- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Multistate Geometry of Shift Current and Polarization

Alexander Avdoshkin1,2,*, Johannes Mitscherling2,3,*, and Joel E. Moore2,4

  • *These authors contributed equally to this work.
PDF

Phys. Rev. Lett. 135, 066901 – Published 5 August, 2025

DOI: https://doi.org/10.1103/w761-8nf7

Abstract

The quantum metric and Berry curvature capture essential properties of nontrivial Bloch states and underpin many fascinating phenomena. However, it becomes increasingly evident that a more comprehensive understanding of quantum state geometry is necessary to explain properties involving Bloch states of multiple bands, such as optical transitions. To this end, we employ quantum state projectors to develop an explicitly gauge-invariant formalism and demonstrate its power with applications to nonlinear optics and the theory of electronic polarization. We provide a simple expression for the shift current that resolves its precise relation to the moments of electronic polarization, clarifies the treatment of band degeneracies, and reveals its decomposition into the sum of the skewness of the occupied states and intrinsically multistate geometry. The projector approach is applied to calculate nonlinear optical properties of transition metal dichalcogenides (TMDs) layers, using previously calculated minimal tight-binding models, and demonstrated analytically on a three-band generalization of the Rice-Mele chain to elucidate the different contributions. We close with comments on further applications of the projector operator approach to multistate geometry.

View figure in article

Physics Subject Headings (PhySH)

See Also

Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals

Johannes Mitscherling, Alexander Avdoshkin, and Joel E. Moore
Phys. Rev. B 112, 085104 (2025)

Article Text

Supplemental Material

References (76)

  1. N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall effect, Rev. Mod. Phys. 82, 1539 (2010).
  2. J. E. Moore, The birth of topological insulators, Nature (London) 464, 194 (2010).
  3. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  4. S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun. 6, 8944 (2015).
  5. J. S. Hofmann, E. Berg, and D. Chowdhury, Superconductivity, pseudogap, and phase separation in topological flat bands, Phys. Rev. B 102, 201112(R) (2020).
  6. S. A. Chen and K. T. Law, Ginzburg-Landau theory of flat-band superconductors with quantum metric, Phys. Rev. Lett. 132, 026002 (2024).
  7. J. Mitscherling and T. Holder, Bound on resistivity in flat-band materials due to the quantum metric, Phys. Rev. B 105, 085154 (2022).
  8. G. Bouzerar and D. Mayou, Quantum transport in flat bands and supermetallicity, Phys. Rev. B 103, 075415 (2021).
  9. K.-E. Huhtinen and P. Törmä, Conductivity in flat bands from the Kubo-Greenwood formula, arXiv:2212.03192.
  10. A. Kruchkov, Quantum transport anomalies in dispersionless quantum states, Phys. Rev. B 107, L241102 (2023).
  11. I. Komissarov, T. Holder, and R. Queiroz, The quantum geometric origin of capacitance in insulators, Nat. Commun. 15, 4621 (2024).
  12. R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).
  13. B. Mera and T. Ozawa, Kähler geometry and Chern insulators: Relations between topology and the quantum metric, Phys. Rev. B 104, 045104 (2021).
  14. P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional Chern insulators, Phys. Rev. B 108, 205144 (2023).
  15. Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  16. B. Mera and J. Mitscherling, Nontrivial quantum geometry of degenerate flat bands, Phys. Rev. B 106, 165133 (2022).
  17. W. Chen and G. von Gersdorff, Measurement of interaction-dressed Berry curvature and quantum metric in solids by optical absorption, SciPost Phys. Core 5, 040 (2022).
  18. T. Kashihara, Y. Michishita, and R. Peters, Quantum metric on the Brillouin zone in correlated electron systems and its relation to topology for Chern insulators, Phys. Rev. B 107, 125116 (2023).
  19. N. Verma and R. Queiroz, Instantaneous response and quantum geometry of insulators, arXiv:2403.07052.
  20. 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).
  21. J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Riemannian geometry of resonant optical responses, Nat. Phys. 18, 290 (2022).
  22. A. Bouhon, A. Timmel, and R.-J. Slager, Quantum geometry beyond projective single bands, arXiv:2303.02180.
  23. A. Avdoshkin and F. K. Popov, Extrinsic geometry of quantum states, Phys. Rev. B 107, 245136 (2023).
  24. A. Avdoshkin, Geometry of degenerate quantum states, configurations of m-planes and invariants on complex Grassmannians, arXiv:2404.03234.
  25. O. Antebi, J. Mitscherling, and T. Holder, The drude weight of a flatband metal, Phys. Rev. B 110, L241111 (2024).
  26. J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000).
  27. B. M. Fregoso, T. Morimoto, and J. E. Moore, Quantitative relationship between polarization differences and the zone-averaged shift photocurrent, Phys. Rev. B 96, 075421 (2017).
  28. P. Zhu and A. Alexandradinata, Anomalous shift and optical vorticity in the steady photovoltaic current, Phys. Rev. B 110, 115108 (2024).
  29. I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000).
  30. T. Holder, D. Kaplan, and B. Yan, Consequences of time-reversal-symmetry breaking in the light-matter interaction: Berry curvature, quantum metric, and diabatic motion, Phys. Rev. Res. 2, 033100 (2020).
  31. D. Kaplan, T. Holder, and B. Yan, Unifying semiclassics and quantum perturbation theory at nonlinear order, SciPost Phys. 14, 082 (2023).
  32. W. J. Jankowski and R.-J. Slager, Quantized shift response in multi-gap topological phases, Phys. Rev. Lett. 133, 186601 (2024).
  33. A. Kruchkov and S. Ryu, Spectral sum rules reflect topological and quantum-geometric invariants, arXiv:2312.17318.
  34. Y. Onishi and L. Fu, Quantum weight, arXiv:2406.06783.
  35. Y. Onishi and L. Fu, Universal relation between energy gap and dielectric constant, Phys. Rev. B 110, 155107 (2024).
  36. Y. Onishi and L. Fu, Fundamental bound on topological gap, Phys. Rev. X 14, 011052 (2024).
  37. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  38. A. M. Essin, A. M. Turner, J. E. Moore, and D. Vanderbilt, Orbital magnetoelectric coupling in band insulators, Phys. Rev. B 81, 205104 (2010).
  39. A. Malashevich, I. Souza, S. Coh, and D. Vanderbilt, Theory of orbital magnetoelectric response, New J. Phys. 12, 053032 (2010).
  40. B. Sturman and V. Fridkin, Photovoltaic and Photo-Refractive Effects in Noncentrosymmetric Materials (Routledge, London, 1992), 10.1201/9780203743416.
  41. S. M. Young and A. M. Rappe, First principles calculation of the shift current photovoltaic effect in ferroelectrics, Phys. Rev. Lett. 109, 116601 (2012).
  42. A. M. Cook, B. M. Fregoso, F. de Juan, S. Coh, and J. E. Moore, Design principles for shift current photovoltaics, Nat Commun 8, 14176 (2017).
  43. A. Alexandradinata, Quantization of intraband and interband berry phases in the shift current, Phys. Rev. B 110, 075159 (2024).
  44. O. Pozo and F. de Juan, Computing observables without eigenstates: Applications to bloch Hamiltonians, Phys. Rev. B 102, 115138 (2020).
  45. A. Graf and F. Piéchon, Berry curvature and quantum metric in N-band systems: An eigenprojector approach, Phys. Rev. B 104, 085114 (2021).
  46. S. Patankar, L. Wu, B. Lu, M. Rai, J. D. Tran, T. Morimoto, D. E. Parker, A. G. Grushin, N. L. Nair, J. G. Analytis, J. E. Moore, J. Orenstein, and D. H. Torchinsky, Resonance-enhanced optical nonlinearity in the Weyl semimetal taas, Phys. Rev. B 98, 165113 (2018).
  47. A. Agarwal, W. J. Jankowski, D. Bennett, and R.-J. Slager, Shift photocurrent vortices from topological polarization textures, npj 2D Mater. Appl. 9, 43 (2025).
  48. G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao, Three-band tight-binding model for monolayers of group-vib transition metal dichalcogenides, Phys. Rev. B 88, 085433 (2013).
  49. J. Mitscherling, A. Avdoshkin, and J. E. Moore, companion paper, Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals, Phys. Rev. B 112, 085104 (2025).
  50. See Supplemental Material at http://link.aps.org/supplemental/10.1103/w761-8nf7 for the details on (i) the symmetry decomposition of the shift current, (ii) the separation of the third-order geometric tensor Cα;βγmn, (iii) the full expression for the summation over band indices of Qβγmn and Cα;βγmn, (iv) further details on the torsion tensor, (v) all details of the application to transition metal dichalcogenides (TMDs), and (vi) the discussion of the Stub Rice-Mele model as a minimal three-band model for skewness and the corresponding interband corrections. The SM includes Refs. [20,32,45,48,49,51,52].
  51. M. J. Rice and E. J. Mele, Elementary excitations of a linearly conjugated diatomic polymer, Phys. Rev. Lett. 49, 1455 (1982).
  52. P. M. Neves, J. P. Wakefield, S. Fang, H. Nguyen, L. Ye, and J. G. Checkelsky, Crystal net catalog of model flat band materials, npj Comput. Mater. 10, 39 (2024).
  53. D. Kaplan, T. Holder, and B. Yan, Twisted photovoltaics at terahertz frequencies from momentum shift current, Phys. Rev. Res. 4, 013209 (2022).
  54. 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).
  55. 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).
  56. L.-K. Shi, D. Zhang, K. Chang, and J. C. W. Song, Geometric photon-drag effect and nonlinear shift current in centrosymmetric crystals, Phys. Rev. Lett. 126, 197402 (2021).
  57. H. Wang, X. Tang, J. Li, and X. Qian, Generalized Wilson loop method for nonlinear light-matter interaction, npj Qantum Mater. 7, 61 (2022).
  58. 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).
  59. R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993).
  60. R. Resta and D. Vanderbilt, Theory of polarization: A modern approach, in Physics of Ferroelectrics: A Modern Perspective (Springer, Berlin, Heidelberg, 2007), pp. 31–68.
  61. J. Ibañez-Azpiroz, F. de Juan, and I. Souza, Assessing the role of interatomic position matrix elements in tight-binding calculations of optical properties, SciPost Phys. 12, 070 (2022).
  62. S. Chen, S. Chaudhary, G. Refael, and C. Lewandowski, Enhancing shift current via virtual multiband transitions, arXiv:2402.06721.
  63. G. E. Topp, G. Jotzu, J. W. McIver, L. Xian, A. Rubio, and M. A. Sentef, Topological Floquet engineering of twisted bilayer graphene, Phys. Rev. Res. 1, 023031 (2019).
  64. G. E. Topp, C. J. Eckhardt, D. M. Kennes, M. A. Sentef, and P. Törmä, Light-matter coupling and quantum geometry in moiré materials, Phys. Rev. B 104, 064306 (2021).
  65. W. T. Tai and M. Claassen, Quantum-geometric light-matter coupling in correlated quantum materials, arXiv:2303.01597.
  66. M. Walicki, C. J. Eckhardt, and M. A. Sentef, Floquet engineering nearly flat bands through quantum-geometric light-matter coupling with surface polaritons, arXiv:2406.01298.
  67. D. Porlles and W. Chen, Quantum geometry of singlet superconductors, Phys. Rev. B 108, 094508 (2023).
  68. M. M. Hirschmann and J. Mitscherling, Symmetry-enforced double Weyl points, multiband quantum geometry, and singular flat bands of doping-induced states at the Fermi level, Phys. Rev. Mater. 8, 014201 (2024).
  69. B. Jorissen, L. Covaci, and B. Partoens, Comparative analysis of tight-binding models for transition metal dichalcogenides, SciPost Phys. Core 7, 004 (2024).
  70. 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).
  71. X.-W. Zhang, Y. Ren, C. Wang, T. Cao, and D. Xiao, Gate-tunable phonon magnetic moment in bilayer graphene, Phys. Rev. Lett. 130, 226302 (2023).
  72. W. Chen, X.-W. Zhang, Y. Su, T. Cao, D. Xiao, and S.-Z. Lin, Gauge theory of giant phonon magnetic moment in doped dirac semimetals, Phys. Rev. B 111, 035126 (2025).
  73. M. Taherinejad and D. Vanderbilt, Adiabatic pumping of Chern-Simons axion coupling, Phys. Rev. Lett. 114, 096401 (2015).
  74. J. Mitscherling, Longitudinal and anomalous hall conductivity of a general two-band model, Phys. Rev. B 102, 165151 (2020).
  75. V. Kozii, A. Avdoshkin, S. Zhong, and J. E. Moore, Intrinsic anomalous Hall conductivity in a nonuniform electric field, Phys. Rev. Lett. 126, 156602 (2021).
  76. Y. Fang, J. Cano, and Sayed Ali Akbar Ghorashi, Quantum geometry induced nonlinear transport in altermagnets, Phys. Rev. Lett. 133, 106701 (2024).

Outline

Information

Phys. Rev. Lett. 135, 066901– Published 5 August, 2025
  • Received 8 October 2024
  • Revised 22 May 2025
  • Accepted 11 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation