Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Riemannian geometry of resonant optical responses


The geometry of quantum states is well established as a basis for understanding the response of electronic systems to static electromagnetic fields, as exemplified by the theory of the quantum and anomalous Hall effects. However, it has been challenging to relate quantum geometry to resonant optical responses. The main obstacle is that optical transitions involve a pair of states, whereas existing geometrical properties are defined for a single state. As a result, a concrete geometric understanding of optical responses has so far been limited to two-level systems, where the Hilbert space is completely determined by a single state and its orthogonal complement. Here, we construct a general theory of Riemannian geometry for resonant optical processes by identifying transition dipole moment matrix elements as tangent vectors. This theory applies to arbitrarily high-order responses, suggesting that optical responses can generally be thought of as manifestations of the Riemannian geometry of quantum states. We use our theory to show that third-order photovoltaic Hall effects are related to the Riemann curvature tensor and demonstrate an experimentally accessible regime where they dominate the response.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Geometry of the cell-periodic Bloch state and optical transitions.
Fig. 2: Third-order photovoltaic Hall conductivity of a Dirac fermion.
Fig. 3: First-principles calculations on massive Dirac materials.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The codes that support the findings of this study are available from the corresponding authors upon reasonable request.


  1. 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).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  6. Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  7. Neupert, T., Chamon, C. & Mudry, C. Measuring the quantum geometry of Bloch bands with current noise. Phys. Rev. B 87, 245103 (2013).

    Article  ADS  Google Scholar 

  8. Peotta, S. & Törmä, P. Superfluidity in topologically nontrivial flat bands. Nat. Commun. 6, 8944 (2015).

    Article  ADS  Google Scholar 

  9. Xie, F., Song, Z., Lian, B. & Bernevig, B. A. Topology-bounded superfluid weight in twisted bilayer graphene. Phys. Rev. Lett. 124, 167002 (2020).

    Article  ADS  Google Scholar 

  10. Lapa, M. F. & Hughes, T. L. Semiclassical wave packet dynamics in nonuniform electric fields. Phys. Rev. B 99, 121111 (2019).

    Article  ADS  Google Scholar 

  11. Gao, Y. & Xiao, D. Nonreciprocal directional dichroism induced by the quantum metric dipole. Phys. Rev. Lett. 122, 227402 (2019).

    Article  ADS  Google Scholar 

  12. Zhao, Y., Gao, Y. & Xiao, D. Electric polarization in inhomogeneous crystals. Phys. Rev. B 104, 144203 (2021).

    Article  ADS  Google Scholar 

  13. Kozii, V., Avdoshkin, A., Zhong, S. & Moore, J. E. Intrinsic anomalous Hall conductivity in non-uniform electric field. Phys. Rev. Lett. 126, 156602 (2021).

    Article  ADS  Google Scholar 

  14. Gao, Y., Yang, S. A. & Niu, Q. Geometrical effects in orbital magnetic susceptibility. Phys. Rev. B 91, 214405 (2015).

    Article  ADS  Google Scholar 

  15. Rhim, J.-W., Kim, K. & Yang, B.-J. Quantum distance and anomalous Landau levels of flat bands. Nature 584, 59–63 (2020).

    Article  ADS  Google Scholar 

  16. Hosur, P. Circular photogalvanic effect on topological insulator surfaces: Berry-curvature-dependent response. Phys. Rev. B 83, 035309 (2011).

    Article  ADS  Google Scholar 

  17. Morimoto, T. & Nagaosa, N. Topological nature of nonlinear optical effects in solids. Sci. Adv. 2, 1501524 (2016).

    Article  ADS  Google Scholar 

  18. Nagaosa, N. & Morimoto, T. Concept of quantum geometry in optoelectronic processes in solids: application to solar cells. Adv. Mater. 29, 1603345 (2017).

    Article  Google Scholar 

  19. Ahn, J., Guo, G.-Y. & Nagaosa, N. Low-frequency divergence and quantum geometry of the bulk photovoltaic effect in topological semimetals. Phys. Rev. X 10, 041041 (2020).

    Google Scholar 

  20. de Juan, F., Grushin, A. G., Morimoto, T. & Moore, J. E. Quantized circular photogalvanic effect in Weyl semimetals. Nat. Commun. 8, 15995 (2017).

    Article  ADS  Google Scholar 

  21. de Juan, F. et al. Difference frequency generation in topological semimetals. Phys. Rev. Res. 2, 012017 (2020).

    Article  Google Scholar 

  22. Flicker, F. et al. Chiral optical response of multifold fermions. Phys. Rev. B 98, 155145 (2018).

    Article  ADS  Google Scholar 

  23. Holder, T., Kaplan, D. & Yan, B. Consequences of time-reversal-symmetry breaking in the light-matter interaction: Berry curvature, quantum metric, and diabatic motion. Phys. Rev. Res. 2, 033100 (2020).

    Article  Google Scholar 

  24. Watanabe, H. & Yanase, Y. Chiral photocurrent in parity-violating magnet and enhanced response in topological antiferromagnet. Phys. Rev. X 11, 011001 (2021).

    Google Scholar 

  25. Sturman, B. I., Fridkin, V. M. & Bradley, J. E. S. The Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials 1st edn, Vol. 8 (Routledge, 1992).

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

    Article  ADS  Google Scholar 

  27. Boyd, R. W. Nonlinear Optics 4th edn (Academic Press, 2020).

  28. Provost, J. P. & Vallee, G. Riemannian structure on manifolds of quantum states. Commun. Math. Phys. 76, 289–301 (1980).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Aversa, C. & Sipe, J. E. Nonlinear optical susceptibilities of semiconductors: results with a length-gauge analysis. Phys. Rev. B 52, 14636–14645 (1995).

    Article  ADS  Google Scholar 

  30. Ventura, G. B., Passos, D. J., Lopes dos Santos, J. M. B., Viana Parente Lopes, J. M. & Peres, N. M. R. Gauge covariances and nonlinear optical responses. Phys. Rev. B 96, 035431 (2017).

    Article  ADS  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  32. Blount, E. I. in Solid State Physics Vol. 13 (eds Seitz, F. & Turnbull, D) 305–373 (Elsevier, 1962).

  33. Bieliavsky, P., Cahen, M., Gutt, S., Rawnsley, J. & Schwachhöfer, L. Symplectic connections. Int. J. Geom. Methods Mod. Phys. 3, 375–420 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  34. Sipe, J. E. & Shkrebtii, A. I. Second-order optical response in semiconductors. Phys. Rev. B 61, 5337–5352 (2000).

    Article  ADS  Google Scholar 

  35. Oka, T. & Aoki, H. Photovoltaic Hall effect in graphene. Phys. Rev. B 79, 081406 (2009).

    Article  ADS  Google Scholar 

  36. Fregoso, B. M. Bulk photovoltaic effects in the presence of a static electric field. Phys. Rev. B 100, 064301 (2019).

    Article  ADS  Google Scholar 

  37. Kim, H. et al. Accurate gap determination in monolayer and bilayer graphene/h-BN Moiré superlattices. Nano Lett. 18, 7732–7741 (2018).

    Article  ADS  Google Scholar 

  38. Acun, A. et al. Germanene: the germanium analogue of graphene. J. Phys. Condens. Matter 27, 443002 (2015).

    Article  Google Scholar 

  39. Xia, Y. et al. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface. Nat. Phys. 5, 398–402 (2009).

    Article  Google Scholar 

  40. Chandra, H. K. & Guo, G.-Y. Quantum anomalous Hall phase and half-metallic phase in ferromagnetic (111) bilayers of 4d and 5d transition metal perovskites. Phys. Rev. B 95, 134448 (2017).

    Article  ADS  Google Scholar 

  41. Hazra, T., Verma, N. & Randeria, M. Bounds on the superconducting transition temperature: applications to twisted bilayer graphene and cold atoms. Phys. Rev. X 9, 031049 (2019).

    Google Scholar 

  42. Verma, N., Hazra, T. & Randeria, M. Optical spectral weight, phase stiffness and Tc bounds for trivial and topological flat band superconductors. Proc. Natl Acad. Sci. USA 118, 2106744118 (2021).

    Article  Google Scholar 

  43. Ahn, J. & Nagaosa, N. Superconductivity-induced spectral weight transfer due to quantum geometry. Phys. Rev. B 104, L100501 (2021).

    Article  ADS  Google Scholar 

  44. Nakahara, M. Geometry, Topology and Physics 2nd edn (CRC Press, 2003).

  45. Allendoerfer, C. B. & Weil, A. The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Am. Math. Soc. 53, 101–129 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  46. Ma, Y.-Q., Gu, S.-J., Chen, S., Fan, H. & Liu, W.-M. The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems. EPL 103, 10008 (2013).

    Article  ADS  Google Scholar 

  47. Tan, X. et al. Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit. Phys. Rev. Lett. 122, 210401 (2019).

    Article  ADS  Google Scholar 

  48. Zhu, Y.-Q. et al. Note on ‘Experimental measurement of quantum metric tensor and related topological phase transition with a superconducting qubit’. Preprint at (2019).

  49. Ma, Y.-Q. Euler characteristic number of the energy band and the reason for its non-integer values. Preprint at (2020).

  50. Ahn, J., Kim, D., Kim, Y. & Yang, B.-J. Band topology and linking structure of nodal line semimetals with Z2 monopole charges. Phys. Rev. Lett. 121, 106403 (2018).

    Article  ADS  Google Scholar 

  51. Ahn, J., Park, S., Kim, D., Kim, Y. & Yang, B.-J. Stiefel-Whitney classes and topological phases in band theory. Chin. Phys. B 28, 117101 (2019).

    Article  ADS  Google Scholar 

  52. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).

    Article  ADS  Google Scholar 

  53. Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).

    Article  ADS  Google Scholar 

  54. Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).

    Article  ADS  Google Scholar 

  55. Wang, X., Yates, J. R., Souza, I. & Vanderbilt, D. Ab initio calculation of the anomalous hall conductivity by Wannier interpolation. Phys. Rev. B 74, 195118 (2006).

    Article  ADS  Google Scholar 

  56. Marzari, N., Mostofi, A. A., Yates, J. R., Souza, I. & Vanderbilt, D. Maximally localized Wannier functions: theory and applications. Rev. Mod. Phys. 84, 1419–1475 (2012).

    Article  ADS  Google Scholar 

  57. Ibañez-Azpiroz, J., Tsirkin, S. S. & Souza, I. Ab initio calculation of the shift photocurrent by Wannier interpolation. Phys. Rev. B 97, 245143 (2018).

    Article  ADS  Google Scholar 

Download references


We appreciate E. Khalaf and D. Parker for helpful discussions and thank M. Christos for useful comments on the manuscript. J.A. was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (grant no. 2020R1A6A3A03037129). J.A. and A.V. were supported by the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the United States Department of Energy Office of Science, Office of Basic Energy Sciences, through the Ames Laboratory under contract no. DE-AC02-07CH11358. G.-Y.G. acknowledges the support from the Ministry of Science and Technology and National Center for Theoretical Sciences in Taiwan and thanks the National Center for High-performance Computing in Taiwan for the computing time. N.N. was supported by Japan Science and Technology Agency CREST grant nos. JPMJCR1874 and JPMJCR16F1 and by Japan Society for the Promotion of Science KAKENHI grant no. 18H03676.

Author information

Authors and Affiliations



J.A. conceived the original idea and performed the theoretical analysis. G.-Y.G. performed first-principles calculations. N.N. and A.V. supervised the project. All authors discussed results and contributed to the formulation of the theory and writing of the manuscript.

Corresponding authors

Correspondence to Junyeong Ahn, Guang-Yu Guo, Naoto Nagaosa or Ashvin Vishwanath.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Peer review information Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Notes 1–4.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ahn, J., Guo, GY., Nagaosa, N. et al. Riemannian geometry of resonant optical responses. Nat. Phys. 18, 290–295 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing