{"id":"6d1883da-b4b2-4b15-a8bf-502d40854742","arxiv_id":"2608.07684","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Optimally embedded tight binding fits orbital positions to a reference response, reproducing shift and injection currents for GaAs and CdS without changing band structures.","lead":"Tight-binding models of materials can now reproduce nonlinear optical responses by tuning the positions assigned to orbitals, instead of only matching energy bands. This makes cheap models useful for computing photocurrents and quantum geometry.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim's scope exceeds the demonstrated span of diagonal embeddings; CdS \\sigma_{xxz} shows the load-bearing premise can fail, so the general claim needs qualification.","rationale":"The reader's weakest_assumption identifies the expressivity of the diagonal embedding ansatz as the load-bearing premise, and I agree: the method's success for a given observable depends on the reference error being representable by the embedding degrees of freedom. The paper itself supplies a concrete counterexample in CdS \\sigma_{xxz}, where the premise fails and the fit cannot simultaneously correct all independent shift-current components. This is a genuine limitation of the central claim as stated in the abstract, which says the method yields models of CdS that \"quantitatively reproduce non-linear optical responses\" without noting that one component is not reproducible. I considered alternative concerns, such as the self-referential validation from using the same Wannier functions for both hoppings and reference responses, and the absence of out-of-sample predictions; these are real but secondary. The self-reference does not invalidate the method as a fitting procedure, and the paper does show some generalization across observables, e.g., the GaAs shift-current fit also improves optical conductivity. The derivations in Sec. III and App. B appear internally consistent, and the paper is honest about the CdS failure and the restrictive span condition. For these reasons, the conditional verdict is appropriate: the work merits publication with scope clarified and with supporting data made available, but the strongest claim needs qualification.","tokens_in":33772,"tokens_out":4569,"duration_ms":48076,"concrete_test":"Apply a span test to CdS: compute the Jacobian of the shift-current tensor with respect to all embedding components at the MLWF-WCC positions, project the reference-minus-geometry-independent error onto the column space of this Jacobian, and compute the relative residual norm. If the residual for \\sigma_{xxz} is large (e.g., greater than 30% of the error norm), the failure is structural rather than due to local minima, and the abstract's \"quantitative reproduction\" claim for CdS must be qualified. Repeat the same projection on a third material with a strong TBA response error to test whether the diagonal-embedding span is typically sufficient or only occasionally so.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim in the abstract is that optimally embedded tight-binding models of GaAs and CdS quantitatively reproduce nonlinear optical responses at no cost to band-structure accuracy. That conclusion rests on the premise, stated in Sec. III C, that the difference between the ab-initio reference response and the geometry-independent tight-binding expression can be approximately represented by varying the diagonal embeddings \\tau_\\alpha: if not, \"we can never hope to get a good fit for the observable with the given hoppings.\" This is a strong and material restriction, because the embedding degrees of freedom are k-independent and scale linearly with the number of orbitals, while optical responses are frequency-dependent functionals of the full position operator, including off-diagonal matrix elements. The premise is shown to fail in the paper itself for one component of CdS (Sec. IV A 2): no embedding choice reproduces all three independent shift-current components, and \\sigma_{xxz} retains a qualitative error above about 6 eV. The abstract nevertheless says the method reproduces nonlinear optical responses for CdS, which overstates what is demonstrated. Since the reference responses are computed from the same Wannier functions that define the hoppings, the fit is also partly self-referential: it corrects TBA error within a model already derived from those Wannier functions, and does not by itself establish predictive power for responses outside the fitted set. The method is useful as a fitting procedure, but the general claim needs to be scoped to cases where the span condition holds.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":33960,"tokens_out":4047,"duration_ms":43282,"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":[{"comment":"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.","section":"Abstract; Sec. IV A 2"},{"comment":"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.","section":"Sec. III C"},{"comment":"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.","section":"Sec. II B; Sec. IV"},{"comment":"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.","section":"Sec. IV; Figs. 2–4"}],"minor_comments":[{"comment":"The notation \tilde{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.","section":"Eq. (16)"},{"comment":"The labels a1, a2, a3 in the irreducible unit cell are not defined in the caption; please explain what these labels denote.","section":"Fig. 2(e)"},{"comment":"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.","section":"Fig. 13 caption"},{"comment":"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.","section":"App. A 1"},{"comment":"The term 'external Berry connection' is introduced without a citation; if this is standard terminology, please add a reference, otherwise define it more explicitly.","section":"Sec. II A 3"},{"comment":"The manuscript does not state data or code availability; for reproducibility, consider depositing the tight-binding models and optimization scripts in a public repository.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid methods contribution with transparent reporting of its own limitations, but the abstract overstates the CdS result and the validation strategy is partially circular. I would support publication after the central claim is qualified and after an out-of-sample or quantitative error assessment is added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a solid, honest contribution. What's actually new is the systematic treatment of orbital embeddings tau_alpha as tuning parameters, with explicit position derivatives for the quantum geometric tensor and a clean geometry decomposition. The GaAs results are striking: fitting embeddings to the shift current also improves the optical conductivity, which is a modest but real out-of-sample signal. The derivations in Sec. III and App. B look correct to me, and the V2O3 metric-trace reshaping is a nice qualitative illustration of how much geometry can change without touching the band structure.\n\nThe soft spots are real but mostly handled well. The CdS xxz shift-current failure is the paper's own honest report, but the abstract still says the method \"quantitatively reproduces nonlinear optical responses\" for CdS, which overstates what is demonstrated. That is worth flagging to the authors. The second issue is that the reference responses are Wannier-interpolated quantities built from the same Wannier functions that define the hoppings, so the validation is partly self-referential. I agree with the reader that this is a fitting procedure rather than a derivation, but the paper is transparent about that framing, and the GaAs conductivity improvement shows the fit can generalize beyond the fitted target.\n\nThe stress-test note worries that the scope exceeds the demonstrated span of diagonal embeddings. I think that is fair but slightly harsh. The paper itself states the span condition explicitly in Sec. III C, and the CdS failure is presented as a case where the TBA should not be applied. The abstract could be more careful, but the body is honest. A more serious practical weakness is that no code or data are shipped, so the quantitative results are not independently reproducible from the manuscript alone.\n\nWho is this for? Anyone constructing tight-binding models for optical responses, and people working on Wannier interpolation or quantum geometry in minimal models. It deserves a serious referee: the method is useful, the derivations are checkable, and the failure case is instructive rather than hidden. My recommendation is to send it to review, with a request to soften the abstract's CdS claim and ideally release the fitting code or the optimized embeddings for both materials.","headline":"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.","tokens_in":34571,"tokens_out":1271,"would_cite":true,"duration_ms":13653,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["orbital embeddings","tight-binding models","geometry-dependent observables","nonlinear optical responses","shift current","injection current","quantum geometric tensor","quantum metric"],"falsifier":"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.","tokens_in":33488,"feed_emoji":"🔬","tokens_out":9071,"duration_ms":86318,"temperature":0.7,"pith_summary":"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.","feed_headline":"Tuned orbital positions fix nonlinear optics in tight-binding models","feed_subtitle":"Fitting a handful of orbital positions reproduces reference responses at no cost to band-structure accuracy.","key_machinery":"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 $H\\tau=y$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the prior failure of diagonal tight binding with Wannier charge centers for the GaAs shift current and provides the practical evaluation method the paper compares against.","marker":"[12]"},{"why":"Supplies the distinction between geometry-dependent and geometry-independent quantities and the relation that the paper generalizes to the multiband Berry connection.","marker":"[6]"},{"why":"Provides the Wannier interpolation machinery and the real-space Hamiltonian and position matrix elements used to build the tight-binding models and reference responses.","marker":"[9]"},{"why":"Defines maximally localized Wannier functions and the Wannier charge centers that serve as the default embedding the paper improves upon.","marker":"[19]"},{"why":"Motivates minimal metric trace embeddings through superfluid weight, one of the applications of the position-derivative machinery.","marker":"[5]"},{"why":"Computes the smooth Wannier-interpolated optical responses used as reference curves for GaAs and CdS.","marker":"[68]"},{"why":"Represents the earlier first-principles Wannier-interpolation approach to nonlinear optical responses against which the optimally embedded tight-binding results are assessed.","marker":"[11]"}],"fun_headline_variants":["Orbital embeddings as tunable parameters fix tight-binding optics","Fitting orbital positions corrects geometry-dependent tight-binding responses","Tight-binding models improved by tuning orbital embeddings","Orbital embeddings tune geometry responses without harming band structure","Quantum metric reshaped by orbital embeddings in tight-binding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Orbital embeddings as tunable parameters fix tight-binding optics","Fitting orbital positions corrects geometry-dependent tight-binding responses","Tight-binding models improved by tuning orbital embeddings","Orbital embeddings tune geometry responses without harming band structure","Quantum metric reshaped by orbital embeddings in tight-binding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000643,"raw_usage":{"total_tokens":2952,"prompt_tokens":935,"completion_tokens":2017,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":1938}},"tokens_in":551,"tokens_out":2017,"duration_ms":14690,"temperature":1.0,"reasoning_tokens":1938,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:24:27.452761+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines maximally localized Wannier functions and the Wannier charge centers that serve as the default embedding the paper improves upon."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Computes the smooth Wannier-interpolated optical responses used as reference curves for GaAs and CdS."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Represents the earlier first-principles Wannier-interpolation approach to nonlinear optical responses against which the optimally embedded tight-binding results are assessed."}],"review_version":1}