- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

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

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

Johannes Mitscherling1,2,*, Alexander Avdoshkin3,2,*, and Joel E. Moore2,4

  • *These authors contributed equally to this work.
PDF

Phys. Rev. B 112, 085104 – Published 5 August, 2025

DOI: https://doi.org/10.1103/qscv-qxqt

Abstract

The importance of simple geometrical invariants, such as the Berry curvature and quantum metric, constructed from the Bloch states of a crystal has become well established over four decades of research. More complex aspects of geometry emerge in properties linking multiple bands, such as optical responses. In the companion work [Phys. Rev. Lett. 135, 066901 (2025)], we introduced multistate geometrical invariants using an explicitly gauge-invariant formalism based on projection operators, which we used to clarify the relation between the shift current and the theory of electronic polarization among other advancements for second-order nonlinear optics. Here, we provide considerably more detail on the projector formalism and the geometrical invariants arising in the vicinity of a specific value of crystal momentum. We combine the introduction to multistate quantum geometry with broadly relevant algebraic relationships and detailed example calculations, enabling extensions toward future applications to topological and geometrical properties of insulators and metals.

View figure in article

Physics Subject Headings (PhySH)

See Also

Multistate Geometry of Shift Current and Polarization

Alexander Avdoshkin, Johannes Mitscherling, and Joel E. Moore
Phys. Rev. Lett. 135, 066901 (2025)

Article Text

References (52)

  1. 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).
  2. M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the hofstadter spectrum, Phys. Rev. Lett. 75, 1348 (1995).
  3. M.-C. Chang and Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands, Phys. Rev. B 53, 7010 (1996).
  4. G. Sundaram and Q. Niu, Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects, Phys. Rev. B 59, 14915 (1999).
  5. J. E. Avron, R. Seiler, and B. Simon, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51, 51 (1983).
  6. G. Provost and J. P. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys. 76, 289 (1980).
  7. A. Avdoshkin, J. Mitscherling, and J. E. Moore, companion paper, Multistate geometry of shift current and polarization, Phys. Rev. Lett. 135, 066901 (2025).
  8. O. Pozo and F. de Juan, Computing observables without eigenstates: Applications to bloch hamiltonians, Phys. Rev. B 102, 115138 (2020).
  9. A. Graf and F. Piéchon, Berry curvature and quantum metric in N-band systems: An eigenprojector approach, Phys. Rev. B 104, 085114 (2021).
  10. B. Mera and T. Ozawa, Kähler geometry and Chern insulators: Relations between topology and the quantum metric, Phys. Rev. B 104, 045104 (2021).
  11. B. Mera and J. Mitscherling, Nontrivial quantum geometry of degenerate flat bands, Phys. Rev. B 106, 165133 (2022).
  12. A. Avdoshkin and F. K. Popov, Extrinsic geometry of quantum states, Phys. Rev. B 107, 245136 (2023).
  13. A. Avdoshkin, Geometry of degenerate quantum states, configurations of m-planes and invariants on complex grassmannians, arXiv:2404.03234.
  14. O. Antebi, J. Mitscherling, and T. Holder, The drude weight of a flatband metal, Phys. Rev. B 110, L241111 (2024).
  15. N. Dupuis, Field Theory of Condensed Matter and Ultracold Gases (World Scientific (Europe), London, 2023).
  16. 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).
  17. 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).
  18. W. T. Tai and M. Claassen, Quantum-geometric light-matter coupling in correlated quantum materials, arXiv:2303.01597.
  19. 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).
  20. 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).
  21. N. Verma and R. Queiroz, Instantaneous response and quantum geometry of insulators, arXiv:2403.07052.
  22. 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).
  23. J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Riemannian geometry of resonant optical responses, Nat. Phys. 18, 290 (2022).
  24. A. Bouhon, A. Timmel, and R.-J. Slager, Quantum geometry beyond projective single bands, arXiv:2303.02180.
  25. 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).
  26. D. Kaplan, T. Holder, and B. Yan, Unifying semiclassics and quantum perturbation theory at nonlinear order, SciPost Phys. 14, 082 (2023).
  27. W. J. Jankowski and R.-J. Slager, Quantized shift response in multi-gap topological phases, Phys. Rev. Lett. 133, 186601 (2024).
  28. A. Kruchkov and S. Ryu, Spectral sum rules reflect topological and quantum-geometric invariants, arXiv:2312.17318.
  29. Y. Onishi and L. Fu, Quantum weight: A fundamental property of quantum many-body systems, Phys. Rev. Res. 7, 023158 (2025).
  30. Y. Onishi and L. Fu, Universal relation between energy gap and dielectric constant, Phys. Rev. B 110, 155107 (2024).
  31. Y. Onishi and L. Fu, Fundamental bound on topological gap, Phys. Rev. X 14, 011052 (2024).
  32. B. Bradlyn and P. Abbamonte, Spectral density and sum rules for second-order response functions, Phys. Rev. B 110, 245132 (2024).
  33. B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, NJ, 2013).
  34. J. Romero, C. A. Velasquez, and J. D. Vergara, N-bein formalism for the parameter space of quantum geometry, J. Phys. A: Math. Theor. 57, 375301 (2024).
  35. R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).
  36. P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vortexability: A unifying criterion for ideal fractional Chern insulators, Phys. Rev. B 108, 205144 (2023).
  37. S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun. 6, 8944 (2015).
  38. 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).
  39. 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).
  40. B. Hetényi and P. Lévay, Fluctuations, uncertainty relations, and the geometry of quantum state manifolds, Phys. Rev. A 108, 032218 (2023).
  41. Y. Onishi, A. Avdoshkin, and L. Fu, Geometric bound on the structure factor and a harmonic condition on band geometry, Phys. Rev. B 112, 035158 (2025).
  42. 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).
  43. 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).
  44. H. Wang, X. Tang, J. Li, and X. Qian, Generalized Wilson loop method for nonlinear light-matter interaction, npj Quantum Mater. 7, 61 (2022).
  45. P. Zhu and A. Alexandradinata, Anomalous shift and optical vorticity in the steady photovoltaic current, Phys. Rev. B 110, 115108 (2024).
  46. J. E. Sipe and A. I. Shkrebtii, Second-order optical response in semiconductors, Phys. Rev. B 61, 5337 (2000).
  47. J. Mitscherling and W. Metzner, Longitudinal conductivity and Hall coefficient in two-dimensional metals with spiral magnetic order, Phys. Rev. B 98, 195126 (2018).
  48. R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993).
  49. 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.
  50. 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).
  51. Y. Jiang, T. Holder, and B. Yan, Revealing quantum geometry in nonlinear quantum materials, Rep. Prog. Phys. 88, 076502 (2025).
  52. L. Gao, Z. Addison, E. J. Mele, and A. M. Rappe, Intrinsic fermi-surface contribution to the bulk photovoltaic effect, Phys. Rev. Res. 3, L042032 (2021).

Outline

Information

Phys. Rev. B 112, 085104– Published 5 August, 2025
  • Received 5 December 2024
  • Accepted 25 June 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 Max Planck Society.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation