- 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

Self-consistent evaluation of the Berry connection for Wannier functions

Martin Thümmler1, Thomas Lettau2, Alexander Croy1, Ulf Peschel3, and Stefanie Gräfe4,5

PDF

Phys. Rev. B 114, 245203 – Published 13 October, 2026

DOI: https://doi.org/10.1103/rbc9-v3jz

Abstract

The Berry connection is a gauge-dependent quantity frequently used to describe the optical response of solids. Its evaluation requires a k derivative with respect to the cell-periodic part of the Bloch functions and is commonly calculated in the Wannier basis by using overlap matrices of cell-periodic parts of Bloch functions at neighboring k points. So far, all proposed interpolation schemes for the Berry connection do not account for the matrix structure of the overlap matrices explicitly but treat the matrix elements as independent, or only distinguish between diagonal and off-diagonal entries. In this work, we propose a self-consistent interpolation scheme based on the matrix logarithm resulting in a strongly improved accuracy. Furthermore, we discuss how the basis set incompleteness of the bands used in the ab initio calculation imposes constraints on the accuracy. We quantify the basis incompleteness based on the singular values of the overlap matrices and relate it to the invariant part of the spread functional ΩI of the Wannier functions. Numerical calculations for monolayer MoS2 and bulk Si demonstrate that the proposed interpolation scheme is much less sensitive to the Wannierization details and leads to an improved quality of the velocity matrix and the optical conductivity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. G. H. Wannier, The structure of electronic excitation levels in insulating crystals, Phys. Rev. 52, 191 (1937).
  2. Y. Noel, C. M. Zicovich-Wilson, B. Civalleri, P. D'Arco, and R. Dovesi, Polarization properties of ZnO and BeO: An ab initio study through the Berry phase and Wannier functions approaches, Phys. Rev. B 65, 014111 (2001).
  3. T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Orbital magnetization in periodic insulators, Phys. Rev. Lett. 95, 137205 (2005).
  4. M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza, Wannier-based calculation of the orbital magnetization in crystals, Phys. Rev. B 85, 014435 (2012).
  5. S. Coh and D. Vanderbilt, Electric polarization in a Chern insulator, Phys. Rev. Lett. 102, 107603 (2009).
  6. X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation, Phys. Rev. B 74, 195118 (2006).
  7. A. A. Soluyanov and D. Vanderbilt, Wannier representation of Z2 topological insulators, Phys. Rev. B 83, 035108 (2011).
  8. D. Gresch, G. Autès, O. V. Yazyev, M. Troyer, D. Vanderbilt, B. A. Bernevig, and A. A. Soluyanov, Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials, Phys. Rev. B 95, 075146 (2017).
  9. F. Giustino, M. L. Cohen, and S. G. Louie, Electron-phonon interaction using Wannier functions, Phys. Rev. B 76, 165108 (2007).
  10. H. Lee, S. Poncé, K. Bushick, S. Hajinazar, J. Lafuente-Bartolome, J. Leveillee, C. Lian, J.-M. Lihm, F. Macheda, H. Mori, H. Paudyal, W. H. Sio, S. Tiwari, M. Zacharias, X. Zhang, N. Bonini, E. Kioupakis, E. R. Margine, and F. Giustino, Electron–phonon physics from first principles using the EPW code, npj Comput. Mater. 9, 156 (2023).
  11. W. Kohn, Analytic properties of Bloch waves and Wannier functions, Phys. Rev. 115, 809 (1959).
  12. D. Monaco, G. Panati, A. Pisante, and S. Teufel, Optimal decay of Wannier functions in Chern and quantum Hall insulators, Commun. Math. Phys. 359, 61 (2018).
  13. H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  14. N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
  15. G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, et al., wannier90 as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  16. S. S. Tsirkin, High performance Wannier interpolation of Berry curvature and related quantities with wannierberri code, npj Comput. Mater. 7, 33 (2021).
  17. N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
  18. I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
  19. A. Damle, L. Lin, and L. Ying, SCDM-k: Localized orbitals for solids via selected columns of the density matrix, J. Comput. Phys. 334, 1 (2017).
  20. A. Damle, A. Levitt, and L. Lin, Variational formulation for Wannier functions with entangled band structure, Multiscale Model. Simul. 17, 167 (2019).
  21. R. Sakuma, Symmetry-adapted Wannier functions in the maximal localization procedure, Phys. Rev. B 87, 235109 (2013).
  22. K. Koepernik, O. Janson, Y. Sun, and J. van den Brink, Symmetry-conserving maximally projected Wannier functions, Phys. Rev. B 107, 235135 (2023).
  23. T. Koretsune, Construction of maximally-localized Wannier functions using crystal symmetry, Comput. Phys. Commun. 285, 108645 (2023).
  24. J.-M. Lihm, M. Ghim, S.-J. Hong, and C.-H. Park, Accurate calculation of Wannier centers, position matrix, and composite operators using translationally equivariant and higher-order finite differences, Phys. Rev. B (2026), doi: 10.1103/wv76-36ny.
  25. T. Cole and D. Vanderbilt, Exact expression for the Berry connection in the projection gauge, Phys. Rev. B 113, 245106 (2026).
  26. E. Blount, Formalisms of band theory, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1962), Vol. 13, pp. 305–373.
  27. D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  28. N. A. Spaldin, A beginner's guide to the modern theory of polarization, J. Solid State Chem. 195, 2 (2012).
  29. J. Ibañez-Azpiroz, S. S. Tsirkin, and I. Souza, Ab initio calculation of the shift photocurrent by Wannier interpolation, Phys. Rev. B 97, 245143 (2018).
  30. J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Spectral and Fermi surface properties from Wannier interpolation, Phys. Rev. B 75, 195121 (2007).
  31. R. E. F. Silva, F. Martín, and M. Ivanov, High harmonic generation in crystals using maximally localized Wannier functions, Phys. Rev. B 100, 195201 (2019).
  32. A. Urru, I. Souza, Ó. P. Ocaña, S. S. Tsirkin, and D. Vanderbilt, Optical spatial dispersion via Wannier interpolation, Phys. Rev. B 112, 045201 (2025).
  33. M. Thümmler, A. Croy, U. Peschel, and S. Gräfe, Modeling high-order harmonic generation in quantum dots using a real-space tight-binding approach, J. Chem. Phys. 164, 094114 (2026).
  34. F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems, Phys. Rev. Lett. 52, 2111 (1984).
  35. M. Stengel and N. A. Spaldin, Accurate polarization within a unified Wannier function formalism, Phys. Rev. B 73, 075121 (2006).
  36. J. Zak, Berry's phase for energy bands in solids, Phys. Rev. Lett. 62, 2747 (1989).
  37. J. J. Esteve-Paredes and J. J. Palacios, A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis, SciPost Phys. Core 6, 002 (2023).
  38. R. Resta, Quantum-mechanical position operator in extended systems, Phys. Rev. Lett. 80, 1800 (1998).
  39. W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math. 7, 649 (1954).
  40. S. Blanes, F. Casas, J. Oteo, and J. Ros, The Magnus expansion and some of its applications, Phys. Rep. 470, 151 (2009).
  41. G. H. Golub and C. F. Van Loan, Matrix Computations (JHU Press, Baltimore, MD, 2013).
  42. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, et al., quantum espresso: A modular and open-source software project for quantum simulations of materials, J. Phys.: Condens. Matter 21, 395502 (2009).
  43. P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, Jr., A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, et al., Advanced capabilities for materials modelling with quantum espresso, J. Phys.: Condens. Matter 29, 465901 (2017).
  44. P. Giannozzi, O. Baseggio, P. Bonfà, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F. Ferrari Ruffino, A. Ferretti, N. Marzari, I. Timrov, A. Urru, and S. Baroni, quantum espresso toward the exascale, J. Chem. Phys. 152, 154105 (2020).
  45. H. Kageshima and K. Shiraishi, Momentum-matrix-element calculation using pseudopotentials, Phys. Rev. B 56, 14985 (1997).
  46. J. Tóbik and A. Dal Corso, Electric fields with ultrasoft pseudo-potentials: Applications to benzene and anthracene, J. Chem. Phys. 120, 9934 (2004).
  47. A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput. 34, C153 (2012).
  48. M. Thümmler, Accurate Berry connection scripts, GitHub, 2026, https://github.com/mathunje/AccurateBerryConnection.
  49. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  50. D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials, Phys. Rev. B 88, 085117 (2013).
  51. T. Sohier, M. Calandra, and F. Mauri, Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene, Phys. Rev. B 96, 075448 (2017).
  52. U. von Barth and C. D. Gelatt, Validity of the frozen-core approximation and pseudopotential theory for cohesive energy calculations, Phys. Rev. B 21, 2222 (1980).
  53. Due to the unitarity of the Fourier transform, the integrals can be equivalently evaluated in real space, where the commutator is translated into a convolution with a maximum support (in a mathematical sense) of twice the Wigner-Seitz cell in each direction.
  54. D. A. Greenwood, The Boltzmann equation in the theory of electrical conduction in metals, Proc. Phys. Soc. 71, 585 (1958).
  55. J. H. Ryoo, C.-H. Park, and I. Souza, Computation of intrinsic spin Hall conductivities from first principles using maximally localized Wannier functions, Phys. Rev. B 99, 235113 (2019).
  56. P. I. Davies and N. J. Higham, A Schur-Parlett algorithm for computing matrix functions, SIAM J. Matrix Anal. Appl. 25, 464 (2003).
  57. R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 (1993).
  58. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, UK, 2012).
  59. G. W. Stewart, Perturbation theory for the singular value decomposition, in SVD and Signal Processing, II: Algorithms, Analysis, and Applications (University of Maryland at College Park, 1990).

Outline

Information

Phys. Rev. B 114, 245203– Published 13 October, 2026
  • Received 23 April 2026
  • Revised 21 July 2026
  • Accepted 24 September 2026
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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation