pith. sign in

arxiv: 2604.21660 · v2 · submitted 2026-04-23 · ❄️ cond-mat.mtrl-sci

Self-consistent evaluation of the Berry connection for Wannier functions

Martin Th\"ummler , Thomas Lettau , Alexander Croy , Ulf Peschel , Stefanie Gr\"afe This is my paper

Pith reviewed 2026-05-09 20:46 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords Berry connectionWannier functionsmatrix logarithminterpolation schemeoptical conductivityvelocity matrixbasis incompletenessspread functional
0
0 comments X

The pith

A matrix-logarithm-based self-consistent scheme improves Berry connection interpolation accuracy for Wannier functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an interpolation method for the Berry connection that uses the matrix logarithm on overlap matrices between neighboring k-points in a self-consistent way. This respects the full matrix structure of the overlaps instead of treating elements independently, resulting in higher accuracy for derived quantities like velocity matrices. The work also shows how to quantify the impact of using a limited number of bands through singular value analysis, connecting it to the invariant spread of the Wannier functions. This matters because it enables more reliable calculations of optical properties in solids without requiring very dense sampling or perfect basis sets.

Core claim

A self-consistent interpolation scheme based on the matrix logarithm applied to overlap matrices of cell-periodic Bloch functions at neighboring k-points yields strongly improved accuracy for the Berry connection. This approach accounts for the full matrix structure rather than treating elements independently and results in improved velocity matrices and optical conductivity. The scheme is less sensitive to Wannierization details, and basis incompleteness is quantified via singular values of the overlaps, related to the invariant spread of the Wannier functions.

What carries the argument

The matrix logarithm of the overlap matrices used in a self-consistent interpolation scheme for the Berry connection.

If this is right

  • The improved interpolation makes velocity matrix elements more reliable.
  • Optical conductivity calculations gain better quality for materials like monolayer MoS2 and bulk silicon.
  • The method reduces dependence on particular choices in the Wannierization process.
  • Constraints from finite band sets can be assessed using singular values of overlaps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach might enable accurate optical response predictions with coarser k-grids in ab initio simulations.
  • Similar matrix-based interpolations could apply to other Berry-phase related quantities in band theory.
  • Further tests on materials with strong correlations or topological features could test the scheme's robustness.

Load-bearing premise

That the matrix logarithm applied to the full overlap matrices produces a physically valid interpolation of the Berry connection while the singular-value analysis correctly captures the accuracy limits imposed by basis incompleteness.

What would settle it

A direct comparison showing that the optical conductivity from the new scheme deviates more from experimental measurements than from standard interpolation methods in a well-characterized material would falsify the claim of improved accuracy.

Figures

Figures reproduced from arXiv: 2604.21660 by Alexander Croy, Martin Th\"ummler, Stefanie Gr\"afe, Thomas Lettau, Ulf Peschel.

Figure 1
Figure 1. Figure 1: Band structure for a) monolayer MoS2 (8 × 8 × 1 k-grid) and b) bulk silicon (8×8×8 k-grid). The DFT calcula￾tion performed with Quantum Espresso is depicted with thick black dots. The gray and orange lines denote the Wannier￾interpolated band structure. For gray, the maximally local￾ized Wannier functions (MLWFs) are used whereas for orange the conduction and valence bands (CB-VB) Wannierizations are stitc… view at source ↗
Figure 2
Figure 2. Figure 2: a) BZ-resolved magnitude of the Berry connec [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mismatch M, see Eq. (52), of the velocity operator for different schemes and ab-initio grid sizes calculated for MoS2: a) MLWF, b) CB-VB, Si: c) MLWF, and d) CB-VB. The dashed lines are guide for the eyes and indicate power scalings. 0.0 0.5 1.0 1.5 0 2 Density MoS2 a) Nk = 8 Nk = 25 8 15 20 25 16 18 20 22 Ω V I,MoS2 [1 /˚A] CB Both VB b) 0 1 2 σ keb spill [˚A] 0 5 Density ×5 Si c) Nk = 5 Nk = 8 Nk = 15 5 … view at source ↗
Figure 4
Figure 4. Figure 4: Quantification of basis spill of the MLWF Wannier [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Real part of the optical conductivity for MoS [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Scheme-dependent relative maximal peak height of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original 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 $\Omega_\mathrm{I}$ of the Wannier functions. Numerical calculations for monolayer MoS$_2$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a self-consistent interpolation scheme for the Berry connection A(k) in the Wannier basis that applies the matrix logarithm directly to the full overlap matrices S_mn(k,k') rather than treating elements independently. It claims this yields substantially higher accuracy in the velocity matrix elements and optical conductivity, with reduced sensitivity to Wannierization details. The authors further relate basis incompleteness (via singular values of the overlaps) to the invariant spread Ω_I and demonstrate the improvement numerically for monolayer MoS2 and bulk Si.

Significance. If the central claim holds, the method would provide a more robust and accurate route to computing gauge-dependent quantities such as the Berry connection and derived optical responses from first-principles Wannier functions. The explicit link between singular-value diagnostics and Ω_I offers a useful diagnostic for basis quality that could be adopted more broadly in computational materials science.

major comments (2)
  1. [matrix-logarithm interpolation section] The central claim that the matrix-logarithm interpolation is physically valid and gauge-consistent rests on the assertion that self-consistency resolves branch-choice issues when singular values of S deviate from unity. However, no derivation is supplied showing how the deviation ||S†S − I|| propagates into the error of the interpolated velocity matrix or optical conductivity (see the section on the matrix-logarithm scheme and the subsequent error discussion).
  2. [numerical results for MoS2 and Si] The numerical improvement is demonstrated only for two materials (MoS2 and Si). Without an explicit error bound, convergence proof for the self-consistent loop, or comparison against exact finite-difference derivatives on a dense k-grid, it remains unclear whether the reported gains are general or specific to the chosen Wannierization parameters.
minor comments (2)
  1. [method section] The notation for the overlap matrices S_mn(k,k') and the precise definition of the self-consistency iteration should be stated more explicitly, ideally with a pseudocode outline.
  2. [figures] Figure captions for the optical-conductivity plots should include the k-grid density and the number of bands retained in the Wannierization to allow direct reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the manuscript to incorporate clarifications and additional analysis where feasible.

read point-by-point responses

  1. Referee: [matrix-logarithm interpolation section] The central claim that the matrix-logarithm interpolation is physically valid and gauge-consistent rests on the assertion that self-consistency resolves branch-choice issues when singular values of S deviate from unity. However, no derivation is supplied showing how the deviation ||S†S − I|| propagates into the error of the interpolated velocity matrix or optical conductivity (see the section on the matrix-logarithm scheme and the subsequent error discussion).

    Authors: We appreciate this observation. The manuscript already relates deviations of the singular values from unity to the invariant spread Ω_I as a diagnostic of basis incompleteness. To directly address the request for propagation of the error, we will add a short derivation in the revised matrix-logarithm section showing that the leading-order error in the interpolated velocity matrix elements scales with ||S†S − I|| and that the self-consistent fixed-point procedure enforces gauge consistency by minimizing the residual branch ambiguity across neighboring k-points. This will be supported by a brief expansion of the matrix logarithm around the identity. revision: yes

  2. Referee: [numerical results for MoS2 and Si] The numerical improvement is demonstrated only for two materials (MoS2 and Si). Without an explicit error bound, convergence proof for the self-consistent loop, or comparison against exact finite-difference derivatives on a dense k-grid, it remains unclear whether the reported gains are general or specific to the chosen Wannierization parameters.

    Authors: We agree that broader validation strengthens the claims. In the revision we will include a direct comparison of the self-consistent interpolated velocities against finite-difference derivatives evaluated on a denser k-grid for both MoS2 and Si. We will also supply an explicit error estimate derived from the singular-value spectrum of the overlap matrices and document the rapid numerical convergence of the self-consistent loop (typically within 3–5 iterations). While the underlying formulation is material-independent and the improvement is tied to the full-matrix treatment rather than specific Wannierization details, we acknowledge that the current numerical evidence is limited to two systems; we will note this scope limitation explicitly. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the matrix-logarithm interpolation for Berry connection

full rationale

The paper proposes a new interpolation scheme for the Berry connection that applies the matrix logarithm directly to the full overlap matrices obtained from ab-initio calculations. This construction is independent of the target quantities (velocity matrix elements and optical conductivity) and is not obtained by fitting parameters to a subset of data that are then relabeled as predictions. No load-bearing uniqueness theorem is imported via self-citation, and the relation between singular values of the overlaps and the invariant spread Ω_I is presented as a separate quantification of basis incompleteness rather than a definitional equivalence. Validation rests on explicit numerical comparisons for MoS₂ and Si, keeping the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the matrix logarithm for overlap matrices and on the established relation between singular values and the invariant spread functional in Wannier theory.

axioms (2)
  • standard math The matrix logarithm is well-defined and yields a valid interpolation when applied to the overlap matrices of cell-periodic Bloch functions.
    Invoked as the basis of the self-consistent scheme.
  • domain assumption Singular values of the overlap matrices quantify basis-set incompleteness and can be related to the invariant part of the spread functional Ω_I.
    Used to discuss accuracy constraints imposed by the ab-initio basis.

pith-pipeline@v0.9.0 · 5520 in / 1320 out tokens · 63781 ms · 2026-05-09T20:46:10.111109+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Accurate calculation of Wannier centers, position matrix, and composite operators using translationally equivariant and higher-order finite differences

    cond-mat.mtrl-sci 2026-04 unverdicted novelty 7.0

    A translationally equivariant and higher-order finite-difference method for Wannier interpolation yields more accurate Wannier centers, position matrices, electric polarization, orbital magnetization, and spin Hall co...

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    As a consequence, it selects the Wannier functions such that the basis spill effects occurring in the log- and sclog-scheme are min- imized

    minimizes ΩI to obtain a reduced number of smooth bands within a certain energy range. As a consequence, it selects the Wannier functions such that the basis spill effects occurring in the log- and sclog-scheme are min- imized. We note that the non-zero value of Ω V I is an inherent property of the finite number of the Wannier functions. IV. NUMERICAL RES...

    work page

  2. [2]

    Using Wannier90 [17], we then computed two Wannierzations for each of the Nk ×N k ×1 grids with 11 bands

    with a plane-wave energy-cutoff of 90 Ry in two di- mensions [52] to obtain the self-consistent ground state density on a 9×9×1k-grid. Using Wannier90 [17], we then computed two Wannierzations for each of the Nk ×N k ×1 grids with 11 bands. We use the maxi- mally localized Wannier functions (MLWFs) of all bands as the first Wannierization. The second one ...

    work page

  3. [3]

    An in- teresting question to investigate next is whether the self- consistent logarithmic interpolation scheme preserves the symmetries of the crystal

    can be combined with our scheme to further improve the quality of the estimated Berry connection. An in- teresting question to investigate next is whether the self- consistent logarithmic interpolation scheme preserves the symmetries of the crystal. Additionally, a band energy- weighted mismatch measure of the velocity operator may enable a more quantitat...

    work page

  4. [4]

    G. H. Wannier, Physical Review52, 191 (1937)

    work page 1937

  5. [5]

    Y. Noel, C. M. Zicovich-Wilson, B. Civalleri, P. D’Arco, and R. Dovesi, Physical Review B65, 014111 (2001)

    work page 2001

  6. [6]

    Thonhauser, D

    T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Physical Review Letters95, 137205 (2005). 12

    work page 2005

  7. [7]

    M. G. Lopez, D. Vanderbilt, T. Thonhauser, and I. Souza, Physical Review B85, 014435 (2012)

    work page 2012

  8. [8]

    Coh and D

    S. Coh and D. Vanderbilt, Physical Review Letters102, 107603 (2009)

    work page 2009

  9. [9]

    X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt, Phys- ical Review B74, 195118 (2006)

    work page 2006

  10. [10]

    A. A. Soluyanov and D. Vanderbilt, Physical Review B 83, 035108 (2011)

    work page 2011

  11. [11]

    Gresch, G

    D. Gresch, G. Aut` es, O. V. Yazyev, M. Troyer, D. Van- derbilt, B. A. Bernevig, and A. A. Soluyanov, Physical Review B95, 075146 (2017)

    work page 2017

  12. [12]

    Giustino, M

    F. Giustino, M. L. Cohen, and S. G. Louie, Physical Re- view B76, 165108 (2007)

    work page 2007

  13. [13]

    H. Lee, S. Ponc´ e, 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, npj Computational Materials 9, 156 (2023)

    work page 2023

  14. [14]

    Kohn, Physical Review115, 809 (1959)

    W. Kohn, Physical Review115, 809 (1959)

    work page 1959

  15. [15]

    Monaco, G

    D. Monaco, G. Panati, A. Pisante, and S. Teufel, Com- munications in Mathematical Physics359, 61 (2018)

    work page 2018

  16. [16]

    H. J. Monkhorst and J. D. Pack, Physical Review B13, 5188 (1976)

    work page 1976

  17. [17]

    Marzari and D

    N. Marzari and D. Vanderbilt, Physical Review B56, 12847 (1997)

    work page 1997

  18. [18]

    Pizzi, V

    G. Pizzi, V. Vitale, R. Arita, S. Bl¨ ugel, F. Freimuth, G. G´ eranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Iba˜ nez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Ponc´ e, T. Pon- weiser, J. Qiao, F. Th¨ ole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbi...

    work page 2020

  19. [19]

    S. S. Tsirkin, npj Computational Materials7, 33 (2021)

    work page 2021

  20. [20]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Reviews of Modern Physics84, 1419 (2012)

    work page 2012

  21. [21]

    Souza, N

    I. Souza, N. Marzari, and D. Vanderbilt, Physical Review B65, 035109 (2001)

    work page 2001

  22. [22]

    Damle, L

    A. Damle, L. Lin, and L. Ying, Journal of Computational Physics334, 1 (2017)

    work page 2017

  23. [23]

    Damle, A

    A. Damle, A. Levitt, and L. Lin, Multiscale Modeling & Simulation17, 167 (2019)

    work page 2019

  24. [24]

    Sakuma, Physical Review B87, 235109 (2013)

    R. Sakuma, Physical Review B87, 235109 (2013)

    work page 2013

  25. [25]

    Koepernik, O

    K. Koepernik, O. Janson, Y. Sun, and J. van den Brink, Physical Review B107, 235135 (2023)

    work page 2023

  26. [26]

    Koretsune, Computer Physics Communications285, 108645 (2023)

    T. Koretsune, Computer Physics Communications285, 108645 (2023)

    work page 2023

  27. [27]

    J.-M. Lihm, M. Ghim, S.-J. Hong, and C.-H. Park, arXiv preprint arXiv:2604.22614 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  28. [28]

    Cole and D

    T. Cole and D. Vanderbilt, arXiv preprint arXiv:2602.18973 (2026)

    work page arXiv 2026

  29. [29]

    Blount, inSolid State Physics, Vol

    E. Blount, inSolid State Physics, Vol. 13 (Elsevier, 1962) pp. 305–373

    work page 1962

  30. [30]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Reviews of Modern Physics82, 1959 (2010)

    work page 1959

  31. [31]

    N. A. Spaldin, Journal of Solid State Chemistry195, 2 (2012)

    work page 2012

  32. [32]

    Iba˜ nez-Azpiroz, S

    J. Iba˜ nez-Azpiroz, S. S. Tsirkin, and I. Souza, Physical Review B97, 245143 (2018)

    work page 2018

  33. [33]

    J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys- ical Review B75, 195121 (2007)

    work page 2007

  34. [34]

    R. E. F. Silva, F. Mart´ ın, and M. Ivanov, Physical Review B100, 195201 (2019)

    work page 2019

  35. [35]

    A. Urru, I. Souza, ´O. P. Oca˜ na, S. S. Tsirkin, and D. Van- derbilt, Physical Review B112, 045201 (2025)

    work page 2025

  36. [36]

    Th¨ ummler, A

    M. Th¨ ummler, A. Croy, U. Peschel, and S. Gr¨ afe, The Journal of Chemical Physics164, 094114 (2026)

    work page 2026

  37. [37]

    Wilczek and A

    F. Wilczek and A. Zee, Physical Review Letters52, 2111 (1984)

    work page 1984

  38. [38]

    Stengel and N

    M. Stengel and N. A. Spaldin, Physical Review B73, 075121 (2006)

    work page 2006

  39. [39]

    Zak, Physical Review Letters62, 2747 (1989)

    J. Zak, Physical Review Letters62, 2747 (1989)

    work page 1989

  40. [40]

    J. J. Esteve-Paredes and J. J. Palacios, SciPost Physics Core6, 002 (2023)

    work page 2023

  41. [41]

    Resta, Physical Review Letters80, 1800 (1998)

    R. Resta, Physical Review Letters80, 1800 (1998)

    work page 1998

  42. [42]

    Magnus, Communications on Pure and Applied Mathematics7, 649 (1954)

    W. Magnus, Communications on Pure and Applied Mathematics7, 649 (1954)

    work page 1954

  43. [43]

    Blanes, F

    S. Blanes, F. Casas, J. Oteo, and J. Ros, Physics Reports 470, 151 (2009)

    work page 2009

  44. [44]

    P. C. Moan and J. Niesen, Foundations of Computational Mathematics8, 291 (2008)

    work page 2008

  45. [45]

    R. A. Horn and C. R. Johnson,Matrix analysis(Cam- bridge university press, 2012)

    work page 2012

  46. [46]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, 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, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. S...

    work page 2009

  47. [47]

    Giannozzi, O

    P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. Buongiorno Nardelli, M. Calandra, R. Car, C. Cavaz- zoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carn- imeo, A. Dal Corso, S. de Gironcoli, P. Delugas, R. A. DiStasio, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Ko...

    work page 2017

  48. [48]

    Giannozzi, O

    P. Giannozzi, O. Baseggio, P. Bonf` a, 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, The Journal of Chemical Physics 152, 154105 (2020)

    work page 2020

  49. [49]

    Kageshima and K

    H. Kageshima and K. Shiraishi, Physical Review B56, 14985 (1997)

    work page 1997

  50. [50]

    T´ obik and A

    J. T´ obik and A. Dal Corso, The Journal of Chemical Physics120, 9934 (2004)

    work page 2004

  51. [51]

    A. H. Al-Mohy and N. J. Higham, SIAM Journal on Sci- entific Computing34, C153 (2012)

    work page 2012

  52. [52]

    Th¨ ummler, Accurate berry connection scripts, https://github.com/mathunje/AccurateBerryConnection (2026)

    M. Th¨ ummler, Accurate berry connection scripts, https://github.com/mathunje/AccurateBerryConnection (2026)

    work page 2026

  53. [53]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Physical Re- view Letters77, 3865 (1996)

    work page 1996

  54. [54]

    D. R. Hamann, Physical Review B88, 085117 (2013). 13

    work page 2013

  55. [55]

    Sohier, M

    T. Sohier, M. Calandra, and F. Mauri, Physical Review B96, 075448 (2017)

    work page 2017

  56. [56]

    von Barth and C

    U. von Barth and C. D. Gelatt, Physical Review B21, 2222 (1980)

    work page 1980

  57. [57]

    Due to the unitarity of the Fourier transform, the inte- grals 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

    work page

  58. [58]

    D. A. Greenwood, Proceedings of the Physical Society 71, 585 (1958)

    work page 1958

  59. [59]

    G. W. Stewart, inSVD and Signal Processing, II : Al- gorithms, Analysis, and Applications(Elsevier Science Pub. Co., 1991)

    work page 1991