Pith. sign in

REVIEW 4 major objections 6 minor 97 references

Optimally embedded tight binding for reproducing geometry dependent observables

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Treating the orbital embedding positions in a tight-binding model as tunable parameters, fixed against a reference response from ab initio computation or experiment, reproduces nonlinear optical responses such as shift and injection…

desk verdict A genuinely useful and honest method paper: treating orbital embeddings as fit parameters fixes nonlinear optical responses in tight binding, with the main caveat being that the span of diagonal embeddings is finite and the paper's own CdS result shows it. read the letter →

arxiv 2608.07684 v1 pith:KUTDVQUQ submitted 2026-08-07 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.str-el

Jonas J Telle , Gunnar F Lange This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.str-el
keywords orbitalembeddingstight-bindingmodelsgeometry-dependentobservablesnonlinearopticalresponsesshiftcurrentinjectionquantumgeometrictensormetric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Tight-binding models usually fix the orbital positions, called embeddings, by Wannier localization, but those positions are not chosen to reproduce physical observables. This paper argues that embeddings should instead be treated as free geometric parameters and fixed by fitting a reference response from ab initio computation or experiment. With hoppings and on-site energies held fixed, the paper derives how any tight-binding observable splits into a geometry-independent part and a geometry-dependent part that depends only on the embeddings, and it gives explicit position derivatives for the quantum geometric tensor. Applied to GaAs and CdS, optimizing the embeddings reproduces shift currents, injection currents, and optical conductivity at essentially the accuracy of full Wannier interpolation, with no loss of band-structure accuracy. The reader should take away that the embedding choice is a genuine model parameter, not a byproduct of Wannier localization.

What carries the argument

The load-bearing object is the geometry decomposition of the Berry connection under a displacement of the orbital embeddings, where $D=e^{-ik\cdot\Delta}$ with $\Delta_{\alpha\beta}=\delta\tau_\alpha\,\delta_{\alpha\beta}$. The paper derives the corresponding displacement of the quantum geometric tensor, whose real part is the Fubini-Study metric and whose imaginary part is the Berry curvature, and then takes first and second position derivatives so that a standard least-squares fit of the embeddings to a reference observable is cheap. The same derivatives allow fast minimization of the metric trace, and for the interspace Berry connection the optimal embeddings are obtained from an explicit pseudoinverse equation $H\tau=y$.

What would settle it

Run the paper's coarse-grid optimal-embedding procedure for every independent component of a nonlinear optical response in a material with strong position-operator effects, then evaluate the fitted model on a dense grid against a Wannier-interpolated reference. The claim predicts quantitative agreement for all components in normal cases and identifies irreducible failure on the CdS $\sigma_{xxz}$ component; finding additional components with irreducible qualitative errors would falsify the general claim that diagonal embeddings are sufficiently expressive.

Watch

Extended reading notes

Core claim

The central claim is that the usual failure of tight-binding models on geometry-dependent responses comes less from the diagonal tight-binding assumption itself than from the default choice of orbital embeddings. Under that assumption the position operator is represented by intra-cell orbital positions $\tau_\alpha$, and the paper shows that changing these positions shifts the Berry connection by a term $\Delta_j^{nm}=\langle u'_n|\Delta_j|u'_m\rangle$ built from a diagonal displacement matrix, while leaving the Hamiltonian and band structure untouched. Any tight-binding observable therefore separates into a geometry-independent part and a geometry-dependent correction, and the correction can be fitted to a reference response by minimizing the mean squared error. For GaAs and CdS, the fitted embeddings quantitatively reproduce the nonlinear optical response components that maximally localized Wannier centers get wrong, and the fit on a coarse grid generalizes to dense-grid responses. The paper also shows that embeddings can reshape the local quantum metric substantially, for example in the Chern insulator V2O3, even when the average metric trace is held fixed.

Load-bearing premise

The fit works only when the part of a reference response that depends on the position operator can be expressed using the diagonal orbital positions that the model allows; if the true position-operator dependence reaches into off-diagonal matrix elements that the model discards, no choice of orbital embeddings can repair the response.

Editorial extensions

If this is right

  • For a fixed Hamiltonian, orbital embeddings are the only free parameters that change geometry-dependent observables, so tight-binding models reporting such observables should state explicitly how the embeddings were fixed.
  • An optimally embedded model reproduces selected nonlinear optical responses without enlarging the Hilbert space or adding off-diagonal position matrix elements, so minimal models remain minimal.
  • The decomposition gives a practical diagnostic: if the reference response minus the geometry-independent term cannot be represented by the embedding-dependent correction, the diagonal tight-binding ansatz is not adequate for that observable, as happens for the $\sigma_{xxz}$ shift current component in CdS.
  • The position derivatives for the quantum geometric tensor extend to observables built from it, including optical conductivity, injection current, shift current, and superfluid weight.
  • Because the average metric trace is bounded while its local shape can be redistributed, embedding choices can produce large local quantum metric peaks without small gaps, which is relevant for flat-band and quantum sensing applications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: experimental discrepancies in tight-binding analyses of quantum weight that are currently attributed to correlation effects may first need an embedding contribution to be separated out.
  • Beyond the paper: the independent-sectors assumption could be tested with a perturbation that changes positions linearly while hoppings change exponentially, predicting different responses in geometry-dependent versus geometry-independent observables.
  • Beyond the paper: the representability criterion could serve as a screening tool, flagging materials whose target response cannot be fitted on a coarse grid as requiring the full position operator rather than diagonal tight binding.
  • Beyond the paper: constraining the fit to symmetry-allowed Wyckoff parameters would preserve crystal symmetries while retaining enough freedom to fit responses; the paper suggests this route but leaves it for future work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper introduces optimally embedded tight binding (OETB), in which the diagonal orbital embeddings τ_α are treated as tunable parameters, fixed by fitting a reference geometry-dependent observable. The authors derive a decomposition of tight-binding observables into geometry-independent and geometry-dependent parts, provide first and second position derivatives for the quantum geometric tensor and related optical responses, and apply the method to GaAs and CdS optical conductivity, injection current, and shift current. They also study qualitative geometry effects in Haldane and QWZ toy models and in the Chern insulator V2O3, showing that embeddings can dramatically reshape the metric trace. The manuscript is transparent about the CdS σ_xxz failure and about residual low-frequency errors, but the abstract's general claim that CdS nonlinear optical responses are quantitatively reproduced overstates what is shown.

Significance. If the method works generally, it is a practically valuable contribution: it provides a cheap way to improve geometry-dependent tight-binding responses without altering band structures, with analytic derivatives that make the optimization efficient. The decomposition and derivative formulas (Sec. III, App. B) appear correct and are useful beyond the specific applications. The paper also gives explicit credit to the limitations it finds: the CdS σ_xxz component cannot be fit simultaneously with the other shift-current components, and low-frequency residuals remain. The qualitative geometry effects in V2O3 and toy models are a nice demonstration of the sensitivity of the quantum metric to embeddings. The main weakness is that the central claim of quantitative reproduction is not matched by a quantitative error analysis or an out-of-sample validation, and the CdS failure is not reflected in the abstract.

major comments (4)
  1. [Abstract; Sec. IV A 2] The abstract claims that optimally embedded tight-binding models of GaAs and CdS 'quantitatively reproduce non-linear optical responses,' but Sec. IV A 2 reports that no single embedding reproduces all three independent shift-current components of CdS, and σ_xxz retains a qualitative error above about 6 eV (Fig. 4(d)). This is a load-bearing limitation of the central claim, and the abstract should be qualified to say which components are reproduced or should present CdS shift current as a partial failure.
  2. [Sec. III C] The key premise—that the difference between the reference response and the geometry-independent tight-binding expression lies in the span of the diagonal embedding variations—is stated but not quantitatively tested. The CdS σ_xxz result shows the premise can fail. I request a diagnostic that lets the reader assess expressibility: report the geometry-independent residual versus the optimized residual, and compare the number of independent embedding degrees of freedom with the number of independent response constraints (frequency points times tensor components). Without such a diagnostic, the scope of the method is unclear.
  3. [Sec. II B; Sec. IV] All reference responses are Wannier-interpolated quantities built from the same Wannier functions that define the hoppings, so fitting embeddings to those references and then showing agreement is partly circular. The paper would be substantially strengthened by an out-of-sample test—for example, fitting only the QGT or only the conductivity and then predicting the shift current, or fitting below a frequency cutoff and predicting above it. Without such a test, the claim that embeddings become genuine model parameters with predictive power is not fully established.
  4. [Sec. IV; Figs. 2–4] The central claim is quantitative reproduction, but the results are presented only as visual comparisons in Figs. 2–4. I ask the authors to report quantitative error metrics—for example, relative L2 errors over the frequency range for each independent component and each embedding choice. This would also make the comparison between WCC, IBC, ABQ, and tailored fits precise rather than impressionistic.
minor comments (6)
  1. [Eq. (16)] The notation ilde{a}^{αβ}_j(τ^α_j) is confusing because the argument is written as if only the α component of the position vector is variable; please clarify that the dependence is on the full set of embedding components.
  2. [Fig. 2(e)] The labels a1, a2, a3 in the irreducible unit cell are not defined in the caption; please explain what these labels denote.
  3. [Fig. 13 caption] The caption uses 'dense' and 'REF' in a way that is easy to confuse; please rephrase to distinguish clearly between the dense-grid true response and the coarse-grid reference response that is being fitted.
  4. [App. A 1] The sentence 'We could not find their band structure for comparison' is vague; please specify which previous work was meant and what comparison was attempted.
  5. [Sec. II A 3] The term 'external Berry connection' is introduced without a citation; if this is standard terminology, please add a reference, otherwise define it more explicitly.
  6. [General] The manuscript does not state data or code availability; for reproducibility, consider depositing the tight-binding models and optimization scripts in a public repository.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: fitted-response plots are explicit fit diagnostics, and the predictive claim is carried by out-of-sample checks (ABQ fit to shift current, coarse-grid to dense-grid generalization).

full rationale

The paper's derivation chain is self-contained rather than circular. The geometry decomposition of Sec. III C (Berry connection A = A' + Delta, and the resulting delta-Q expression in Eq. 19) is an internal algebraic result, derived from the TBA position-operator definition, not from the optical-response targets. The optimization procedure (Sec. III B) openly defines optimal embeddings as the minimizers of mean-squared error against a chosen reference, and the plots of SCT/OCT/ICT fits against REF are presented as fit diagnostics ('responses obtained by fitting the orbital embeddings to optimally reproduce ...'), not as independent predictions. The genuinely predictive evidence is out-of-sample: fitting only the Abelian QGT (ABQ, a gauge-invariant geometric object) substantially improves the shift current, and embeddings fitted on a coarse q-grid generalize to dense-grid responses. The CdS sigma_xxz failure (Sec. IV A 2) demonstrates that the diagonal-embedding ansatz has limited span and that agreement is not automatic, further showing the fits are not vacuous. The only author self-citation, Ref. [77] for standard k-local QGT bounds, is non-load-bearing and supported by independent bounds [75,76] and elementary QGT positivity. The use of Wannier-interpolated references from the same Wannier functions that define the hoppings limits external grounding but is not circularity: it is an intentional comparison of the TBA approximation to the full Wannier interpolation within a common DFT baseline. No step reduces a claimed prediction to its fitting input by construction, so the paper does not exhibit significant circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the diagonal tight-binding ansatz, on the availability of a reference response from Wannier interpolation of the same Wannier functions used for the hoppings, and on the fitted orbital embeddings. No invented physical entities are introduced. The main free parameters are the orbital positions themselves, plus minor numerical settings.

free parameters (3)
  • Orbital embeddings tau_alpha^j = Not tabulated in the text; plotted in Fig. 2(e) for GaAs and described for CdS in Sec. IV A 2
    The orbital positions are the tunable parameters fitted against reference responses (interspace Berry connection, QGT, or optical response tensors). The paper's central results depend on these fitted values.
  • Quantum weight bounding value for V2O3 constraint = BZ average of 1000 random samples; explicit value not stated
    In Sec. V B 2 and App. A 3, the equipotential bound for the constrained optimization is set by sampling; the specific bound is an arbitrary numerical choice for the qualitative study.
  • Relaxation time tau for injection current = 1 fs (arbitrary scaling, Fig. 4 caption)
    Injection current magnitudes are scaled by an unspecified phenomenological relaxation time, so the plotted absolute values are not ab initio predictions.
assumptions (5)
  • domain assumption Diagonal tight-binding assumption: the position operator in the orbital basis is diagonal with diagonal values tau_alpha.
    Core ansatz of the paper, introduced in Sec. II A 3, Eq. (TBA). The entire optimization operates within this assumption.
  • domain assumption Wannier interpolation of the reference response from the same Wannier functions used for the tight-binding hoppings is a valid proxy for the true ab initio response.
    Used throughout Sec. IV; the paper calls DFT the ground truth but actually compares against Wannier-interpolated quantities computed with Wannier90/WannierBerri.
  • standard math The shift current expression from Ibanez-Azpiroz et al. (2018) applies, involving the optical QGT and shift vector.
    Sec. III B 3, Eq. for sigma_kij; standard nonlinear response theory used without modification.
  • standard math The QGT of an isolated occupied subspace can be computed from Wannier-interpolated Berry connection.
    Sec. III B 2 and App. B; standard Kubo-type formula for the QGT.
  • standard math Bloch theorem and exponential localizability of Wannier functions for the projected spaces considered.
    Background for the model, Sec. II B; uses the Brouder et al. localization result and Marzari-Vanderbilt Wannierization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimally embedded tight binding for reproducing geometry dependent observables." pith.science (2026). https://pith.science/paper/KUTDVQUQ

@misc{pith2026260807684,
  author       = {Pith},
  title        = {Pith review of: Optimally embedded tight binding for reproducing geometry dependent observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KUTDVQUQ}},
  note         = {Machine review of arXiv:2608.07684}
}
read the original abstract

In tight-binding models, the position operator is reduced to intra-cell orbital positions (embeddings). While accurately reproducing band structures, such models often fail for geometry dependent responses depending on the position operator. To address this, we investigate the role of these embeddings and introduce the general framework of optimally embedded tight binding. Treating the embeddings as geometric tuning parameters to be fixed against a reference response (obtained from ab-initio computation or experiment), we obtain tight-binding models of GaAs and CdS which quantitatively reproduce non-linear optical responses at no cost to band structure accuracy. The optimal embeddings are determined efficiently using position derivatives obtained from decomposing tight-binding observables into a geometry independent and dependent part, and explicit derivatives are provided for key quantities such as the quantum geometric tensor. The decomposition reveals where geometric effects dominate, and we show in both toy models and in the Chern insulator V2O3 how geometry can dramatically alter the local metric trace. Our results highlight that orbital embeddings should be treated as a genuine model parameter which should be explicitly fixed against physical data to get accurate minimal models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

97 extracted references · 50 canonical work pages

  1. [1]

    II B 3, we noted that disentanglement and the TBA can break symmetries

    Symmetry considerations In Sec. II B 3, we noted that disentanglement and the TBA can break symmetries. Additionally, numerical op- timisation can generally cause minor symmetry breaking, such as maximal localisation slightly opening Kramers- degenerate band touchings in systems with time-reversal symmetry. When symmetries are of particular impor- tance, ...

  2. [2]

    Optical responses for CdS The results for CdS (wurtzite structure) are shown in Fig. 4. In panels (a)–(b) we show the independent com- ponentsς xx andς zz of the optical conductivity. Note that ML WF WCCs display a reasonable quantitative er- ror, counter to expectations expressed in prior literature that first order responses are accurately captured in t...

  3. [3]

    Physical interpretation When viewing diagonal tight binding as an approxi- mation of a more complete Wannier picture, the optimal embeddings can be interpreted simply as a free parame- ter in the minimal model which should be fixed to min- imize the error in the approximation. The orbitals|αR⟩ are still considered to correspond to Wannier functions, and t...

  4. [4]

    As an example, our results in Sec

    Band gap dependence Beyond the binary subdivision of geometry dependent and independent quantities, one may consider the degree of geometry dependence. As an example, our results in Sec. IV A indicate that the low frequency optical conduc- tivity is only weakly geometry dependent. By considering how the band gap enters geometric correction terms, we can s...

  5. [5]

    To test the ex- tent of this effect, we are interested in making the metric trace large locally ink-space without altering the hop- pings

    Bounds The discussion above indicates that materials with rel- atively flat band gaps might have a surprisingly large metric peak, compared to what one would expect from the geometry independent contribution. To test the ex- tent of this effect, we are interested in making the metric trace large locally ink-space without altering the hop- pings. However, ...

  6. [6]

    Geometry dependence in toy models The geometry dependence of the metric trace in a Hal- dane model [38, 41] is illustrated in Fig. 5. In panel (a), we show the band gap over the BZ. Panels (b)–(c) show, respectively, the BZ average metric trace (the quantum weight) and the maximal value of the metric trace over the BZ, corresponding to its highest peak. T...

  7. [7]

    Geometry dependence in V2O3 The results for maximising the metric trace, within the bounds defined by an equipotential curve of the quantum weight, at theM 1-point for monolayer V 2O3 are shown in Fig. 6. The first panel shows the band gap over the BZ, and the second shows the ML WF WCC metric trace. Notice the slight symmetry breaking in the WCC metric t...

  8. [8]

    This is the idea under- pinning critical quantum sensing, which employs this en- hanced sensitivity for improved quantum sensors

    Engineering the quantum metric One reason why the quantum metric has received sig- nificant interest is because of its relationship to the quan- tum Fisher information, which signals where a system is sensitive to external responses. This is the idea under- pinning critical quantum sensing, which employs this en- hanced sensitivity for improved quantum se...

Show all 97 references
  1. [9]

    In a physical platform, we may con- ceivably have some control over the Wannier functions, but changing these will generally alter both positional and energetic DOFs

    Experimental feasibility of tuning geometry It is important to highlight that, while the position DOFs (orbital embeddings) and the energetic DOFs (hoppings) are completely independent in a tight-binding model, they are related physically to matrix elements of Wannier function...

  2. [10]

    Symmetry-projected geometry dependence It is generally true that, if there are fixed maximal Wyckoff positions for a model, then the minimal quan- tum weight will be attained at these [5]. However, with free Wyckoff parameters, there is no obvious choice for embeddings, and th...

  3. [11]

    C. Wang, X. Liu, L. Kang, B.-L. Gu, Y. Xu, and W. Duan, First-principles calculation of nonlinear optical responses by Wannier interpolation, Physical Review B 96, 115147 (2017)

    2017

  4. [12]

    Iba˜ nez-Azpiroz, S

    J. Iba˜ nez-Azpiroz, S. S. Tsirkin, and I. Souza, Ab initio calculation of the shift photocurrent by Wannier interpo- lation, Physical Review B97, 245143 (2018)

    2018

  5. [13]

    DFT and W annierisation First-principle calculations were performed within the framework of plane-wave density functional theory (DFT) as implemented in theQuantumESPRESSOpack- age [57, 58] andWannier90was used for Wannierisation [9]. Note that the accuracy of the ab-initio co...

  6. [14]

    We use a frequency spacing of 0.03 eV

    Fitting optical responses We useWannierBerrito compute smooth high reso- lution plots for optical responses for GaAs and CdS. We use a frequency spacing of 0.03 eV. For linear optical con- ductivity, ak-grid of 85 3 was used with a fixed smearing of 0.1. For second order respo...

  7. [15]

    Qualitative effects of geometry For 2D two-band toy models, it is a simple matter to compute relevant observables on a 2D grid correspond- ing to all orbital separation vectors. In testing different parameters for QWZ and Haldane models, we find in- dications that as flat a pl...

  8. [16]

    While great for numerical implementation, it is not particularly readable, so it is worth discussing the nota- tion explicitly

    Position derivatives The computations below are simple but index inten- sive. While great for numerical implementation, it is not particularly readable, so it is worth discussing the nota- tion explicitly. Matrices are denoted by non-italic capi- tal letters and can depend on ...

  9. [17]

    with the Python packageVaspBandUnfolding[73]

    The QGT from DFT As an aside, we note that for plane-wave DFT such as implemented inVASP[72], the single-particle wave func- tions are given as|ψ n⟩= P G CnG |k+G⟩, where the coefficientsC nG can be retrieved from theWAVECAR, e.g. with the Python packageVaspBandUnfolding[73]. ...

  10. [18]

    In 2D, we have a single Berry curvature component Ω xy = Ω and three metric compo- nents, which we denote bygx,g y for diagonal components andg xy for the off-diagonal

    T opological lower bound There are well-known lower bounds on geometry in terms of topology [75–77]. In 2D, we have a single Berry curvature component Ω xy = Ω and three metric compo- nents, which we denote bygx,g y for diagonal components andg xy for the off-diagonal. We have...

  11. [19]

    Optical upper bound Finally, we provide for completeness a simple upper bound for the quantum weight, as in [78]. By integrating out the delta-function in (17) and moving constants, one finds that the average metric trace satisfies Z k trg= 2ℏ e2 Z ∞ Ωg Re trς ω dω(F3) ≤ 2ℏ e2...

  12. [20]

    Ahn, G.-Y

    J. Ahn, G.-Y. Guo, N. Nagaosa, and A. Vishwanath, Rie- mannian geometry of resonant optical responses, Nature Physics18, 290 (2022)

    2022

  13. [21]

    Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, Cambridge, 2018)

    2018

  14. [22]

    Qiang, T

    X.-B. Qiang, T. Liu, H.-Z. Lu, and X. C. Xie, Quan- tum geometric origin of orbital magnetization, Applied Physics Letters128, 010501 (2026)

    2026

  15. [23]

    Peotta and P

    S. Peotta and P. T¨ orm¨ a, Superfluidity in topologically nontrivial flat bands, Nature Communications6, 8944 (2015)

    2015

  16. [24]

    Huhtinen, J

    K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, and P. T¨ orm¨ a, Revisiting flat band supercon- ductivity: Dependence on minimal quantum metric and band touchings, Physical Review B106, 014518 (2022)

    2022

  17. [25]

    S. H. Simon and M. S. Rudner, Contrasting lattice ge- ometry dependent versus independent quantities: Rami- fications for Berry curvature, energy gaps, and dynamics, Physical Review B102, 165148 (2020)

    2020

  18. [26]

    Fuchs and F

    J.-N. Fuchs and F. Pi´ echon, Orbital embedding and topology of one-dimensional two-band insulators, Physi- cal Review B104, 235428 (2021)

    2021

  19. [27]

    Lim, J.-N

    L.-K. Lim, J.-N. Fuchs, and G. Montambaux, Geome- try of bloch states probed by St¨ uckelberg interferometry, Physical Review A92, 063627 (2015)

    2015

  20. [28]

    A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions, Computer Physics Communications 185, 2309 (2014)

    2014

  21. [29]

    This is sometimes calleddiagonaltight binding

  22. [30]

    This shift-invariance holds also for the multi-band case, as follows from (19) noting that ∆ nm j =c jδnm forc j a constant is zero always when summing over complimen- tary subspacesn̸=m

  23. [31]

    Nastos and J

    F. Nastos and J. E. Sipe, Optical rectification and current injection in unbiased semiconductors, Physical Review B 82, 235204 (2010)

    2010

  24. [32]

    Ghosh, A

    A. Ghosh, A. M. Schankler, and A. M. Rappe, Choos- ing tight-binding models for accurate optoelectronic re- sponses, Physical Review B111, 125203 (2025)

    2025

  25. [33]

    X. Wang, J. R. Yates, I. Souza, and D. Vanderbilt,Ab initiocalculation of the anomalous Hall conductivity by Wannier interpolation, Physical Review B74, 195118 (2006)

    2006

  26. [34]

    Dobardˇ zi´ c, M

    E. Dobardˇ zi´ c, M. Dimitrijevi´ c, and M. V. Milovanovi´ c, Generalized Bloch theorem and topological characteriza- tion, Physical Review B91, 125424 (2015)

    2015

  27. [35]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized wannier functions: Theory and applications, Reviews of Modern Physics84, 1419 (2012)

    2012

  28. [36]

    Cole and D

    T. Cole and D. Vanderbilt, Exact expression for the Berry connection in the projection gauge, Physical Review B 113, 245106 (2026)

    2026

  29. [37]

    Brouder, G

    C. Brouder, G. Panati, M. Calandra, C. Mourougane, and N. Marzari, Exponential Localization of Wannier Functions in Insulators, Physical Review Letters98, 046402 (2007)

    2007

  30. [38]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized Wannier functions for composite energy bands, Physical Review B56, 12847 (1997)

    1997

  31. [39]

    Onishi and L

    Y. Onishi and L. Fu, Quantum weight: A fundamental property of quantum many-body systems, Physical Re- view Research7, 023158 (2025)

    2025

  32. [40]

    Souza, N

    I. Souza, N. Marzari, and D. Vanderbilt, Maximally local- ized wannier functions for entangled energy bands, Phys- ical Review B65, 035109 (2001)

    2001

  33. [41]

    Sakuma, Symmetry-adapted wannier functions in the maximal localization procedure, Physical Review B87, 235109 (2013)

    R. Sakuma, Symmetry-adapted wannier functions in the maximal localization procedure, Physical Review B87, 235109 (2013)

    2013

  34. [42]

    This is vastly stronger than the criterion for exponential localisation, being only that the Chern num- ber (theBZ average band traceof the Berry connection) vanishes

    The condition that they do commute is exceptionally strong, namely that all components of the multi-band Berry connection for the projected space are everywhere zero [19]. This is vastly stronger than the criterion for exponential localisation, being only that the Chern num- b...

  35. [43]

    S. Kim, Y. Chung, Y. Qian, S. Park, C. Jozwiak, E. Rotenberg, A. Bostwick, K. S. Kim, and B.-J. Yang, Direct measurement of the quantum metric tensor in solids, Science388, 1050 (2025)

    2025

  36. [44]

    Lihm and C.-H

    J.-M. Lihm and C.-H. Park, Comprehensive theory of second-order spin photocurrents, Physical Review B105, 045201 (2022)

    2022

  37. [45]

    J. E. Sipe and A. I. Shkrebtii, Second-order optical re- sponse in semiconductors, Physical Review B61, 5337 (2000)

    2000

  38. [46]

    Implementations are directly available in WannierBerri

  39. [47]

    A. Jain, W. J. Jankowski, M. Mehraeen, and R.-J. Slager, Nonlinear odd viscoelastic effect, Physical Review Letters 137, 036302 (2026)

    2026

  40. [48]

    For the QGT, our fits typically take less than a second on a laptop for simple real materials, but do note that cost landscapes can vary a lot depending on the observable

  41. [49]

    J. C. Slater and G. F. Koster, Simplified LCAO Method for the Periodic Potential Problem, Physical Review94, 1498 (1954)

    1954

  42. [50]

    Iba˜ nez-Azpiroz, F

    J. Iba˜ nez-Azpiroz, F. De Juan, and I. Souza, Assessing the role of interatomic position matrix elements in tight- binding calculations of optical properties, SciPost Physics 12, 070 (2022)

    2022

  43. [51]

    The behaviour with noise here is expected, since it is know from (F2) that the area under the optical response is related to the quantum weight, which blows up with the metric trace as the orbitals move apart

  44. [52]

    M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Won- dratschek, Bilbao Crystallographic Server: I. Databases and crystallographic computing programs, Zeitschrift f¨ ur Kristallographie - Crystalline Materials221, 15 (2006)

    2006

  45. [53]

    are similarly geometry independent, at least given orbital displacements that do not break symmetries on which the quantisation relies. Similarly, it would also be interesting to investigate how changing the orbital posi- tions can affect other model parameters frequently de- ...

  46. [54]

    Komissarov, T

    I. Komissarov, T. Holder, and R. Queiroz, The quan- tum geometric origin of capacitance in insulators, Nature Communications15, 4621 (2024)

    2024

  47. [55]

    ACKNOWLEDGMENTS We thank Wojciech J

    appeared, which independently suggests geometric engineering and discusses orbital embeddings in linear optical responses of a toy model, and [56] appeared which discusses the interplay of minimal orbital embeddings and symmetries. ACKNOWLEDGMENTS We thank Wojciech J. Jankowsk...

  48. [56]

    Mellaerts, R

    S. Mellaerts, R. Meng, M. Menghini, V. Afanasiev, J. W. Seo, M. Houssa, and J.-P. Locquet, Two dimensional V2O3 and its experimental feasibility as robust room- temperature magnetic Chern insulator, npj 2D Materials and Applications5, 65 (2021). 27

    2021

  49. [57]

    Claassen, C

    M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux, Position-momentum duality and fractional quantum hall effect in chern insulators, Physical Review Letters114, 236802 (2015)

    2015

  50. [58]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact landau level description of geometry and interaction in a flatband, Physical Review Letters127, 246403 (2021)

    2021

  51. [59]

    T. S. Jackson, G. M¨ oller, and R. Roy, Geometric stability of topological lattice phases, Nature Communications6, 8629 (2015)

    2015

  52. [60]

    S. Lee, S. H. Lee, and B.-J. Yang, Embedding indepen- dent length scale of flat bands (2025), arXiv:2511.02240

    2025

  53. [61]

    Ba lut, B

    D. Ba lut, B. Bradlyn, and P. Abbamonte, Quantum en- tanglement and quantum geometry measured with in- elastic x-ray scattering, Physical Review B111, 125161 (2025)

    2025

  54. [62]

    F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”, Physical Review Letters61, 2015 (1988)

    1988

  55. [63]

    J. C. Budich and E. J. Bergholtz, Non-hermitian topolog- ical sensors, Physical Review Letters125, 180403 (2020)

    2020

  56. [64]

    Chen, S ¸

    W. Chen, S ¸. Kaya ”Ozdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature548, 192 (2017)

    2017

  57. [65]

    Y. R. Kartik, J.-S. You, and H. H. Jen, Scaling of quan- tum geometry near non-hermitian topological phase tran- sitions, Physical Review B112, 195121 (2025)

    2025

  58. [66]

    Sarkar, C

    S. Sarkar, C. Mukhopadhyay, A. Alase, and A. Bayat, Free-Fermionic Topological Quantum Sensors, Physical Review Letters129, 090503 (2022)

    2022

  59. [67]

    R. Liu, Y. Chen, M. Jiang, X. Yang, Z. Wu, Y. Li, H. Yuan, X. Peng, and J. Du, Experimental critical quan- tum metrology with the Heisenberg scaling, npj Quan- tum Information7, 170 (2021)

    2021

  60. [68]

    A. R. Botello-M´ endez, J. C. Obeso-Jureidini, and G. G. Naumis, Toward an Accurate Tight-Binding Model of Graphene’s Electronic Properties under Strain, The Jour- nal of Physical Chemistry C122, 15753 (2018)

    2018

  61. [69]

    Less interestingly, sector specific perturbations should be achievable in analogue simulators

  62. [70]

    Ba lut, B

    D. Ba lut, B. Bradlyn, M. D. Collins, and P. Abbamonte, Fundamental Tests of Quantum Geometric Bounds in Ionic and Covalent Insulators using Inelastic X-Ray Scat- tering (2026), arXiv:2601.19054

    2026

  63. [71]

    de Juan, A

    F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in weyl semimetals, Nature Communications8, 15995 (2017)

    2017

  64. [72]

    W. J. Jankowski and R.-J. Slager, Quantized integrated shift effect in multigap topological phases, Physical Re- view Letters133, 186601 (2024)

    2024

  65. [73]

    L. F. C´ ardenas-Castillo, S. Zhang, F. L. Freire, D. Kochan, and W. Chen, Detecting the spread of valence-band Wannier functions by optical sum rules, Physical Review B110, 075203 (2024)

    2024

  66. [74]

    geun Oh and S

    C. geun Oh and S. Murakami, Orbital embedding and the physical definition of quantum geometry (2026), arXiv:2607.21882

    2026 arXiv

  67. [75]

    Z. Wang, C. Raca, and S. H. Simon, Symmetry and quan- tum geometry in bloch bands (2026), arXiv:2607.21581

    2026 arXiv

  68. [76]

    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. Ge...

    2017

  69. [77]

    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. ...

    2009

  70. [78]

    H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Physical Review B13, 5188 (1976)

    1976

  71. [79]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters77, 3865 (1996)

    1996

  72. [80]

    Kresse and D

    G. Kresse and D. Joubert, From ultrasoft pseudopoten- tials to the projector augmented-wave method, Physical Review B59, 1758 (1999)

    1999

  73. [81]

    Dal Corso, Pseudopotentials periodic table: From H to Pu, Computational Materials Science95, 337 (2014)

    A. Dal Corso, Pseudopotentials periodic table: From H to Pu, Computational Materials Science95, 337 (2014)

    2014

  74. [82]

    M. K. Horton, P. Huck, R. X. Yang, J. M. Munro, S. Dwaraknath, A. M. Ganose, R. S. Kingsbury, M. Wen, J. X. Shen, T. S. Mathis, A. D. Kaplan, K. Berket, J. Riebesell, J. George, A. S. Rosen, E. W. C. Spotte- Smith, M. J. McDermott, O. A. Cohen, A. Dunn, M. C. Kuner, G.-M. Rign...

    2025

  75. [83]

    Materials Data on CdS by Materials Project 10.17188/1272700 (2020)

    2020 doi

  76. [84]

    Biswas, S

    A. Biswas, S. R. Meher, and D. K. Kaushik, Electronic and Band Structure Calculation of Wurtzite CdS Using GGA and GGA+U functionals, Journal of Physics: Con- ference Series2267, 012155 (2022)

    2022

  77. [85]

    K. J. Chang, S. Froyen, and M. L. Cohen, Electronic band structures for zinc-blende and wurtzite CdS, Phys- ical Review B28, 4736 (1983)

    1983

  78. [86]

    S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spec- tra and the structural stability of nickel oxide: An 28 LSDA+U study, Physical Review B57, 1505 (1998)

    1998

  79. [87]

    S. S. Tsirkin, High performance Wannier interpolation of Berry curvature and related quantities with Wannier- Berri code, npj Computational Materials7, 33 (2021)

    2021

  80. [88]

    Nastos and J

    F. Nastos and J. E. Sipe, Optical rectification and shift currents in GaAs and GaP response: Below and above the band gap, Physical Review B74, 035201 (2006)

    2006

  81. [89]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, I. Po- lat, Y....

    2020

  82. [90]

    A. R. Conn, N. I. M. Gould, and P. L. Toint,Trust Region Methods(SIAM, 2000)

    2000

  83. [91]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Physical Review B54, 11169 (1996)

    1996

  84. [92]

    Zheng, VaspBandUnfolding (2017)

    Q. Zheng, VaspBandUnfolding (2017)

    2017

  85. [93]

    Mitscherling, A

    J. Mitscherling, A. Avdoshkin, and J. E. Moore, Gauge- invariant projector calculus for quantum state geometry and applications to observables in crystals, Physical Re- view B112, 085104 (2025)

    2025

  86. [94]

    F. Xie, Z. Song, B. Lian, and B. A. Bernevig, Topology- bounded superfluid weight in twisted bilayer graphene, Physical Review Letters124, 167002 (2020)

    2020

  87. [95]

    Bouhon, A

    A. Bouhon, A. Timmel, and R.-J. Slager, Quan- tum geometry beyond projective single bands (2023), arXiv:2303.02180

    2023 arXiv

  88. [96]

    W. J. Jankowski, R.-J. Slager, and G. F. Lange, Quantum geometric bounds in spinful systems with trivial band topology, Physical Review Research7, L042011 (2025)

    2025

  89. [97]

    Onishi and L

    Y. Onishi and L. Fu, Fundamental Bound on Topological Gap, Physical Review X14, 011052 (2024)

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.