. H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L. Tschirhart, M. Serlin, K. Watanabe, T. Taniguchi, A. H. MacDonald et al., Electrical switching of magnetic order in an orbital Chern insulator, Nature (London) 588, 66 (2020).
  44. Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael et al., Electronic correlations in twisted bilayer graphene near the magic angle, Nat. Phys. 15, 1174 (2019).
  45. Y. Cao, D. Rodan-Legrain, J. M. Park, N. F. Q. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, Nematicity and competing orders in superconducting magic-angle graphene, Science 372, 264 (2021).
  46. M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363, 1059 (2019).
  47. Y.-H. Zhang, H. C. Po, and T. Senthil, Landau level degeneracy in twisted bilayer graphene: Role of symmetry breaking, Phys. Rev. B 100, 125104 (2019).
  48. J. Ahn, S. Park, and B.-J. Yang, Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional systems With Space-time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle, Phys. Rev. X 9, 021013 (2019).
  49. J. Kang and O. Vafek, Strong Coupling Phases of Partially Filled Twisted Bilayer Graphene Narrow Bands, Phys. Rev. Lett. 122, 246401 (2019).
  50. K. Seo, V. N. Kotov, and B. Uchoa, Ferromagnetic Mott State in Twisted Graphene Bilayers at the Magic Angle, Phys. Rev. Lett. 122, 246402 (2019).
  51. C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Ferromagnetism in Narrow Bands of Moiré Superlattices, Phys. Rev. Lett. 124, 187601 (2020).
  52. N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 166601 (2020).
  53. R. Pons, A. Mielke, and T. Stauber, Flat-band ferromagnetism in twisted bilayer graphene, Phys. Rev. B 102, 235101 (2020).
  54. Y. Zhang, K. Jiang, Z. Wang, and F. Zhang, Correlated insulating phases of twisted bilayer graphene at commensurate filling fractions: A Hartree-Fock study, Phys. Rev. B 102, 035136 (2020).
  55. J. Liu and X. Dai, Theories for the correlated insulating states and quantum anomalous Hall effect phenomena in twisted bilayer graphene, Phys. Rev. B 103, 035427 (2021).
  56. M. Xie and A. H. MacDonald, Nature of the Correlated Insulator States in Twisted Bilayer Graphene, Phys. Rev. Lett. 124, 097601 (2020).
  57. N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath, and M. P. Zaletel, Ground State and Hidden Symmetry of Magic-Angle Graphene at Even Integer Filling, Phys. Rev. X 10, 031034 (2020).
  58. S. Liu, E. Khalaf, J. Y. Lee, and A. Vishwanath, Nematic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality, Phys. Rev. Research 3, 013033 (2021).
  59. B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. I. Matrix elements, approximations, perturbation theory, and a k·p two-band model, Phys. Rev. B 103, 205411 (2021).
  60. Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Twisted bilayer graphene. II. Stable symmetry anomaly, Phys. Rev. B 103, 205412 (2021).
  61. B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. III. Interacting Hamiltonian and exact symmetries, Phys. Rev. B 103, 205413 (2021).
  62. B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yazdani, and B. A. Bernevig, Twisted bilayer graphene. IV. Exact insulator ground states and phase diagram, Phys. Rev. B 103, 205414 (2021).
  63. B. A. Bernevig, B. Lian, A. Cowsik, F. Xie, N. Regnault, and Z.-D. Song, Twisted bilayer graphene. V. Exact analytic many-body excitations in Coulomb Hamiltonians: Charge gap, goldstone modes, and absence of cooper pairing, Phys. Rev. B 103, 205415 (2021).
  64. F. Xie, A. Cowsik, Z.-D. Song, B. Lian, B. A. Bernevig, and N. Regnault, Twisted bilayer graphene. VI. An exact diagonalization study at nonzero integer filling, Phys. Rev. B 103, 205416 (2021).
  65. E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vishwanath, Magic angle hierarchy in twisted graphene multilayers, Phys. Rev. B 100, 085109 (2019).
  66. C. Mora, N. Regnault, and B. A. Bernevig, Flatbands and Perfect Metal in Trilayer Moiré Graphene, Phys. Rev. Lett. 123, 026402 (2019).
  67. X. Li, F. Wu, and A. H. MacDonald, Electronic structure of single-twist trilayer graphene, arXiv:1907.12338.
  68. Z. Zhu, S. Carr, D. Massatt, M. Luskin, and E. Kaxiras, Twisted Trilayer Graphene: A Precisely Tunable Platform for Correlated Electrons, Phys. Rev. Lett. 125, 116404 (2020).
  69. K.-T. Tsai, X. Zhang, Z. Zhu, Y. Luo, S. Carr, M. Luskin, E. Kaxiras, and K. Wang, Correlated insulating states and transport signature of superconductivity in twisted trilayer graphene moiré of moiré superlattices, arXiv:1912.03375.
  70. S. Carr, C. Li, Z. Zhu, E. Kaxiras, S. Sachdev, and A. Kruchkov, Ultraheavy and ultrarelativistic dirac quasiparticles in sandwiched graphenes, Nano Lett. 20, 3030 (2020).
  71. D. Calugaru, F. Xie, Z.-D. Song, B. Lian, N. Regnault, and B. A. Bernevig, Twisted symmetric trilayer graphene: Single-particle and many-body Hamiltonians and hidden nonlocal symmetries of trilayer moiré systems with and without displacement field, Phys. Rev. B 103, 195411 (2021).
  72. J. Shin, B. L. Chittari, and J. Jung, Stacking and gate tunable topological flat bands, gaps and anisotropic strip patterns in twisted trilayer graphene, Phys. Rev. B 104, 045413 (2021).
  73. A. Ramires and J. L. Lado, Emulating Heavy Fermions in Twisted Trilayer Graphene, Phys. Rev. Lett. 127, 026401 (2021).
  74. A. Lopez-Bezanilla and J. L. Lado, Electrical band flattening, valley flux, and superconductivity in twisted trilayer graphene, Phys. Rev. Research 2, 033357 (2020).
  75. C. Lei, L. Linhart, W. Qin, F. Libisch, and A. H. MacDonald, Mirror symmetry breaking and lateral stacking shifts in twisted trilayer graphene, Phys. Rev. B 104, 035139 (2021).
  76. M. Christos, S. Sachdev, and M. S. Scheurer, Correlated insulators, semimetals, and superconductivity in twisted trilayer graphene, arXiv:2106.02063.
  77. E. Lake and T. Senthil, Re-entrant superconductivity through a quantum Lifshitz transition in twisted trilayer graphene, Phys. Rev. B 104, 174505 (2021).
  78. W. Qin and A. H. MacDonald, In-Plane Critical Magnetic Fields in Magic-Angle Twisted Trilayer Graphene, Phys. Rev. Lett. 127, 097001 (2021).
  79. A. Fischer, Z. A. H. Goodwin, A. A. Mostofi, J. Lischner, D. M. Kennes, and L. Klebl, Unconventional superconductivity in magic-angle twisted trilayer graphene, npj Quantum Materials 7, 5 (2022).
  80. Y.-Z. Chou, F. Wu, J. D. Sau, and S. D. Sarma, Correlation-Induced Triplet Pairing Superconductivity in Graphene-Based Moiré Systems, Phys. Rev. Lett. 127, 217001 (2021).
  81. J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature (London) 590, 249 (2021).
  82. Z. Hao, A. M. Zimmerman, P. Ledwith, E. Khalaf, D. H. Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Electric field-tunable superconductivity in alternating-twist magic-angle trilayer graphene, Science 371, 1133 (2021).
  83. Y. Cao, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Large Pauli limit violation and reentrant superconductivity in magic-angle twisted trilayer graphene, arXiv:2103.12083.
  84. N. N. T. Nam and M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Phys. Rev. B 96, 075311 (2017).
  85. M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Maximally Localized Wannier Orbitals and the Extended Hubbard Model for Twisted Bilayer Graphene, Phys. Rev. X 8, 031087 (2018).
  86. G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto, A. Reina, J. Kong, and E. Y. Andrei, Observation of Van Hove singularities in twisted graphene layers, Nat. Phys. 6, 109 (2009).
  87. D. Wong, Y. Wang, J. Jung, S. Pezzini, A. M. DaSilva, H.-Z. Tsai, H. S. Jung, R. Khajeh, Y. Kim, J. Lee, S. Kahn, S. Tollabimazraehno, H. Rasool, K. Watanabe, T. Taniguchi, A. Zettl, S. Adam, A. H. MacDonald, and M. F. Crommie, Local spectroscopy of moiré-induced electronic structure in gate-tunable twisted bilayer graphene, Phys. Rev. B 92, 155409 (2015).
  88. K. Kim, A. DaSilva, S. Huang, B. Fallahazad, S. Larentis, T. Taniguchi, K. Watanabe, B. J. LeRoy, A. H. MacDonald, and E. Tutuc, Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene, Proc. Natl. Acad. Sci. USA 114, 3364 (2017).
  89. A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, C. Dean et al., Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature (London) 572, 95 (2019).
  90. S. Wu, Z. Zhang, K. Watanabe, T. Taniguchi, and E. Y. Andrei, Chern insulators, Van Hove singularities and topological flat bands in magic-angle twisted bilayer graphene, Nat. Mater. 20, 488 (2021).
  91. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L012013 for details of the calculations.
  92. G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of Magic Angles in Twisted Bilayer Graphene, Phys. Rev. Lett. 122, 106405 (2019).
  93. P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional Chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Research 2, 023237 (2020).
  94. S. Becker, M. Embree, J. Wittsten, and M. Zworski, Spectral characterization of magic angles in twisted bilayer graphene, Phys. Rev. B 103, 165113 (2021).
  95. J. Wang, Y. Zheng, A. J. Millis, and J. Cano, Chiral approximation to twisted bilayer graphene: Exact intravalley inversion symmetry, nodal structure, and implications for higher magic angles, Phys. Rev. Research 3, 023155 (2021).
  96. Y. Ren, Q. Gao, A. H. MacDonald, and Q. Niu, WKB Estimate of Bilayer Graphene's Magic Twist Angles, Phys. Rev. Lett. 126, 016404 (2021).
  97. A. Alexandradinata and L. Glazman, Semiclassical theory of Landau levels and magnetic breakdown in topological metals, Phys. Rev. B 97, 144422 (2018).

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Phys. Rev. Research 4, L012013– Published 9 February, 2022
  • Received 8 July 2021
  • Revised 16 January 2022
  • Accepted 21 January 2022
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