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Nonlinear Thermoelectric Effects Driven by Electron Quantum Geometry

Xu Yang1,2 and Brian Skinner1

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PRX Energy 5, 033013 – Published 26 August, 2026

DOI: https://doi.org/10.1103/f659-cgsd

Abstract

Nonlinear thermoelectric effects offer the potential to enable new energy technologies, such as voltage-controlled thermal switching and thermoelectric rectification. In this work, we examine how the quantum geometry of the electron bands gives rise to nonlinear thermoelectric responses in materials with suitable symmetries. We derive a series of nonlinear thermoelectric effects governed by the Berry curvature dipole and the quantum metric dipole, respectively. Among them, we identify a particularly interesting quantized thermoelectric response that directly measures the total chirality of Weyl points below the Fermi level. For general nonlinear responses, we derive the nonlinear analogs of the Wiedemann-Franz law and Mott’s formula. These provide a means to estimate the magnitude of nonlinear thermoelectric responses based on existing nonlinear Hall measurements. Our estimates suggest that these effects should be observable in several candidate materials, with In-doped Pb1−xSnxTe standing out as the most promising. Our work offers new insights into the experimental study of quantum geometry through nonlinear thermoelectric measurements.

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Popular Summary

The thermoelectric effect is the conversion of heat into electric current accomplished by electrons flowing through a material. For example, heating one side of a material causes electrons to diffuse from the hot side to the cold side, thereby generating an electric current. Here, the authors study a family of nonlinear thermoelectric effects, in which either an electric current or thermal current is generated by the product of a temperature gradient and an applied electric field. These effects are driven by the quantum geometry of the electron system and in particular by the Berry curvature dipole and quantum metric dipole of the electron bands. The effects can enable a variety of new thermoelectric technologies, including voltage-controlled thermal switching and thermoelectric rectification. The authors highlight particularly promising material candidates and identify a new kind of quantized thermoelectric response.

Article Text

References (98)

  1. A. Shakouri, Recent developments in semiconductor thermoelectric physics and materials, Annu. Rev. Mater. Res. 41, 399 (2011).
  2. Q. Yan and M. G. Kanatzidis, High-performance thermoelectrics and challenges for practical devices, Nat. Mater. 21, 503 (2022).
  3. R. Freer and A. V. Powell, Realising the potential of thermoelectric technology: A roadmap, J. Mater. Chem. C 8, 441 (2020).
  4. X. Zhang and L.-D. Zhao, Thermoelectric materials: Energy conversion between heat and electricity, J. Materiomics 1, 92 (2015).
  5. C. Fu, Y. Sun, and C. Felser, Topological thermoelectrics, APL Mater. 8, 040913 (2020).
  6. H. Adachi, F. Ando, T. Hirai, R. Modak, M. A. Grayson, and K.-I. Uchida, Fundamentals and advances in transverse thermoelectrics, Appl. Phys. Express 18, 090101 (2025).
  7. B. Skinner, P. Chakraborty, J. Scales, and J. P. Heremans, Design principles for topological thermoelectrics, arXiv:2509.25371.
  8. C. Zeng, S. Nandy, and S. Tewari, Fundamental relations for anomalous thermoelectric transport coefficients in the nonlinear regime, Phys. Rev. Res. 2, 032066 (2020).
  9. H. Varshney and A. Agarwal, Intrinsic nonlinear Nernst and Seebeck effect, arXiv:2409.11108.
  10. J.-W. Chen, T. Ishii, S. Pu, and N. Yamamoto, Nonlinear chiral transport phenomena, Phys. Rev. D 93, 125023 (2016).
  11. C. Zeng, S. Nandy, A. Taraphder, and S. Tewari, Nonlinear Nernst effect in bilayer WTe2, Phys. Rev. B 100, 245102 (2019).
  12. C. Zeng, S. Nandy, and S. Tewari, Nonlinear transport in Weyl semimetals induced by Berry curvature dipole, Phys. Rev. B 103, 245119 (2021).
  13. R. Nakai and N. Nagaosa, Nonreciprocal thermal and thermoelectric transport of electrons in noncentrosymmetric crystals, Phys. Rev. B 99, 115201 (2019).
  14. H. Arisawa, Y. Fujimoto, T. Kikkawa, and E. Saitoh, Observation of nonlinear thermoelectric effect in MoGe/Y3Fe5O12, Nat. Commun. 15, 6912 (2024).
  15. T. Nomoto, A. Kikkawa, K. Nakazawa, T. Yamaguchi, and F. Kagawa, Observation of the nonlinear chiral thermoelectric Hall effect in tellurium, Nat. Phys. 21, 1920 (2025).
  16. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  17. G. Wehmeyer, T. Yabuki, C. Monachon, J. Wu, and C. Dames, Thermal diodes, regulators, and switches: Physical mechanisms and potential applications, Appl. Phys. Rev. 4, 041304 (2017).
  18. C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Solid-state thermal rectifier, Science 314, 1121 (2006).
  19. B. Li, L. Wang, and G. Casati, Thermal diode: Rectification of heat flux, Phys. Rev. Lett. 93, 184301 (2004).
  20. A. Kirilyuk, A. V. Kimel, and T. Rasing, Ultrafast optical manipulation of magnetic order, Rev. Mod. Phys. 82, 2731 (2010).
  21. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceed. R. Soc. London, Ser. A 392, 45 (1984).
  22. J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  23. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  24. E. Rossi, Quantum metric and correlated states in two-dimensional systems, Curr. Opin. Solid State Mater. Sci. 25, 100952 (2021).
  25. P. Törmä, Essay: Where can quantum geometry lead us?, Phys. Rev. Lett. 131, 240001 (2023).
  26. N. Verma, P. J. Moll, T. Holder, and R. Queiroz, Quantum geometry: Revisiting electronic scales in quantum matter, arXiv:2504.07173
  27. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
  28. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  29. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  30. F. D. M. Haldane, Berry curvature on the Fermi surface: Anomalous Hall effect as a topological Fermi-liquid property, Phys. Rev. Lett. 93, 206602 (2004).
  31. A. A. Zyuzin and A. A. Burkov, Topological response in Weyl semimetals and the chiral anomaly, Phys. Rev. B 86, 115133 (2012).
  32. D. Son and B. Spivak, Chiral anomaly and classical negative magnetoresistance of Weyl metals, Phys. Rev. B 88, 104412 (2013).
  33. S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun. 6, 8944 (2015).
  34. A. Srivastava and A. Imamoğlu, Signatures of Bloch-band geometry on excitons: Nonhydrogenic spectra in transition-metal dichalcogenides, Phys. Rev. Lett. 115, 166802 (2015).
  35. Y. Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Phys. Rev. B 91, 214405 (2015).
  36. F. Piéchon, A. Raoux, J.-N. Fuchs, and G. Montambaux, Geometric orbital susceptibility: Quantum metric without Berry curvature, Phys. Rev. B 94, 134423 (2016).
  37. J. E. Moore and J. Orenstein, Confinement-induced Berry phase and helicity-dependent photocurrents, Phys. Rev. Lett. 105, 026805 (2010).
  38. 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).
  39. Q. Ma, S.-Y. Xu, H. Shen, D. MacNeill, V. Fatemi, T.-R. Chang, A. M. Mier Valdivia, S. Wu, Z. Du, C.-H. Hsu et al., Observation of the nonlinear Hall effect under time-reversal-symmetric conditions, Nature (London) 565, 337 (2019).
  40. K. Kang, T. Li, E. Sohn, J. Shan, and K. F. Mak, Nonlinear anomalous Hall effect in few-layer WTe2, Nat. Mater. 18, 324 (2019).
  41. Z. Z. Du, H.-Z. Lu, and X. C. Xie, Nonlinear Hall effects, Nat. Rev. Phys. 3, 744 (2021).
  42. C. Wang, Y. Gao, and D. Xiao, Intrinsic nonlinear Hall effect in antiferromagnetic tetragonal CuMnAs, Phys. Rev. Lett. 127, 277201 (2021).
  43. H. Liu, J. Zhao, Y.-X. Huang, W. Wu, X.-L. Sheng, C. Xiao, and S. A. Yang, Intrinsic second-order anomalous Hall effect and its application in compensated antiferromagnets, Phys. Rev. Lett. 127, 277202 (2021).
  44. 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).
  45. X.-Q. Yu, Z.-G. Zhu, J.-S. You, T. Low, and G. Su, Topological nonlinear anomalous Nernst effect in strained transition metal dichalcogenides, Phys. Rev. B 99, 201410 (2019).
  46. Y. D. Wang, Z.-G. Zhu, and G. Su, Quantum theory of nonlinear thermal response, Phys. Rev. B 106, 035148 (2022).
  47. T. Yamaguchi, K. Nakazawa, and A. Yamakage, Microscopic theory of nonlinear Hall effect induced by electric field and temperature gradient, Phys. Rev. B 109, 205117 (2024).
  48. S. Tsirkin and I. Souza, On the separation of Hall and ohmic nonlinear responses, SciPost Phys. Core 5, 039 (2022).
  49. D. E. Parker, T. Morimoto, J. Orenstein, and J. E. Moore, Diagrammatic approach to nonlinear optical response with application to Weyl semimetals, Phys. Rev. B 99, 045121 (2019).
  50. R. Karplus and J. M. Luttinger, Hall effect in ferromagnetics, Phys. Rev. 95, 1154 (1954).

  51. We note that the first terms of the RHS of Eqs. (42) and (55) in Ref. [13] do not appear to incorporate the minus sign associated with the temperature gradient as defined in Eq. (40). After correcting these, their expressions are in exact agreement with our expressions Eqs. (7) and (8).

  52. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt-Saunders, New York, 1976).
  53. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford university press, New York, 2001).
  54. D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase effect in anomalous thermoelectric transport, Phys. Rev. Lett. 97, 026603 (2006).
  55. 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).
  56. L. Smrcka and P. Streda, Transport coefficients in strong magnetic fields, J. Phys. C 10, 2153 (1977).
  57. T. Qin, Q. Niu, and J. Shi, Energy magnetization and the thermal Hall effect, Phys. Rev. Lett. 107, 236601 (2011).
  58. D. L. Bergman and V. Oganesyan, Theory of dissipationless nernst effects, Phys. Rev. Lett. 104, 066601 (2010).
  59. C. Zhang, S. Tewari, V. M. Yakovenko, and S. Das Sarma, Anomalous Nernst effect from a chiral d-density-wave state in underdoped cuprate superconductors, Phys. Rev. B 78, 174508 (2008).
  60. J. P. Jay-Gerin, Thermoelectric power of semiconductors in the extreme quantum limit. I. The “electron-diffusion” contribution., J. Phys. Chem. Solids 35, 81 (1974).
  61. 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).
  62. J. M. Luttinger, Theory of thermal transport coefficients, Phys. Rev. 135, A1505 (1964).
  63. N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Thermoelectric response of an interacting two-dimensional electron gas in a quantizing magnetic field, Phys. Rev. B 55, 2344 (1997).
  64. A. Shitade, Heat transport as torsional responses and Keldysh formalism in a curved spacetime, Prog. Theor. Exp. Phys. 2014, 123I01 (2014).
  65. A. Kapustin and L. Spodyneiko, Thermal hall conductance and a relative topological invariant of gapped two-dimensional systems, Phys. Rev. B 101, 045137 (2020).
  66. O. Matsyshyn and I. Sodemann, Nonlinear Hall acceleration and the quantum rectification sum rule, Phys. Rev. Lett. 123, 246602 (2019).
  67. N. Peshcherenko, C. Felser, and Y. Zhang, Quantized nonlinear Hall effect from chiral monopole, Phys. Rev. B 110, 155143 (2024).
  68. F. De Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in Weyl semimetals, Nat. Commun. 8, 15995 (2017).
  69. D. Rees, K. Manna, B. Lu, T. Morimoto, H. Borrmann, C. Felser, J. E. Moore, D. H. Torchinsky, and J. Orenstein, Helicity-dependent photocurrents in the chiral weyl semimetal RhSi, Sci. Adv. 6, eaba0509 (2020).
  70. Z. Ni, K. Wang, Y. Zhang, O. Pozo, B. Xu, X. Han, K. Manna, J. Paglione, C. Felser, A. G. Grushin, F. de Juan, E. J. Mele, and L. Wu, Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal cosi, Nat. Commun. 12, 154 (2021).
  71. F. Flicker, F. De Juan, B. Bradlyn, T. Morimoto, M. G. Vergniory, and A. G. Grushin, Chiral optical response of multifold fermions, Phys. Rev. B 98, 155145 (2018).
  72. A. Avdoshkin, V. Kozii, and J. E. Moore, Interactions remove the quantization of the chiral photocurrent at weyl points, Phys. Rev. Lett. 124, 196603 (2020).
  73. F. D. M. Haldane, Anomalous Hall effect, dissipationless currents, and quasi-particle berry phases: A new topological ingredient in the fermi-liquid theory of metals, in Invited Talk at the Yukawa International Seminar 2004 (YKIS 2004) (Yukawa Institute for Fundamental Physics, Kyoto University, Kyoto, Japan, 2004).
  74. S. Wang and J. D. Sau, Interaction robustness of the chiral anomaly in Weyl semimetals and Luttinger liquids from a mixed anomaly approach, Phys. Rev. B 113, 075141 (2026).
  75. S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, T.-R. Chang, B. K. Wang, N. Alidoust, G. Bian, M. Neupane, D. Sanchez, H. Zheng, H.-T. Jeng, A. Bansil, T. Neupert, H. Lin, and M. Z. Hasan, New type of Weyl semimetal with quadratic double weyl fermions, Proceed. Natl. Acad. Sci. 113, 1180 (2016).
  76. D. Takane, Z. Wang, S. Souma, K. Nakayama, T. Nakamura, H. Oinuma, Y. Nakata, H. Iwasawa, C. Cacho, T. Kim, K. Horiba, H. Kumigashira, T. Takahashi, Y. Ando, and T. Sato, Observation of chiral fermions with a large topological charge and associated fermi-arc surface states in cosi, Phys. Rev. Lett. 122, 076402 (2019).
  77. J. I. Facio, D. Efremov, K. Koepernik, J.-S. You, I. Sodemann, and J. Van Den Brink, Strongly enhanced berry dipole at topological phase transitions in BiTeI, Phys. Rev. Lett. 121, 246403 (2018).
  78. Z.-F. Zhang, Z.-G. Zhu, and G. Su, Symmetry dictionary on charge and spin nonlinear responses for all magnetic point groups with nontrivial topological nature, Natl. Sci. Rev. 10, nwad104 (2023).
  79. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  80. X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  81. N. Armitage, E. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  82. N. Wang, D. Kaplan, Z. Zhang, T. Holder, N. Cao, A. Wang, X. Zhou, F. Zhou, Z. Jiang, C. Zhang, S. Ru, H. Cai, K. Watanabe, T. Taniguchi, B. Yan, and W. Gao, Quantum-metric-induced nonlinear transport in a topological antiferromagnet, Nature (London) 621, 487 (2023).
  83. Y. Zhang, Y. Sun, and B. Yan, Berry curvature dipole in weyl semimetal materials: An Ab Initio study, Phys. Rev. B 97, 041101 (2018).
  84. C.-L. Zhang, T. Liang, M. S. Bahramy, N. Ogawa, V. Kocsis, K. Ueda, Y. Kaneko, M. Kriener, and Y. Tokura, Berry curvature generation detected by nernst responses in ferroelectric weyl semimetal, Proceed. Natl. Acad. Sci. 118, e2111855118 (2021).
  85. C.-L. Zhang, T. Liang, Y. Kaneko, N. Nagaosa, and Y. Tokura, Giant berry curvature dipole density in a ferroelectric weyl semimetal, npj Quantum Mater. 7, 1 (2022).
  86. M. Hirschberger, R. Chisnell, Y. S. Lee, and N. P. Ong, Thermal Hall effect of spin excitations in a kagome magnet, Phys. Rev. Lett. 115, 106603 (2015).
  87. D. D. Vu, R. A. Nelson, B. L. Wooten, J. Barker, J. E. Goldberger, and J. P. Heremans, Magnon gap mediated lattice thermal conductivity in MnBi2Te4, Phys. Rev. B 108, 144402 (2023).
  88. B. L. Wooten, R. Iguchi, P. Tang, J. S. Kang, K.-I. Uchida, G. E. W. Bauer, and J. P. Heremans, Electric field–dependent phonon spectrum and heat conduction in ferroelectrics, Sci. Adv. 9, eadd7194 (2023).
  89. L. Heywood, J. Wen, X. Yang, T. Song, Y. Zhao, B. Skinner, J. E. Goldberger, B. Yan, and J. P. Heremans, Anomalous heat current driven by a berry curvature dipole in (Pb1−xSnx)1−yInyTe, in APS Global Physics Summit 2026 (APS, Denver, CO, 2026), Session MAR-C30, https://meetings-archive.aps.org/smt/2026/mar-c30/12/ (unpublished).
  90. H. Watanabe, A. Daido, and Y. Yanase, Nonreciprocal optical response in parity-breaking superconductors, Phys. Rev. B 105, 024308 (2022).
  91. H. Tanaka, H. Watanabe, and Y. Yanase, Nonlinear optical responses in noncentrosymmetric superconductors, Phys. Rev. B 107, 024513 (2023).
  92. X. Hu, J. H. Han, and Y. Ran, Supercurrent-induced anomalous thermal Hall effect as a new probe to superconducting gap anisotropy, Phys. Rev. B 108, L041106 (2023).
  93. O. Matsyshyn, G. Vignale, and J. C. Song, Superconducting berry curvature dipole, arXiv:2410.21363.
  94. R. Matsumoto and S. Murakami, Theoretical prediction of a rotating magnon wave packet in ferromagnets, Phys. Rev. Lett. 106, 197202 (2011).
  95. A. Mook, J. Henk, and I. Mertig, Tunable magnon Weyl points in ferromagnetic pyrochlores, Phys. Rev. Lett. 117, 157204 (2016).
  96. Y. Su and X. R. Wang, Chiral anomaly of Weyl magnons in stacked honeycomb ferromagnets, Phys. Rev. B 96, 104437 (2017).
  97. T. Liu and Z. Shi, Magnon quantum anomalies in Weyl ferromagnets, Phys. Rev. B 99, 214413 (2019).
  98. X. Yang, Dataset for Figures in “Nonlinear thermoelectric effects driven by electron quantum geometry” (Xu Yang and Brian Skinner) (2026).

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PRX Energy 5, 033013– Published 26 August, 2026
  • Received 24 November 2025
  • Revised 12 June 2026
  • Accepted 15 July 2026
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