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Nonlinear Hall effects


The Hall effects comprise one of the oldest but most vital fields in condensed matter physics, and they persistently inspire new findings, such as quantum Hall effects and topological phases of matter. The recently discovered nonlinear Hall effect is a new member of the family of Hall effects. It is characterized as a transverse Hall voltage in response to two longitudinal currents in the Hall measurement, but it does not require time-reversal symmetry to be broken. It has deep connections to symmetry and topology and, thus, opens new avenues by which to probe the spectral, symmetry and topological properties of emergent quantum materials and phases of matter. In this Perspective, we present an overview of the recent progress regarding the nonlinear Hall effect. We discuss the open problems, the prospects of the use of the nonlinear Hall effect in spectroscopic and device applications, and generalizations to other nonlinear transport effects.

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Fig. 1: The nonlinear Hall effect and its generalizations.
Fig. 2: Experimental studies of the nonlinear Hall effect in WTe2.
Fig. 3: Known mechanisms of the nonlinear Hall effect.
Fig. 4: Device applications of the nonlinear Hall effect.


  1. 1.

    Hall, E. H. et al. On a new action of the magnet on electric currents. Am. J. Math. 2, 287–292 (1879).

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Hall, E. H. XVIII. On the “Rotational Coefficient” in nickel and cobalt. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 12, 157–172 (1881).

    Article  Google Scholar 

  3. 3.

    Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    ADS  Article  Google Scholar 

  4. 4.

    Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensional magnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562 (1982).

    ADS  Article  Google Scholar 

  5. 5.

    Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983).

    ADS  Article  Google Scholar 

  6. 6.

    Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    ADS  Article  Google Scholar 

  7. 7.

    Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    ADS  Article  Google Scholar 

  8. 8.

    König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    ADS  Article  Google Scholar 

  9. 9.

    Yu, R. et al. Quantized anomalous Hall effect in magnetic topological insulators. Science 329, 61–64 (2010).

    ADS  Article  Google Scholar 

  10. 10.

    Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    ADS  Article  Google Scholar 

  11. 11.

    Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    ADS  Article  Google Scholar 

  12. 12.

    Niu, Q., Thouless, D. J. & Wu, Y.-S. Quantized Hall conductance as a topological invariant. Phys. Rev. B 31, 3372–3377 (1985).

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    Kohmoto, M. Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160, 343–354 (1985).

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Wen, X. G. & Niu, Q. Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces. Phys. Rev. B 41, 9377–9396 (1990).

    ADS  Article  Google Scholar 

  15. 15.

    Wen, X.-G. Topological orders and edge excitations in fractional quantum Hall states. Adv. Phys. 44, 405–473 (1995).

    ADS  Article  Google Scholar 

  16. 16.

    Wilczek, F. Quantum mechanics of fractional-spin particles. Phys. Rev. Lett. 49, 957–959 (1982).

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractional quantized Hall states. Phys. Rev. Lett. 52, 1583–1586 (1984).

    ADS  Article  Google Scholar 

  18. 18.

    Arovas, D., Schrieffer, J. R. & Wilczek, F. Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722–723 (1984).

    ADS  Article  Google Scholar 

  19. 19.

    Jain, J. K. Composite-fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. 63, 199–202 (1989).

    ADS  Article  Google Scholar 

  20. 20.

    von Klitzing, K. Essay: Quantum Hall effect and the new international system of units. Phys. Rev. Lett. 122, 200001 (2019).

    Article  Google Scholar 

  21. 21.

    Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abelian anyons and topological quantum computation. Rev. Mod. Phys. 80, 1083 (2008).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Cage, M. E. et al. The Quantum Hall Effect (Springer, 2012).

  23. 23.

    Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539–1592 (2010).

    ADS  Article  Google Scholar 

  24. 24.

    Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212–215 (2015).

    ADS  Article  Google Scholar 

  25. 25.

    Machida, Y., Nakatsuji, S., Onoda, S., Tayama, T. & Sakakibara, T. Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order. Nature 463, 210–213 (2010).

    ADS  Article  Google Scholar 

  26. 26.

    Yasuda, K. et al. Geometric Hall effects in topological insulator heterostructures. Nat. Phys. 12, 555–559 (2016).

    Article  Google Scholar 

  27. 27.

    Onsager, L. Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426 (1931).

    ADS  MATH  Article  Google Scholar 

  28. 28.

    Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects. Rev. Mod. Phys. 87, 1213–1260 (2015).

    ADS  Article  Google Scholar 

  29. 29.

    Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

    ADS  Article  Google Scholar 

  30. 30.

    Yao, W., Xiao, D. & Niu, Q. Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B 77, 235406 (2008).

    ADS  Article  Google Scholar 

  31. 31.

    Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in monolayers of MoS2 and other group-VI dichalcogenides. Phys. Rev. Lett. 108, 196802 (2012).

    ADS  Article  Google Scholar 

  32. 32.

    Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).

    ADS  Article  Google Scholar 

  33. 33.

    Sodemann, I. & Fu, L. Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials. Phys. Rev. Lett. 115, 216806 (2015).

    ADS  Article  Google Scholar 

  34. 34.

    Low, T., Jiang, Y. & Guinea, F. Topological currents in black phosphorus with broken inversion symmetry. Phys. Rev. B 92, 235447 (2015).

    ADS  Article  Google Scholar 

  35. 35.

    Facio, J. I. et al. Strongly enhanced Berry dipole at topological phase transitions in BiTeI. Phys. Rev. Lett. 121, 246403 (2018).

    ADS  Article  Google Scholar 

  36. 36.

    You, J.-S., Fang, S., Xu, S.-Y., Kaxiras, E. & Low, T. Berry curvature dipole current in the transition metal dichalcogenides family. Phys. Rev. B 98, 121109 (2018).

    ADS  Article  Google Scholar 

  37. 37.

    Zhang, Y., van den Brink, J., Felser, C. & Yan, B. Electrically tuneable nonlinear anomalous Hall effect in two-dimensional transition-metal dichalcogenides WTe2 and MoTe2. 2D Mater. 5, 044001 (2018).

    Article  Google Scholar 

  38. 38.

    Zhang, Y., Sun, Y. & Yan, B. Berry curvature dipole in Weyl semimetal materials: an ab initio study. Phys. Rev. B 97, 041101 (2018).

    ADS  Article  Google Scholar 

  39. 39.

    Du, Z. Z., Wang, C. M., Lu, H.-Z. & Xie, X. C. Band signatures for strong nonlinear Hall effect in bilayer WTe2. Phys. Rev. Lett. 121, 266601 (2018).

    ADS  Article  Google Scholar 

  40. 40.

    Ma, Q. et al. Observation of the nonlinear Hall effect under time-reversal-symmetric conditions. Nature 565, 337–342 (2019).

    ADS  Article  Google Scholar 

  41. 41.

    Kang, K., Li, T., Sohn, E., Shan, J. & Mak, K. F. Observation of the nonlinear anomalous Hall effect in few-layer WTe2. Nat. Mater. 18, 324–328 (2019).

    ADS  Article  Google Scholar 

  42. 42.

    Hamamoto, K., Ezawa, M., Kim, K. W., MorimotoMorimoto, T. & Nagaosa, N. Nonlinear spin current generation in noncentrosymmetric spin-orbit coupled systems. Phys. Rev. B 95, 224430 (2017).

    ADS  Article  Google Scholar 

  43. 43.

    Araki, Y. Strain-induced nonlinear spin Hall effect in topological Dirac semimetal. Sci. Rep. 8, 15236 (2018).

    ADS  Article  Google Scholar 

  44. 44.

    König, E. J., Dzero, M., Levchenko, A. & Pesin, D. A. Gyrotropic Hall effect in Berry-curved materials. Phys. Rev. B 99, 155404 (2019).

    ADS  Article  Google Scholar 

  45. 45.

    Papaj, M. & Fu, L. Magnus Hall effect. Phys. Rev. Lett. 123, 216802 (2019).

    ADS  Article  Google Scholar 

  46. 46.

    Yu, X.-Q., Zhu, Z.-G., You, J.-S., Low, T. & Su, G. Topological nonlinear anomalous Nernst effect in strained transition metal dichalcogenides. Phys. Rev. B 99, 201410 (2019).

    ADS  Article  Google Scholar 

  47. 47.

    Zeng, C., Nandy, S., Taraphder, A. & Tewari, S. Nonlinear Nernst effect in bilayer WTe2. Phys. Rev. B 100, 245102 (2019).

    ADS  Article  Google Scholar 

  48. 48.

    Tokura, Y. & Nagaosa, N. Nonreciprocal responses from non-centrosymmetric quantum materials. Nat. Commun. 9, 3740 (2018).

    ADS  Article  Google Scholar 

  49. 49.

    Du, L. et al. Engineering symmetry breaking in 2D layered materials. Nat. Rev. Phys. 3, 193–206 (2021).

    Article  Google Scholar 

  50. 50.

    Xiao, D., Chang, M. C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959–2007 (2010).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Shvetsov, O. O., Esin, V. D., Timonina, A. V., Kolesnikov, N. N. & Deviatov, E. V. Nonlinear Hall effect in three-dimensional Weyl and Dirac semimetals. JETP Lett. 109, 715–721 (2019).

    ADS  Article  Google Scholar 

  52. 52.

    Dzsaber, S. et al. Giant spontaneous Hall effect in a nonmagnetic Weyl–Kondo semimetal. Proc. Natl Acad. Sci. USA 118, e2013386118 (2021).

    Article  Google Scholar 

  53. 53.

    Qin, M.-S. et al. Strain tunable Berry curvature dipole, orbital magnetization and nonlinear Hall effect in WSe2 monolayer. Chin. Phys. Lett. 38, 017301 (2021).

    ADS  Article  Google Scholar 

  54. 54.

    Ho, S.-C. et al. Hall effects in artificially corrugated bilayer graphene without breaking time-reversal symmetry. Nat. Electron. 4, 116–125 (2021).

    Article  Google Scholar 

  55. 55.

    Huang, M. et al. Giant nonlinear Hall effect in twisted WSe2. Preprint at arXiv (2020).

  56. 56.

    Tiwari, A. et al. Giant c-axis nonlinear anomalous Hall effect in Td-MoTe2 and WTe2. Nat. Commun. 12, 2049 (2021).

    ADS  Article  Google Scholar 

  57. 57.

    Kiswandhi, A. & Osada, T. Observation of possible nonlinear anomalous Hall effect in organic two-dimensional Dirac fermion system. Preprint at arXiv (2021).

  58. 58.

    He, P. et al. Quantum frequency doubling in the topological insulator Bi2Se3. Nat. Commun. 12, 698 (2021).

    ADS  Article  Google Scholar 

  59. 59.

    Kumar, D. et al. Room-temperature nonlinear Hall effect and wireless radiofrequency rectification in Weyl semimetal TaIrTe4. Nat. Nanotechnol. 16, 421–425 (2021).

    ADS  Article  Google Scholar 

  60. 60.

    Shen, S.-Q. Topological Insulators 2nd edn (Springer, 2017).

    MATH  Book  Google Scholar 

  61. 61.

    Landau, L. D., Pitaevskii, L. P. & Lifshitz, E. M. Electrodynamics of Continuous Media 2nd edn vol. 8 (Elsevier, 2008).

  62. 62.

    Kubo, R. Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957).

    ADS  MathSciNet  Article  Google Scholar 

  63. 63.

    Evans, D. J., Cohen, E. G. D. & Morriss, G. P. Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404 (1993).

    ADS  MATH  Article  Google Scholar 

  64. 64.

    Evans, D. J. & P Morriss, G. Statistical Mechanics of Nonequilbrium Liquids (Cambridge Univ. Press, 2008).

    Book  Google Scholar 

  65. 65.

    Esposito, M., Harbola, U. & Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665–1702 (2009).

    ADS  MathSciNet  MATH  Article  Google Scholar 

  66. 66.

    Campisi, M., Hänggi, P. & Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys. 83, 771–791 (2011).

    ADS  MATH  Article  Google Scholar 

  67. 67.

    Morimoto, T. & Nagaosa, N. Nonreciprocal current from electron interactions in noncentrosymmetric crystals: roles of time reversal symmetry and dissipation. Sci. Rep. 8, 2973 (2018).

    ADS  Article  Google Scholar 

  68. 68.

    Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A Math. Phys. Eng. Sci. 392, 45–57 (1984).

    ADS  MathSciNet  MATH  Google Scholar 

  69. 69.

    Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q. & Zwanziger, J. The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics (Springer, 2003).

    MATH  Book  Google Scholar 

  70. 70.

    Karplus, R. & Luttinger, J. M. Hall effect in ferromagnetics. Phys. Rev. 95, 1154–1160 (1954).

    ADS  MATH  Article  Google Scholar 

  71. 71.

    Xu, S.-Y. et al. Electrically switchable Berry curvature dipole in the monolayer topological insulator WTe2. Nat. Phys. 14, 900–906 (2018).

    Article  Google Scholar 

  72. 72.

    Son, J., Kim, K.-H., Ahn, Y. H., Lee, H.-W. & Lee, J. Strain engineering of the Berry curvature dipole and valley magnetization in monolayer MoS2. Phys. Rev. Lett. 123, 036806 (2019).

    ADS  Article  Google Scholar 

  73. 73.

    Battilomo, R., Scopigno, N. & Ortix, C. Berry curvature dipole in strained graphene: a Fermi surface warping effect. Phys. Rev. Lett. 123, 196403 (2019).

    ADS  MathSciNet  Article  Google Scholar 

  74. 74.

    Chen, C., Wang, H., Wang, D. & Zhang, H. Strain-engineered nonlinear Hall effect in HgTe. Spin 9, 1940017 (2019).

    ADS  Article  Google Scholar 

  75. 75.

    Zhou, B. T., Zhang, C.-P. & Law, K. T. Highly tunable nonlinear Hall effects induced by spin-orbit couplings in strained polar transition-metal dichalcogenides. Phys. Rev. Appl. 13, 024053 (2020).

    ADS  Article  Google Scholar 

  76. 76.

    Singh, S., Kim, J., Rabe, K. M. & Vanderbilt, D. Engineering Weyl phases and nonlinear Hall effects in Td-MoTe2. Phys. Rev. Lett. 125, 046402 (2020).

    ADS  Article  Google Scholar 

  77. 77.

    Xiao, R.-C., Shao, D.-F., Zhang, Z.-Q. & Jiang, H. Two-dimensional metals for piezoelectriclike devices based on Berry-curvature dipole. Phys. Rev. Appl. 13, 044014 (2020).

    ADS  Article  Google Scholar 

  78. 78.

    Samal, S. S., Nandy, S. & Saha, K. Nonlinear transport without spin-orbit coupling or warping in two-dimensional Dirac semimetals. Phys. Rev. B 103, L201202 (2021).

    ADS  Article  Google Scholar 

  79. 79.

    Hu, J.-X., Zhang, C.-P., Xie, Y.-M. & Law, K. T. Nonlinear Hall effects in strained twisted bilayer WSe2. Preprint at arXiv (2020).

  80. 80.

    Zhang, C.-P. et al. Giant nonlinear Hall effect in strained twisted bilayer graphene. Preprint at arXiv (2020).

  81. 81.

    Pantaleón, P. A., Low, T. & Guinea, F. Tunable large Berry dipole in strained twisted bilayer graphene. Phys. Rev. B 103, 205403 (2021).

    ADS  Article  Google Scholar 

  82. 82.

    He, Z. & Weng, H. Giant nonlinear Hall effect in twisted bilayer WTe2. Preprint at arXiv (2021).

  83. 83.

    Smit, J. The spontaneous Hall effect in ferromagnetics I. Physica 21, 877–887 (1955).

    ADS  Article  Google Scholar 

  84. 84.

    Smit, J. The spontaneous Hall effect in ferromagnetics II. Physica 24, 39–51 (1958).

    ADS  Article  Google Scholar 

  85. 85.

    Berger, L. Side-jump mechanism for the Hall effect of ferromagnets. Phys. Rev. B 2, 4559–4566 (1970).

    ADS  Article  Google Scholar 

  86. 86.

    Crépieux, A. & Bruno, P. Theory of the anomalous Hall effect from the Kubo formula and the Dirac equation. Phys. Rev. B 64, 014416 (2001).

    ADS  Article  Google Scholar 

  87. 87.

    Sinitsyn, N. Semiclassical theories of the anomalous Hall effect. J. Phys. Condens. Matter 20, 023201 (2008).

    ADS  Article  Google Scholar 

  88. 88.

    Sinitsyn, N., MacDonald, A., Jungwirth, T., Dugaev, V. & Sinova, J. Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach. Phys. Rev. B 75, 045315 (2007).

    ADS  Article  Google Scholar 

  89. 89.

    Tian, Y., Ye, L. & Jin, X. Proper scaling of the anomalous Hall effect. Phys. Rev. Lett. 103, 087206 (2009).

    ADS  Article  Google Scholar 

  90. 90.

    Hou, D. et al. Multivariable scaling for the anomalous Hall effect. Phys. Rev. Lett. 114, 217203 (2015).

    ADS  Article  Google Scholar 

  91. 91.

    Yue, D. & Jin, X. Towards a better understanding of the anomalous Hall effect. J. Phys. Soc. Jpn. 86, 011006 (2016).

    ADS  Article  Google Scholar 

  92. 92.

    Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Saunders, 1976).

    MATH  Google Scholar 

  93. 93.

    Du, Z. Z., Wang, C. M., Li, S., Lu, H.-Z. & Xie, X. C. Disorder-induced nonlinear Hall effect with time-reversal symmetry. Nat. Commun. 10, 3047 (2019).

    ADS  Article  Google Scholar 

  94. 94.

    Pancharatnam, S. Generalized theory of interference and its applications. Proc. Indian Acad. Sci. Sect A 44, 398–417 (1956).

    MathSciNet  Article  Google Scholar 

  95. 95.

    Sinitsyn, N. A., Niu, Q. & MacDonald, A. H. Coordinate shift in the semiclassical Boltzmann equation and the anomalous Hall effect. Phys. Rev. B 73, 075318 (2006).

    ADS  Article  Google Scholar 

  96. 96.

    Resta, R. Linear and nonlinear Hall conductivity in presence of interaction and disorder. Preprint at arXiv (2021).

  97. 97.

    Nandy, S. & Sodemann, I. Symmetry and quantum kinetics of the nonlinear Hall effect. Phys. Rev. B 100, 195117 (2019).

    ADS  Article  Google Scholar 

  98. 98.

    Xiao, C., Du, Z. Z. & Niu, Q. Theory of nonlinear Hall effects: modified semiclassics from quantum kinetics. Phys. Rev. B 100, 165422 (2019).

    ADS  Article  Google Scholar 

  99. 99.

    Du, Z. Z., Wang, C. M., Sun, H.-P., Lu, H.-Z. & Xie, X. C. Quantum theory of the nonlinear Hall effect. Preprint at arXiv (2020).

  100. 100.

    Gao, Y., Zhang, F. & Zhang, W. Second-order nonlinear Hall effect in Weyl semimetals. Phys. Rev. B 102, 245116 (2020).

    ADS  Article  Google Scholar 

  101. 101.

    König, E. J. & Levchenko, A. Quantum kinetics of anomalous and nonlinear Hall effects in topological semimetals. Preprint at arXiv (2021).

  102. 102.

    Culcer, D., Sekine, A. & MacDonald, A. H. Interband coherence response to electric fields in crystals: Berry-phase contributions and disorder effects. Phys. Rev. B 96, 035106 (2017).

    ADS  Article  Google Scholar 

  103. 103.

    Zhao, L. et al. Evidence of an odd-parity hidden order in a spin–orbit coupled correlated iridate. Nat. Phys. 12, 32–36 (2016).

    Article  Google Scholar 

  104. 104.

    Zhao, L. et al. A global inversion-symmetry-broken phase inside the pseudogap region of YBa2Cu3Oy. Nat. Phys. 13, 250–254 (2017).

    Article  Google Scholar 

  105. 105.

    Xiao, R.-C., Shao, D.-F., Huang, W. & Jiang, H. Electrical detection of ferroelectriclike metals through the nonlinear Hall effect. Phys. Rev. B 102, 024109 (2020).

    ADS  Article  Google Scholar 

  106. 106.

    Rostami, H. & Juričić, V. Probing quantum criticality using nonlinear Hall effect in a metallic Dirac system. Phys. Rev. Res. 2, 013069 (2020).

    Article  Google Scholar 

  107. 107.

    Shao, D.-F., Zhang, S.-H., Gurung, G., Yang, W. & Tsymbal, E. Y. Nonlinear anomalous Hall effect for Néel vector detection. Phys. Rev. Lett. 124, 067203 (2020).

    ADS  Article  Google Scholar 

  108. 108.

    Xiao, J. et al. Berry curvature memory through electrically driven stacking transitions. Nat. Phys. 16, 1028–1034 (2020).

    Article  Google Scholar 

  109. 109.

    Isobe, H., Xu, S.-Y. & Fu, L. High-frequency rectification via chiral Bloch electrons. Sci. Adv. 6, eaay2497 (2020).

    ADS  Article  Google Scholar 

  110. 110.

    Kim, K. W., Morimoto, T. & Nagaosa, N. Shift charge and spin photocurrents in Dirac surface states of topological insulator. Phys. Rev. B 95, 035134 (2017).

    ADS  Article  Google Scholar 

  111. 111.

    Bhalla, P., MacDonald, A. H. & Culcer, D. Resonant photovoltaic effect in doped magnetic semiconductors. Phys. Rev. Lett. 124, 087402 (2020).

    ADS  Article  Google Scholar 

  112. 112.

    Zhang, Y. & Fu, L. Terahertz detection based on nonlinear Hall effect without magnetic field. Proc. Natl Acad. Sci. USA 118, e2100736118 (2021).

    Article  Google Scholar 

  113. 113.

    Nakai, R. & Nagaosa, N. Nonreciprocal thermal and thermoelectric transport of electrons in noncentrosymmetric crystals. Phys. Rev. B 99, 115201 (2019).

    ADS  Article  Google Scholar 

  114. 114.

    Zeng, C., Nandy, S. & Tewari, S. Fundamental relations for anomalous thermoelectric transport coefficients in the nonlinear regime. Phys. Rev. Res. 2, 032066 (2020).

    Article  Google Scholar 

  115. 115.

    Mandal, D., Das, K. & Agarwal, A. Magnus Nernst and thermal Hall effect. Phys. Rev. B 102, 205414 (2020).

    ADS  Article  Google Scholar 

  116. 116.

    Das, S. K., Nag, T. & Nandy, S. Topological Magnus responses in two and three dimensional systems. Preprint at arXiv (2021).

  117. 117.

    Toshio, R., Takasan, K. & Kawakami, N. Anomalous hydrodynamic transport in interacting noncentrosymmetric metals. Phys. Rev. Res. 2, 032021 (2020).

    Article  Google Scholar 

  118. 118.

    Zhang, C.-P., Gao, X.-J., Xie, Y.-M., Po, H. C. & Law, K. T. Higher-order nonlinear anomalous Hall effects induced by Berry curvature multipoles. Preprint at arXiv (2020).

  119. 119.

    He, P. et al. Bilinear magnetoelectric resistance as a probe of three-dimensional spin texture in topological surface states. Nat. Phys. 14, 495–499 (2018).

    Article  Google Scholar 

  120. 120.

    He, P. et al. Observation of out-of-plane spin texture in a SrTiO3(111) two-dimensional electron gas. Phys. Rev. Lett. 120, 266802 (2018).

    ADS  Article  Google Scholar 

  121. 121.

    He, P. et al. Nonlinear magnetotransport shaped by Fermi surface topology and convexity. Nat. Commun. 10, 1290 (2019).

    ADS  Article  Google Scholar 

  122. 122.

    He, P. et al. Nonlinear planar Hall effect. Phys. Rev. Lett. 123, 016801 (2019).

    ADS  Article  Google Scholar 

  123. 123.

    Zhang, S. S.-L. & Vignale, G. Theory of bilinear magneto-electric resistance from topological-insulator surface states. Spintronics XI 10732, 1073215 (2018).

    Google Scholar 

  124. 124.

    Dyrdał, A., Barnaś, J. & Fert, A. Spin-momentum-locking inhomogeneities as a source of bilinear magnetoresistance in topological insulators. Phys. Rev. Lett. 124, 046802 (2020).

    ADS  Article  Google Scholar 

  125. 125.

    Zyuzin, A. A., Silaev, M. & Zyuzin, V. A. Nonlinear chiral transport in Dirac semimetals. Phys. Rev. B 98, 205149 (2018).

    ADS  Article  Google Scholar 

  126. 126.

    Zeng, C., Nandy, S. & Tewari, S. Chiral anomaly induced nonlinear Nernst and thermal Hall effects in Weyl semimetals. Preprint at arXiv (2020).

  127. 127.

    Li, R.-H., Heinonen, O. G., Burkov, A. A. & Zhang, S. S.-L. Nonlinear Hall effect in Weyl semimetals induced by chiral anomaly. Phys. Rev. B 103, 045105 (2021).

    ADS  Article  Google Scholar 

  128. 128.

    Esin, V. D., Timonina, A. V., Kolesnikov, N. N. & Deviatov, E. V. Second-harmonic voltage response for the magnetic Weyl semimetal Co3Sn2S2. JETP Lett. 111, 685–689 (2020).

    ADS  Article  Google Scholar 

  129. 129.

    Watanabe, H. & Yanase, Y. Nonlinear electric transport in odd-parity magnetic multipole systems: application to Mn-based compounds. Phys. Rev. Res. 2, 043081 (2020).

    Article  Google Scholar 

  130. 130.

    Boyd, R. W. Nonlinear Optics (Academic, 1992).

    Google Scholar 

  131. 131.

    Flensberg, K., Hu, B. Y.-K., Jauho, A.-P. & Kinaret, J. M. Linear-response theory of Coulomb drag in coupled electron systems. Phys. Rev. B 52, 14761–14774 (1995).

    ADS  Article  Google Scholar 

  132. 132.

    Kamenev, A. & Oreg, Y. Coulomb drag in normal metals and superconductors: diagrammatic approach. Phys. Rev. B 52, 7516–7527 (1995).

    ADS  Article  Google Scholar 

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We are grateful for the helpful discussions with Huimei Liu, Suyang Xu, Hyunsoo Yang, Zhi-Min Liao, Kin Fai Mak, Ning Wang, Zefei Wu, Meizhen Huang, Tse-Ming Chen, Silke Paschen, Sami Dzsaber, A. Kiswandhi, Archana Tiwari and A. W. Tsen. This work was supported by the National Natural Science Foundation of China (12004157 and 11925402), the National Basic Research Program of China (2015CB921102), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB28000000), Guangdong Province (2020KCXTD001 and 2016ZT06D348), Shenzhen High-level Special Fund (G02206304 and G02206404) and the Science, Technology and Innovation Commission of Shenzhen Municipality (ZDSYS20170303165926217, JCYJ20170412152620376 and KYTDPT20181011104202253).

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Correspondence to Hai-Zhou Lu.

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Nature Reviews Physics thanks Kam Tuen Law, Dimitrie Culcer and the other, anonymous, reviewers for their contribution to the peer review of this work.

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Du, Z.Z., Lu, HZ. & Xie, X.C. Nonlinear Hall effects. Nat Rev Phys (2021).

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