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Anomalous geometric transport signatures of topological Euler class

Ashwat Jain1,*,†, Wojciech J. Jankowski2,*,‡, and Robert-Jan Slager2,3,§

  • 1Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, United Kingdom
  • 2Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
  • 3Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

  • *These authors contributed equally to this work.
  • †Contact author: jain.ashwat@gmail.com
  • ‡Contact author: wjj25@cam.ac.uk
  • §Contact author: rjs269@cam.ac.uk
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Phys. Rev. B 111, 235149 – Published 27 June, 2025

DOI: https://doi.org/10.1103/5gmg-q1z5

Abstract

We investigate Riemannian quantum-geometric structures in semiclassical transport features of two-dimensional multigap topological phases. In particular, we study nonlinear Hall-like bulk electric current responses and, accordingly, semiclassical equations of motion induced by the presence of a topological Euler invariant. We provide analytic understanding of these quantities by phrasing them in terms of momentum-space geodesics and geodesic deviation equations and further corroborate these insights with numerical solutions. Within this framework, we moreover uncover anomalous bulk dynamics associated with the second- and third-order nonlinear Hall conductivities induced by a patch Euler invariant. As a main finding, our results show how one can reconstruct the Euler invariant by coupling to electric fields at nonlinear order and from the gradients of the electric fields. Furthermore, we comment on the possibility of deducing the nontrivial non-Abelian Euler class invariant specifically in second-order nonlinear ballistic conductance measurements within a triple-contact setup, which was recently proposed to probe the Euler characteristics of more general Fermi surfaces. Generally, our results provide a route for deducing the topology in real materials that exhibit the Euler invariant by analyzing bulk electrical currents.

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References (85)

  1. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  2. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  3. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  4. R. Arouca, A. Cappelli, and T. H. Hansson, Quantum field theory anomalies in condensed matter physics, SciPost Phys. Lect. Notes 62 (2022).
  5. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  6. E. Fradkin, E. Dagotto, and D. Boyanovsky, Physical realization of the parity anomaly in condensed matter physics, Phys. Rev. Lett. 57, 2967 (1986).
  7. M. M. Otrokov, I. I. Klimovskikh, H. Bentmann, D. Estyunin, A. Zeugner, Z. S. Aliev, S. Gaß, A. U. B. Wolter, A. V. Koroleva, A. M. Shikin et al., Prediction and observation of an antiferromagnetic topological insulator, Nature (London) 576, 416 (2019).
  8. N. H. Jo, L.-L. Wang, R.-J. Slager, J. Yan, Y. Wu, K. Lee, B. Schrunk, A. Vishwanath, and A. Kaminski, Intrinsic axion insulating behavior in antiferromagnetic MnBi6 Te10, Phys. Rev. B 102, 045130 (2020).
  9. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  10. H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nat. Phys. 5, 438 (2009).
  11. D. E. Kharzeev, The chiral magnetic effect and anomaly-induced transport, Prog. Part. Nucl. Phys. 75, 133 (2014).
  12. S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, and A. Vishwanath, Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals, Phys. Rev. X 4, 031035 (2014).
  13. J. Behrends, A. G. Grushin, T. Ojanen, and J. H. Bardarson, Visualizing the chiral anomaly in Dirac and Weyl semimetals with photoemission spectroscopy, Phys. Rev. B 93, 075114 (2016).
  14. M. M. Vazifeh and M. Franz, Electromagnetic response of Weyl semimetals, Phys. Rev. Lett. 111, 027201 (2013).
  15. A. Cortijo, Y. Ferreirós, K. Landsteiner, and M. A. H. Vozmediano, Elastic gauge fields in Weyl semimetals, Phys. Rev. Lett. 115, 177202 (2015).
  16. J. Gooth, A. C. Niemann, T. Meng, A. G. Grushin, K. Landsteiner, B. Gotsmann, F. Menges, M. Schmidt, C. Shekhar, V. Süß et al., Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP, Nature (London) 547, 324 (2017).
  17. R.-J. Slager, The translational side of topological band insulators, J. Phys. Chem. Solids 128, 24 (2019).
  18. Y. Ran, Y. Zhang, and A. Vishwanath, One-dimensional topologically protected modes in topological insulators with lattice dislocations., Nat. Phys. 5, 298 (2009).
  19. O. Parrikar, T. L. Hughes, and R. G. Leigh, Torsion, parity-odd response, and anomalies in topological states, Phys. Rev. D 90, 105004 (2014).
  20. Y. Ferreiros, Y. Kedem, E. J. Bergholtz, and J. H. Bardarson, Mixed axial-torsional anomaly in Weyl semimetals, Phys. Rev. Lett. 122, 056601 (2019).
  21. J. Nissinen, Emergent spacetime and gravitational Nieh-Yan anomaly in chiral p+ip Weyl superfluids and superconductors, Phys. Rev. Lett. 124, 117002 (2020).
  22. J. Nissinen, Emergent geometry, torsion and anomalies in non-relativistic topological matter, J. Phys.: Conf. Ser. 2531, 012002 (2023).
  23. M. Stone, Gravitational anomalies and thermal Hall effect in topological insulators, Phys. Rev. B 85, 184503 (2012).
  24. J. Nissinen and G. E. Volovik, Elasticity tetrads, mixed axial-gravitational anomalies, and (3+1)-d quantum Hall effect, Phys. Rev. Res. 1, 023007 (2019).
  25. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8, 50 (2017).
  26. R.-J. Slager, A. Mesaros, V. Juričić, and J. Zaanen, The space group classification of topological band-insulators, Nat. Phys. 9, 98 (2013).
  27. J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Topological classification of crystalline insulators through band structure combinatorics, Phys. Rev. X 7, 041069 (2017).
  28. B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature (London) 547, 298 (2017).
  29. K. Shiozaki and M. Sato, Topology of crystalline insulators and superconductors, Phys. Rev. B 90, 165114 (2014).
  30. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  31. A. Bouhon, T. Bzdusek, and R.-J. Slager, Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102, 115135 (2020).
  32. Z. Davoyan, W. J. Jankowski, A. Bouhon, and R.-J. Slager, Three-dimensional PT-symmetric topological phases with a Pontryagin index, Phys. Rev. B 109, 165125 (2024).
  33. W. J. Jankowski, M. Noormandipour, A. Bouhon, and R.-J. Slager, Disorder-induced topological quantum phase transitions in multigap Euler semimetals, Phys. Rev. B 110, 064202 (2024).
  34. W. J. Jankowski, A. S. Morris, Z. Davoyan, A. Bouhon, F. N. Ünal, and R.-J. Slager, Non-Abelian Hopf-Euler insulators, Phys. Rev. B 110, 075135 (2024).
  35. W. J. Jankowski, A. S. Morris, A. Bouhon, F. N. Ünal, and R.-J. Slager, Optical manifestations and bounds of topological Euler class, Phys. Rev. B 111, L081103 (2025).
  36. W. J. Jankowski and R.-J. Slager, Quantized integrated shift effect in multigap topological phases, Phys. Rev. Lett. 133, 186601 (2024).
  37. A. Bouhon, A. M. Black-Schaffer, and R.-J. Slager, Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry, Phys. Rev. B 100, 195135 (2019).
  38. 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).
  39. A. Bouhon, Q. Wu, R.-J. Slager, H. Weng, O. V. Yazyev, and T. Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nat. Phys. 16, 1137 (2020).
  40. Q. Wu, A. A. Soluyanov, and T. Bzdusek, Non-Abelian band topology in noninteracting metals, Science 365, 1273 (2019).
  41. F. N. Ünal, A. Bouhon, and R.-J. Slager, Topological Euler class as a dynamical observable in optical lattices, Phys. Rev. Lett. 125, 053601 (2020).
  42. W. Zhao, Y.-B. Yang, Y. Jiang, Z. Mao, W. Guo, L. Qiu, G. Wang, L. Yao, L. He, Z. Zhou, Y. Xu, and L. Duan, Quantum simulation for topological Euler insulators, Commun. Phys. 5, 223 (2022).
  43. O. Breach, R.-J. Slager, and F. N. Ünal, Interferometry of non-Abelian band singularities and Euler class topology, Phys. Rev. Lett. 133, 093404 (2024).
  44. B. Jiang, A. Bouhon, Z.-K. Lin, X. Zhou, B. Hou, F. Li, R.-J. Slager, and J.-H. Jiang, Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nat. Phys. 17, 1239 (2021).
  45. B. Jiang, A. Bouhon, S.-Q. Wu, Z.-L. Kong, Z.-K. Lin, R.-J. Slager, and J.-H. Jiang, Observation of an acoustic topological Euler insulator with meronic waves, Sci. Bull. 69, 1653 (2024).
  46. B. Peng, A. Bouhon, B. Monserrat, and R.-J. Slager, Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nat. Commun. 13, 423 (2022).
  47. B. Peng, A. Bouhon, R.-J. Slager, and B. Monserrat, Multigap topology and non-Abelian braiding of phonons from first principles, Phys. Rev. B 105, 085115 (2022).
  48. R.-J. Slager, A. Bouhon, and F. N. Ünal, Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems, Nat. Commun. 15, 1144 (2024).
  49. V. Könye, A. Bouhon, I. C. Fulga, R.-J. Slager, J. van den Brink, and J. I. Facio, Chirality flip of Weyl nodes and its manifestation in strained MoTe2, Phys. Rev. Res. 3, L042017 (2021).
  50. S. H. Lee, Y. Qian, and B.-J. Yang, Euler band topology in spin-orbit coupled magnetic systems, Phys. Rev. B 111, 245127 (2025).
  51. T. B. Wahl, W. J. Jankowski, A. Bouhon, G. Chaudhary, and R.-J. Slager, Exact projected entangled pair ground states with topological Euler invariant, Nat. Commun. 16, 284 (2025).
  52. J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  53. J. Ahn, G.-Y. Guo, and N. Nagaosa, Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals, Phys. Rev. X 10, 041041 (2020).
  54. J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Riemannian geometry of resonant optical responses, Nat. Phys. 18, 290 (2022).
  55. A. Bouhon, A. Timmel, and R.-J. Slager, Quantum geometry beyond projective single bands, arXiv:2303.02180.
  56. E. Kretschmann, Über die prinzipielle Bestimmbarkeit der berechtigten Bezugssysteme beliebiger Relativitätstheorien (i), (ii), Annalen der Physik (Leipzig) 353, 907 (1915).
  57. B. M. Fregoso, R. A. Muniz, and J. E. Sipe, Jerk current: A novel bulk photovoltaic effect, Phys. Rev. Lett. 121, 176604 (2018).
  58. Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of Bloch electrons and its dynamical implications, Phys. Rev. Lett. 112, 166601 (2014).
  59. A. Gao, Y.-F. Liu, J.-X. Qiu, B. Ghosh, T. V. Trevisan, Y. Onishi, C. Hu, T. Qian, H.-J. Tien, S.-W. Chen et al., Quantum metric nonlinear Hall effect in a topological antiferromagnetic heterostructure, Science 381, 181 (2023).
  60. M. F. Lapa and T. L. Hughes, Semiclassical wave packet dynamics in nonuniform electric fields, Phys. Rev. B 99, 121111(R) (2019).
  61. S. Lai, H. Liu, Z. Zhang, J. Zhao, X. Feng, N. Wang, C. Tang, Y. Liu, K. S. Novoselov, S. A. Yang, and W.-B. Gao, Third-order nonlinear Hall effect induced by the Berry-connection polarizability tensor, Nat. Nanotechnol. 16, 869 (2021).
  62. J. Mi, J. Li, M. Li, S. Xu, S. Yu, Z. Li, X. Fan, H. Zhu, Q. Tao, L. Li, and Z. Xu, Third order nonlinear transport properties in topological chiral antiferromagnetic semimetal CoNb3S6, arXiv:2312.05824.
  63. Y. Gao, Z.-Q. Zhang, and K.-H. Ding, Third-order nonlinear Hall effect in two-dimensional Dirac systems, New J. Phys. 25, 053013 (2023).
  64. D. Mandal, S. Sarkar, K. Das, and A. Agarwal, Quantum geometry induced third-order nonlinear transport responses, Phys. Rev. B 110, 195131 (2024).
  65. H. Li, C. Zhang, C. Zhou, C. Ma, X. Lei, Z. Jin, H. He, B. Li, K. T. Law, and J. Wang, Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4, Nat. Commun. 15, 7779 (2024).
  66. Ó. P. Ocaña and I. Souza, Multipole theory of optical spatial dispersion in crystals, SciPost Phys. 14, 118 (2023).
  67. Y. Gao, Semiclassical dynamics and nonlinear charge current, Front. Phys. 14, 33404 (2019).
  68. T. B. Smith, L. Pullasseri, and A. Srivastava, Momentum-space gravity from the quantum geometry and entropy of Bloch electrons, Phys. Rev. Res. 4, 013217 (2022).
  69. Y. Gao and D. Xiao, Nonreciprocal directional dichroism induced by the quantum metric dipole, Phys. Rev. Lett. 122, 227402 (2019).
  70. N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang et al., Quantum-metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).
  71. D. Kaplan, T. Holder, and B. Yan, Unification of nonlinear anomalous Hall effect and nonreciprocal magnetoresistance in metals by the quantum geometry, Phys. Rev. Lett. 132, 026301 (2024).
  72. C. Kane, Quantized nonlinear conductance in ballistic metals, Phys. Rev. Lett. 128, 076801 (2022).
  73. M. Kang, S. Kim, Y. Qian, P. M. Neves, L. Ye, J. Jung, D. Puntel, F. Mazzola, S. Fang, C. Jozwiak et al., Measurements of the quantum geometric tensor in solids, Nat. Phys. 21, 110 (2024).
  74. C. Mondal, R. Ghadimi, and B.-J. Yang, Non-Abelian charge conversion in bilayer binary honeycomb lattice systems, arXiv:2411.06724.
  75. E. J. Nicol and J. P. Carbotte, Optical conductivity of bilayer graphene with and without an asymmetry gap, Phys. Rev. B 77, 155409 (2008).
  76. A. Bouhon and R.-J. Slager, Multi-gap topological conversion of Euler class via band-node braiding: Minimal models, PT-linked nodal rings, and chiral heirs, arXiv:2203.16741.
  77. K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  78. G. Xu, B. Lian, and S.-C. Zhang, Intrinsic quantum anomalous Hall effect in the kagome lattice Cs2LiMn3F12, Phys. Rev. Lett. 115, 186802 (2015).
  79. C. Guo, C. Putzke, S. Konyzheva, X. Huang, M. Gutierrez-Amigo, I. Errea, D. Chen, M. G. Vergniory, C. Felser, M. H. Fischer, T. Neupert, and P. J. W. Moll, Switchable chiral transport in charge-ordered kagome metal CsV3Sb5, Nature (London) 611, 461 (2022).
  80. I. Sodemann and L. Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials, Phys. Rev. Lett. 115, 216806 (2015).
  81. 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).
  82. A. S. Morris, A. Bouhon, and R.-J. Slager, Andreev reflection in Euler materials, New J. Phys. 26, 023014 (2024).
  83. B. Hetényi and P. Lévay, Fluctuations, uncertainty relations, and the geometry of quantum state manifolds, Phys. Rev. A 108, 032218 (2023).
  84. H. Liu, J. Zhao, Y.-X. Huang, X. Feng, C. Xiao, W. Wu, S. Lai, W.-B. Gao, and S. A. Yang, Berry connection polarizability tensor and third-order Hall effect, Phys. Rev. B 105, 045118 (2022).
  85. L. Xiang, C. Zhang, L. Wang, and J. Wang, Third-order intrinsic anomalous Hall effect with generalized semiclassical theory, Phys. Rev. B 107, 075411 (2023).

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Phys. Rev. B 111, 235149– Published 27 June, 2025
  • Received 4 December 2024
  • Revised 9 May 2025
  • Accepted 16 June 2025
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