{"id":"7e0eae03-1086-42e4-9847-d8241e07ba6e","arxiv_id":"2607.20117","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Adding nuclear-velocity phases (electron-translation factors) to atomic orbitals makes tight-binding and Dirac models satisfy the all-electron vibrational sum rules.","lead":"This paper adds a quantum correction to simple tight-binding models so that electrons 'follow' moving atoms, fixing a known failure in predicting how materials respond to vibrations. The corrected models now satisfy the same charge-sum rules as first-principles calculations, tested on gapped graphene and a topological Haldane model.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Delta-like orbital assumption is load-bearing for the exact tight-binding sum rule and is not quantified for finite-width orbitals.","rationale":"The reader identified the same weakest assumption and the same condition. The paper's internal derivation is consistent within the delta-like orbital model, and the one ab initio comparison for gapped graphene is a useful sanity check, but no code, data tables, or formal verification are provided. The finite-width concern is load-bearing because it affects the exactness of the claimed all-electron sum-rule restoration in realistic tight-binding bases. Since the reader already conditions the verdict on clarifying this assumption and providing reproducibility, the conditional verdict remains appropriate: I do not see grounds to accept unconditionally, nor to reject outright.","tokens_in":30183,"tokens_out":29404,"duration_ms":289381,"concrete_test":"Re-derive Eq. (84) without invoking Eq. (34), keeping the full matrix elements ⟨φ_bj|(r−R_b)e^{i(α_b'−α_b)}|φ_b'i⟩ and the non-diagonal velocity-dependent overlap S^{R,ẋ}. Check whether the extra terms cancel in the sublattice sum; if they do not, compute the same two-site or gapped-graphene model with Gaussian/Slater orbitals and compare the exact LCAO Σ_s Z* against Eq. (84) with the delta-limit velocity vertex. A deviation beyond a few percent would require a finite-width generalization of the hopping phases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central sum-rule result, Eq. (84), is derived under the diagonal-position assumption Eq. (34) and the assumption of a velocity-independent diagonal overlap in Sec. I D. The paper itself concedes (Sec. II C, after Eq. (82)) that Eq. (34) is compatible only with delta-like orbitals. For finite-width orbitals the derivation changes at three points: (i) the phase factor in Eq. (38) becomes an operator, so the hopping in Eq. (39) is replaced by matrix elements of e^{i(α_b'−α_b)} times H; (ii) the overlap S^{R,ẋ} is no longer diagonal, reintroducing Pulay velocity terms (Eqs. (30)-(31)) that are dropped; and (iii) the acceleration terms in Eq. (23) do not vanish, so the on-site form Eq. (35) misses m R̈·(r−R) contributions. These corrections enter precisely the velocity vertex f^JI and the optical conductivity σ used in the sum rule. The paper does not bound their size, and the single ab initio comparison (Fig. 1) is one material. Thus the exact equality in Eq. (84) is not established for general tight-binding bases; the restoration of the all-electron sum rule is conditional on delta-like orbitals being quantitatively negligible.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper derives the effect of nuclear velocity on atomic orbitals (electron-translation factors) in LCAO and tight-binding frameworks. Using an Ehrenfest Lagrangian with velocity-including orbitals, the authors obtain an effective tight-binding Hamiltonian with nuclear-velocity-dependent Peierls-like phases in the hoppings and a velocity-dependent mass correction on the on-site energies. They show that these corrections modify the first- and second-order nuclear-displacement derivatives (electron-phonon couplings) by adding velocity vertices proportional to the band velocity, and that they restore the all-electron frequency-dependent sum rules for Born effective charges (Eq. (84)) and for the force-constant matrix (Eqs. (87)-(88)). The theory is applied to gapped graphene and the Haldane model, with a benchmark against ab initio calculations for metallic gapped graphene.","tokens_in":30554,"tokens_out":8006,"duration_ms":72903,"significance":"If correct, this work resolves a long-standing qualitative failure of tight-binding and low-energy models: the vanishing of the non-adiabatic vibrational sum rules. The central derivation is analytic, and the nuclear-velocity phase is a physical input (an electron-translation factor), not a fudge designed to reproduce the sum rule; Eq. (84) follows algebraically from the derived Hamiltonian under the stated delta-like orbital assumption. The paper provides an explicit, self-contained formalism, including second-order derivatives, and a nontrivial ab initio benchmark for the Born-effective-charge sum rule in metallic gapped graphene (Fig. 1(d)). The low-energy Dirac result—a simple velocity vertex proportional to the band velocity—is elegant and should be directly useful for model studies of chiral phonons, vibrational circular dichroism, and non-adiabatic effects in topological systems. The main caveat is the diagonal-position/delta-like orbital assumption, which limits the generality of the exact sum rule and is not quantitatively controlled.","major_comments":[{"comment":"The exact sum rule (84) is derived under the diagonal position-operator assumption Eq. (34), which the paper itself states is compatible only with delta-like orbitals (text after Eq. (82)). For finite-width orbitals, (i) Eq. (38) becomes a matrix element of exp[i α(r)], so the hopping form Eq. (39) is no longer exact; (ii) the overlap S^{R,dot R} is no longer diagonal, reintroducing the Pulay velocity terms Eqs. (30)-(31); and (iii) the acceleration terms in Eq. (23) do not vanish, so the on-site form Eq. (35) misses m Rddot·(r-R) contributions. These terms enter the same vertices f^JI and σ used in Eq. (84). The authors provide no estimate of their magnitude, and the single ab initio comparison (Fig. 1(d)) is one material and one sum rule. The central claim that tight-binding models restore the all-electron sum rules is therefore established only within the delta-like orbital model; a g","section":"§II C, Eqs. (82)-(84) and Eq. (34)"},{"comment":"The numerical validation is limited to the Born-effective-charge sum rule in metallic gapped graphene (Fig. 1(d)) and to model calculations for the Haldane model and phonon lifetimes. There is no first-principles benchmark for the force-constant sum rules Eqs. (87)-(88) or for the off-diagonal Hall-type charge response Eq. (98). While this does not affect the internal consistency, it leaves the practical accuracy of the delta-like approximation for finite-width orbital bases untested outside the single graphene-derived system. A test on a material with more diffuse orbitals would strengthen the claim.","section":"Sec. III A and III B"}],"minor_comments":[{"comment":"The text states that 'Both computations are performed with a Gaussian smearing of ∼0.068 eV, corresponding to room temperature in the Fermi statistics', while the caption of Fig. 1 states a temperature of T=27 meV. Room temperature corresponds to k_B T ≈ 26 meV, not 0.068 eV; please reconcile the smearing parameter and the temperature.","section":"Sec. III A 1"},{"comment":"The vertical axis labels in Fig. 1(c,d) are not defined in the text; please specify the units and the definition of the plotted Born-effective-charge components (e.g., in units of the electron charge) and the sum-rule quantity.","section":"Fig. 1"},{"comment":"The electric-field derivative in Eq. (52) is diagonal in the orbital basis, which is a direct consequence of Eq. (34). This should be stated explicitly when introducing Eq. (52), as it is the same assumption that later controls the sum-rule derivation.","section":"Sec. I D 5"},{"comment":"Several cited items are arXiv preprints or in press; please update any that have appeared in journals by the time of publication.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the delta-like orbital assumption: the exact sum-rule restoration is established only under Eq. (34), and the paper does not quantify finite-width corrections. I would ask the authors to either prove the sum rule for general tight-binding/Wannier bases with the full position operator or provide a quantitative estimate of the corrections, and to expand the numerical validation beyond a single sum rule in one material. The derivation and the Dirac-model result are otherwise strong and likely to be influential."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper is worth engaging with. It resolves a known qualitative failure of tight-binding and low-energy Dirac models: the frequency-dependent vibrational sum rules, which normally vanish, are restored by adding a nuclear-velocity-dependent Peierls-like phase to the hopping. That phase is the electron-translation factor, a physical input with a long history in atomic collisions and VCD, so the result is not obtained by fitting. The derivation is analytic and internally consistent; Eq. (84) follows from the stated assumptions. The correction has a clean interpretation in the Dirac limit as a velocity vertex, and the benchmark against ab initio for metallic gapped graphene shows excellent agreement for the Drude weight sum rule.\n\nThe main soft spot is the delta-like orbital assumption, Eq. (34), which the paper concedes is required for the diagonal position operator. The stress-test note is right that this assumption is load-bearing: for finite-width orbitals, the phase factor in Eq. (38) becomes an operator, the overlap is no longer diagonal, and the Pulay velocity terms and acceleration terms do not vanish. These corrections would enter exactly the vertex that carries the sum rule. The paper does not quantify them. The sum rule is therefore exact only within the delta-orbital tight-binding model, not for general localised bases. Since the Peierls phase is standard practice in tight-binding, the result is still physically meaningful, but the paper should bound the deviations or at least discuss them. The single ab initio test is encouraging but one material. The reader's request for code/data or numerical tables is reasonable; the figures are not independently reproducible as it stands.\n\nThe citation pattern looks fair. The underlying machinery is in Refs. 28-30 (including the authors' own Ref. 30) and the molecular VCD literature; self-citation there is not a red flag, though independent verification of Ref. 30's formalism would help.\n\nOverall, the central argument holds up under its own assumptions. This is a paper for anyone doing tight-binding calculations of phonon linewidths, non-adiabatic responses, or chiral phonons. It deserves a serious referee: the derivation is solid, the claim is concrete, and the limitations are fixable in revision. I would send it out, with a request that the authors clarify the delta-like assumption and provide reproducibility details.","headline":"Tight-binding vibrational sum rules restored by a velocity phase — a sound derivation, with the exact sum rule conditional on the delta-like orbital assumption.","tokens_in":30998,"tokens_out":3272,"would_cite":true,"duration_ms":33420,"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":"Tight-binding vibrational responses can be made to obey all-electron sum rules by adding nuclear-velocity-dependent Peierls phases to the hopping.","keywords":["tight-binding","electron-phonon coupling","Born effective charges","non-adiabatic vibrational response","sum rules","nuclear velocity correction","electron translation factor","Haldane model"],"falsifier":"Construct a tight-binding model with Slater-type or other finite-width orbitals, compute its frequency-dependent summed Born effective charges, and compare them to −iω(m/e)σ_{αβ}(ω) computed from the same model. Any deviation that grows with orbital overlap or width would show the exact sum-rule restoration is an artifact of the delta-orbital assumption.","tokens_in":30097,"feed_emoji":"⚛️","tokens_out":2983,"duration_ms":33453,"temperature":0.7,"pith_summary":"This paper argues that standard tight-binding and Dirac model Hamiltonians, built from atomic orbitals rigidly attached to nuclei, give qualitatively wrong frequency-dependent vibrational responses: their sums over atoms of Born effective charges and force constants vanish, while all-electron theory requires nonzero values tied to the optical conductivity. The missing ingredient, the paper claims, is the nuclear-velocity-dependent phase that moving nuclei imprint on atomic orbitals, known as the electron-translation factor. When this phase is included, the hopping gains Peierls-like velocity-dependent phases that act as a velocity vertex in the electron-phonon coupling, restoring the all-electron sum rules without fitting. The corrected models reproduce first-principles results for metallic gapped graphene and produce qualitatively different behavior for the Haldane model, including topological-state-dependent corrections.","feed_headline":"Nuclear velocity phases make tight-binding models obey sum rules","feed_subtitle":"Velocity-dependent hopping phases restore all-electron vibrational response, matching ab initio in doped gapped graphene.","key_machinery":"The central object is the velocity-including atomic orbital, |ϕ^{Ṙ,R}⟩ = e^{iα(ˆr)}|ϕ^R⟩ with α(ˆr) = (m/ℏ) Ṙ·(ˆr−R), an electron-translation factor that makes the orbital follow the moving nucleus. In the tight-binding approximation, this phase produces nuclear-velocity-dependent Peierls-like phases on inter-site hoppings, modifying the electron-phonon coupling vertex as ∂H/∂R → ∂H/∂R − iω ∂H/∂Ṙ. In the Dirac low-energy limit this velocity vertex is simply proportional to the band velocity. The key identity is Eq. (84), which uses these velocity terms plus the delta-like orbital assumption to convert the sum of Born effective charges into the optical conductivity, restoring the all-electron","core_discovery":"The paper's central claim is that including the nuclear-velocity phase in localized atomic orbitals changes the tight-binding Hamiltonian so that the frequency-dependent Born effective charges and force-constant matrix satisfy the same non-adiabatic sum rules as all-electron calculations. Concretely, the sum over all atoms of the Born effective charges becomes −iω(m/e)σ_{αβ}(ω), exactly the all-electron result, where the entire contribution comes from the velocity derivative of the Hamiltonian. The same mechanism restores the sum rules involving the force-constant matrix and the electronic susceptibility. The authors test this in metallic gapped graphene, where the corrected sum rule agrees","pith_inferences":["Because the exact sum-rule restoration relies on delta-like orbitals, models with finite-width orbitals will need the full matrix elements of e^{iα(ˆr)}; deviations from Eq. (84) would then diagnose how much physics the tight-binding parametrization misses.","The same velocity vertex should appear in other low-energy models beyond the Dirac honeycomb case, such as continuum models of twisted or one-dimensional systems; weakly coupled materials with small electron-phonon coupling are the most sensitive testbeds.","A cheap empirical check of any tight-binding model: compare its atomic-summed Born effective charges to measured or first-principles optical conductivity using Eq. (84); disagreement would flag missing nonlocal or velocity-dependent physics.","The mechanism that makes the wavefunction complex and enables currents in TB models should also produce nonzero phonon magnetic moments and vibrational circular dichroism in models that previously gave zero, a directly testable prediction."],"forward_implications":["Tight-binding and Dirac model calculations of frequency-dependent Born effective charges and force constants will no longer have identically vanishing sum rules; they will match all-electron non-adiabatic behavior without additional fitting parameters.","The electron-phonon coupling vertex acquires an additional velocity-dependent term, which in low-energy Dirac models is simply proportional to the phonon frequency and band velocity.","In metallic gapped graphene, the corrected Born effective charges reproduce ab initio Drude-weight results; corrections can reach roughly 50% at high doping.","The Haldane model's off-diagonal Born effective charges acquire velocity corrections proportional to the Hall conductivity, yielding a nonzero value in the topological off-resonant limit.","The corrections affect the optical phonon linewidth at resonance, especially when the two sublattice masses are unequal, and are larger for weak electron-phonon coupling."],"fun_headline_variants":["Velocity-phase fix restores tight-binding sum rules","Phase corrections make tight-binding match ab initio","Nuclear velocity phases fix vibrational sum rules","Tight-binding gets sum rules right with velocity phases","Velocity-dependent phases restore vibrational response rules"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the diagonal position operator assumption (Eq. 34), that localized orbitals behave as delta-like functions centered at atomic sites; if orbitals have finite width, the exact equality in Eq. (84) between the summed Born effective charges and the optical conductivity is no longer guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Velocity-phase fix restores tight-binding sum rules","Phase corrections make tight-binding match ab initio","Nuclear velocity phases fix vibrational sum rules","Tight-binding gets sum rules right with velocity phases","Velocity-dependent phases restore vibrational response rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1027,"prompt_tokens":709,"completion_tokens":318,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":250}},"tokens_in":453,"tokens_out":318,"duration_ms":3165,"temperature":1.0,"reasoning_tokens":250,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T10:43:02.455500+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a tight-binding model with Slater-type or other finite-width orbitals, compute its frequency-dependent summed Born effective charges, and compare them to −iω(m/e)σ_{αβ}(ω) computed from the same model. Any deviation that grows with orbital overlap or width would show the exact sum-rule restoration is an artifact of the delta-orbital assumption.","supporting_citations":[],"review_version":1}