{"id":"6fa65bbf-30e5-4b89-bc31-570bc6bc25d2","arxiv_id":"2607.25690","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A fitting-free Wannier downfolding scheme yields a dynamical proximity operator that reproduces the low-energy spectrum of graphene heterostructures and exposes virtual-hybridization, intervalley, and bond-resolved couplings.","lead":"The paper derives a frequency-and-momentum-dependent proximity operator directly from density functional theory, replacing fitted model parameters with a downfolded first-principles coupling. It shows that in graphene on cobalt, more than 99% of the proximity exchange comes from virtual hybridization through substrate states.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The PH0Q=0 assumption is satisfied for graphene; the load-bearing risk is numerical convergence of the α-shifted resolvent and Padé continuation in Eqs. (2)–(3), which underpins the quantitative 0.24 meV vs <1 μeV and >99% claims.","rationale":"The reader's conditional verdict is appropriate, but the specific weakest assumption identified—PH0Q=0—is not the most load-bearing concern for the three demonstrations, since graphene's p_z manifold is an exact invariant subspace of H0. The more concrete risk is that the 'exact' spectrum and the dramatic direct-versus-virtual contrasts are produced by a finite-α shifted-resolvent calculation followed by Padé continuation, neither of which is convergence-controlled in the main text. The quantitative claims differ by two to three orders of magnitude, so even a μeV-scale numerical leakage could change the headline ratios. The proposed check—an α/Padé convergence scan plus a direct comparison to the exact P-projected Green's function—would settle this. It does not invalidate the method; it keeps the paper conditional on reported convergence. No code or data are released, and the Supplemental Material [45] is not part of the submitted text, reinforcing the need for an independent numerical check. The verdict therefore remains CONDITIONAL, unchanged from the reader.","tokens_in":11528,"tokens_out":10789,"duration_ms":112061,"concrete_test":"Repeat the graphene/WSe2 calculation at the converged settings with α doubled and quadrupled, and with two independent Padé orders (e.g., 8 and 12 Matsubara points); extract λ_R^bond(ω=0) from Eq. (8) at each setting. If the full value shifts by more than 0.05 meV or the direct projection PVP rises above 10 μeV, the claimed 0.24 meV versus <1 μeV separation is not numerically converged. Additionally, for the chosen α and Padé order, compare the spectral function from Eq. (4) with the exact P-projected Green's function P(ω+i0^+−H)^−1P computed by direct diagonalization of the supercell DFT Hamiltonian; agreement of the Dirac-point spectral function to within the broadening η would close the loop on both the α-leakage and the PH0Q=0 assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The formal Löwdin identity is not in dispute. For graphene's p_z manifold, the Wannier subspace is an exact invariant subspace of H0 by mirror symmetry, so PH0Q=0 holds, and the reader's weakest assumption is unlikely to break the central argument. The actual soft spot is numerical: Eq. (2) replaces QHQ by H+αP, so the P-space component of the response decays only as 1/α, and the real-frequency V(ω) is recovered by Padé continuation from a sparse Matsubara grid. The main text reports no value of α, no Padé order, no convergence criteria, and no error bars; these are deferred to Supplemental Material [45]. The headline quantities—the 0.24 meV bond Rashba versus the below 1 μeV direct projection in graphene/WSe2, and the >99% hybridization-generated exchange in graphene/hBN/Co—are evaluated at real frequency after these two approximations. If the residual P-space leakage at finite α is not small compared with the 1 μeV direct baseline, the direct/virtual decomposition is contaminated and the claimed contrast changes. This is a numerical-correctness risk rather than a formal flaw, but it directly bears on the central quantitative message.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a downfolding scheme that derives a dynamical proximity operator V(k,k';ω) from the Kohn-Sham Hamiltonians of a target layer (H0) and a heterostructure (H), using Löwdin partitioning onto a fixed Wannier subspace P. The effective coupling separates a direct term PVP from a virtual-hybridization term PVQ(z−QHQ)^{-1}QVP, and the authors solve the latter numerically via an α-shifted resolvent on the Matsubara axis followed by Padé continuation to real frequencies. The method is demonstrated on three graphene heterostructures: graphene/hBN/Co(0001), where virtual hybridization is claimed to generate more than 99% of the proximity exchange; graphene/PtSe2, where a sublattice-selective intervalley Kekulé coupling of 4.78 meV is found; and graphene/WSe2, where a bond-resolved Rashba coupling of 0.24 meV is reported against a direct projection below 1 μeV. The paper argues that static, in-subspace projections miss the dominant virtual-hybridization contribution and that the scheme provides a fitting-free route to low-energy models.","tokens_in":11766,"tokens_out":7105,"duration_ms":71217,"significance":"If the numerical implementation is sound, the paper makes a useful methodological contribution: it gives a physically transparent separation between direct and virtual-hybridization contributions, retains frequency and momentum dependence, and produces concrete falsifiable predictions (the 0.24 meV bond Rashba coupling, the 4.78 meV Kekulé intervalley amplitude, and the >99% hybridization share of the exchange). The comparison of the projected k·p parameters with earlier first-principles fits, e.g. λR=0.35 meV for graphene/WSe2, is an encouraging consistency check. The formal core, the Löwdin identity, is standard and exact; the main risk is numerical, not formal. The manuscript would be strengthened by explicit convergence data and by softening the word 'exactly' where the implementation relies on finite-α leakage and Padé continuation.","major_comments":[{"comment":"The central quantitative claims are evaluated after two numerical approximations whose convergence is not reported in the main text: the α-shifted resolvent in Eq. (2) and the Padé continuation to real frequency. Equation (2) reproduces (z−QHQ)^{-1}QV|w> only up to a P-space leakage proportional to (iω−PHP−α)^{-1}PHQ, as the text itself states ('up to a leakage controlled by α'). The main text gives no value of α, no test of leakage, no number of Matsubara points, and no Padé order. This matters because the headline contrast in graphene/WSe2 is 0.24 meV total versus below 1 μeV direct projection, and the hBN/Co exchange is claimed to be >99% hybridization-generated; a leakage contribution at the few-μeV level would contaminate the direct/virtual decomposition and change the message. Please report α, the convergence of results in α, the sparse-grid and Padé parameters, and error estimates, and qualify the word 'exactly' in the abstract accordingly.","section":"Method, Eq. (2) and following paragraph"},{"comment":"The derivation of Eq. (1) assumes PH0Q=0, stated without proof. For the planar graphene target this condition is in fact satisfied by mirror symmetry: the p_z orbitals are odd and the in-plane σ manifold is even under z→−z, so the present applications are safe. However, the paper claims general applicability to any disentangled low-energy manifold [40], for which exact invariance of the selected subspace under H0 is not automatic. The condition should be stated explicitly as an assumption with its symmetry justification, and the effect of a residual PH0Q on the separation between direct and virtual-hybridization contributions should be discussed.","section":"Method, paragraph after Eq. (1)"}],"minor_comments":[{"comment":"The phrase 'reproducing its spectrum exactly within that subspace' overstates the numerical implementation, which involves an α-shifted resolvent and Padé continuation; 'exactly' should be replaced by 'up to controlled numerical convergence' or similar.","section":"Abstract"},{"comment":"Equation (2) should define z explicitly and state the sign convention: with z=iω_ℓ, the desired resolvent is (iω_ℓ−QHQ)^{-1}; the current display mixes iω and αP in a way that is easy to misread. A brief derivation of the P-space leakage term would remove ambiguity.","section":"Method, Eq. (2)"},{"comment":"The band reproduction in the figures is a consistency check rather than an independent validation: the P-projected spectrum is reproduced by construction when Eq. (1) is evaluated exactly. The text should frame these panels as checks of the numerical implementation, not as tests of the method; the independent evidence is the comparison of projected k·p parameters with previous calculations.","section":"Applications, Figs. 1(c), 2(c), 3(a)-(b)"},{"comment":"The statement that the direct projection contributes 0.02 meV to the exchange splitting, against an 'order-10 meV' full splitting, should specify how the exchange splitting is extracted from the diagonal elements V_mm(K,K;ω) and how the 0.02 meV is defined (e.g., difference of direct projections on the two spin channels).","section":"Graphene/hBN/Co, Fig. 1(d)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of cond-mat.mes-hall and is likely suitable for a Letters-style publication once the numerical convergence of the α-shifted resolvent and Padé continuation is documented. The formal construction is sound; the main obstacle is the gap between the abstract's 'exactly' and the approximate numerical implementation. No concerns about citation practice or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good to see this. The paper gives a practical downfolding recipe to compute a frequency-dependent proximity operator V(k,k';ω) directly from DFT, instead of fitting static parameters. That's genuinely useful. The new content is the resolvent correction that resums virtual hybridization through all eliminated states, keeping momentum and real-space structure; the three demonstrations each isolate a feature: energy dependence and exchange origin in graphene/hBN/Co, intervalley Kekulé coupling in graphene/PtSe2, and bond-resolved Rashba texture in graphene/WSe2. The WSe2 result — 0.24 meV bond Rashba vs below 1 μeV direct projection — makes the point that static in-subspace projections can miss essentially all of the proximity SOC. The graphene/PtSe2 Kekulé amplitude is a concrete new prediction. This reads like a tool paper with three solid proofs of concept, not a new physics discovery, and that's fine.\n\nOn the formal side: the Löwdin identity is standard, and the reproducibility of the P-projected spectrum is guaranteed, so that is not an independent test. The reader worried about PH0Q=0; for graphene's p_z manifold that assumption actually holds by mirror symmetry, so I do not think it is the weak link. The weak link is numerical. Eq. (2) replaces QHQ by H+αP, so the P-space component decays only as 1/α; the real-frequency V(ω) is recovered by Padé continuation from a sparse Matsubara grid. The headline numbers — 99%, 0.24 meV vs below 1 μeV — are real-frequency outputs of that two-step approximation, and the main text does not report α, Padé order, or convergence criteria, all deferred to Supplemental. The abstract's 'exactly' overstates what is actually numerical. This is addressable but it is load-bearing for the quantitative message. The DFT numerics were not independently checked, and no code or data are shipped, which limits reproducibility but is normal for a methods Letter.\n\nOverall the method looks right, the demos are internally consistent, and the conclusions follow. The authors should be asked to move the convergence data into the main text or a clearly accessible supplement, state the α leakage bound, and soften 'exactly' to 'to numerical precision.' With that, it is a solid contribution. I would send it to review.","headline":"A genuinely useful downfolding tool that replaces fitted static proximity parameters with a frequency-dependent operator, with solid demos; the main risk is numerical convergence reporting, not the formal framework.","tokens_in":12327,"tokens_out":1766,"would_cite":true,"duration_ms":16204,"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":"A dynamical proximity operator from DFT reproduces what static fits miss.","keywords":["proximity effect","Wannier downfolding","virtual hybridization","dynamical effective Hamiltonian","graphene heterostructures","proximity spin-orbit coupling","Löwdin partitioning","exchange interaction"],"falsifier":"Recompute the graphene/WSe$_2$ downfolding with the target Wannier subspace enlarged to include the W and Se states that produce the poles of $\\mathcal{V}(\\omega)$, so $P$ is closer to an exact invariant subspace. If $PVP$ then rises from below 1 $\\mu$eV toward the 0.24 meV scale, the claim that virtual hybridization carries essentially all proximity spin-orbit coupling is an artifact of the minimal $p_z$ subspace; if the spectrum in the enlarged subspace stops matching the full DFT bands, then the $\\alpha$-leakage or Pad\\'e continuation is not controlled.","tokens_in":11327,"feed_emoji":"🧲","tokens_out":6353,"duration_ms":52416,"temperature":0.7,"pith_summary":"The paper claims that proximity effects in layered heterostructures are inherently dynamical: the coupling between the target layer and its neighbors is carried largely by virtual hybridization with states outside the low-energy subspace, so static parameters fitted to DFT bands discard the dominant physics. It constructs a frequency-dependent proximity operator $\\mathcal{V}(\\mathbf{k},\\mathbf{k}'';\\omega)$ by downfolding the Kohn-Sham Hamiltonian of the full heterostructure onto a fixed Wannier subspace of the target layer, keeping the direct projection and the resolvent renormalization separate. In three graphene-based systems the construction reproduces the DFT spectrum in the low-energy subspace and exposes effects invisible to static fits: more than 99% of the exchange from Co $d$ states, a sublattice-selective intervalley Kekul\\'e coupling in graphene/PtSe$_2$, and a bond-resolved Rashba coupling in graphene/WSe$_2$ that is 0.24 meV versus below 1 $\\mu$eV from the direct projection alone. A sympathetic reader would care because this supplies a fitting-free microscopic basis for low-energy models, spin-relaxation theory, and transport calculations.","feed_headline":"Static fits discard 99% of proximity exchange","feed_subtitle":"A DFT-derived Wannier downfolding exposes virtual hybridization as the dominant proximity channel.","key_machinery":"The central object is the energy-dependent effective Hamiltonian obtained by L\\\"owdin partitioning, $\\mathcal{V}(z) = PVP + PVQ(z-QHQ)^{-1}QVP$, where $P$ is the projector onto the fixed graphene $p_z$ maximally localized Wannier subspace and $Q=1-P$. Because the target subspace is invariant under the isolated-layer Hamiltonian ($PH_0Q=0$), the inter-sector coupling is carried entirely by the proximity perturbation $V$. The second term, evaluated through a shifted resolvent on the Matsubara axis with a multishift Krylov solver and Pad\\'e continuation, resums virtual hybridization to all orders in $V$ and provides the frequency, momentum, and real-space structure of the proximity coupling.","core_discovery":"The central discovery is that the proximity coupling in a weakly hybridizing target layer is, to leading order, not the static in-subspace matrix element $PVP$ but the resolvent-mediated renormalization $PVQ(z-QHQ)^{-1}QVP$ through all eliminated states. The paper derives this operator from first principles and shows that in graphene/hBN/Co(0001) virtual hybridization accounts for more than 99% of the proximity exchange (direct projection 0.02 meV versus an order-10 meV full coupling), in graphene/PtSe$_2$ it produces a sublattice-diagonal intervalley Kekul\\'e coupling $\\lambda_K = 4.78$ meV with a $\\sqrt{3}\\times\\sqrt{3}$ charge modulation, and in graphene/WSe$_2$ the bond Rashba coupling is 0.24 meV compared with below 1 $\\mu$eV from $PVP$ alone. The spectrum of the downfolded operator coincides with the DFT bands in the target subspace, so the construction is presented as exact within that subspace. The method therefore claims to replace parameter fitting with a derived, energy-, momentum-, and spatially resolved proximity operator.","pith_inferences":["An implication the authors do not draw: if virtual hybridization dominates in these three graphene systems, then existing static model parameters fitted to DFT bands elsewhere may be silently absorbing the same omitted renormalization, so spin-relaxation and transport predictions built on them should be re-examined case by case.","The frequency-dependent $\\mathcal{V}(\\omega)$ could be used directly as an energy-dependent self-energy or scattering vertex for quasiparticle lifetimes and $T$-matrix transport calculations, a natural next step that is not demonstrated here.","The sublattice-diagonal Kekul\\'e coupling in graphene/PtSe$_2$ suggests a mechanism for valley-coherent superlattices without bond-centered order; a testable extension would be to probe the $\\sqrt{3}\\times\\sqrt{3}$ charge modulation by scanning tunneling microscopy or to compute its effect on intervalley scattering times."],"forward_implications":["Static in-subspace projections systematically underestimate proximity spin-orbit and exchange couplings, so low-energy models built from static fits miss the dominant virtual-hybridization channel.","The downfolded operator supplies fitting-free $k\\cdot p$ parameters, such as the staggered potential, intrinsic spin-orbit couplings, and Rashba coupling, that reproduce the downfolded bands.","The finite-momentum structure of $\\mathcal{V}(\\mathbf{k},\\mathbf{k}'';\\omega)$ captures intervalley couplings, including the sublattice-selective Kekul\\'e term in graphene/PtSe$_2$, that static symmetry-inspired models cannot represent.","The construction applies to any target layer with a disentangled low-energy manifold, extending to twisted stacks, magnetic and superconducting contacts, defect ensembles, and multilayer graphene."],"supporting_citations":[{"why":"Supplies the maximally localized Wannier function formalism for entangled energy bands that selects the fixed low-energy target subspace $P$.","marker":"[40]"},{"why":"Provides the theory and construction of maximally localized Wannier functions used to build the $p_z$ reference basis.","marker":"[41]"},{"why":"Supplies the L\\\"owdin partitioning identity that yields the energy-dependent proximity operator with direct and resolvent-renormalized terms.","marker":"[42]"},{"why":"Supplies the shifted-resolvent technique used to evaluate $(z-QHQ)^{-1}$ without explicit inversion.","marker":"[44]"},{"why":"Supplemental Material containing the downfolding derivation, finite-momentum block structure, and convergence in the shift $\\alpha$ that the numerical exactness relies on.","marker":"[45]"},{"why":"Supplies the low-energy proximity spin-orbit Hamiltonian $H_\\tau(\\mathbf{q})$ of graphene on transition-metal dichalcogenides onto which the downfolded couplings are projected.","marker":"[8]"},{"why":"Supplies the hBN/Co graphene system and its previous exchange-splitting values, the baseline that the downfolded Co resonance must reproduce.","marker":"[10]"},{"why":"Supplies the Pad\\'e approximant continuation that maps the Matsubara-axis coupling to real frequencies in the reported spectra.","marker":"[50]"}],"fun_headline_variants":["Virtual hybridization drives 99% of proximity exchange","DFT downfolding reveals true proximity mechanism","Proximity effects: static fits miss the virtual part","99% of proximity exchange is virtual, not static","First-principles proximity operator beats static fits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on the assumption that the chosen low-energy subspace of the bare target layer is exactly closed under the bare layer's Hamiltonian, so that the only couplings between that subspace and the rest of the Hilbert space are those induced by the proximity perturbation; if the bare layer itself hybridizes these states, the split between direct and virtual contributions changes and the downfolding is no longer exact.","fun_headline_variants_meta":{"raw":{"variants":["Virtual hybridization drives 99% of proximity exchange","DFT downfolding reveals true proximity mechanism","Proximity effects: static fits miss the virtual part","99% of proximity exchange is virtual, not static","First-principles proximity operator beats static fits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1275,"prompt_tokens":1012,"completion_tokens":263,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":628,"completion_tokens_details":{"reasoning_tokens":190}},"tokens_in":628,"tokens_out":263,"duration_ms":3028,"temperature":1.0,"reasoning_tokens":190,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:25:31.880946+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the graphene/WSe$_2$ downfolding with the target Wannier subspace enlarged to include the W and Se states that produce the poles of $\\mathcal{V}(\\omega)$, so $P$ is closer to an exact invariant subspace. If $PVP$ then rises from below 1 $\\mu$eV toward the 0.24 meV scale, the claim that virtual hybridization carries essentially all proximity spin-orbit coupling is an artifact of the minimal $p_z$ subspace; if the spectrum in the enlarged subspace stops matching the full DFT bands, then the $\\alpha$-leakage or Pad\\'e continuation is not controlled.","supporting_citations":[{"cited_title":"Souza, N","cited_arxiv_id":null,"evidence_quote":"Supplies the maximally localized Wannier function formalism for entangled energy bands that selects the fixed low-energy target subspace $P$."},{"cited_title":"L¨ owdin, A note on the quantum-mechanical per- turbation theory, J","cited_arxiv_id":null,"evidence_quote":"Supplies the L\\\"owdin partitioning identity that yields the energy-dependent proximity operator with direct and resolvent-renormalized terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplemental Material containing the downfolding derivation, finite-momentum block structure, and convergence in the shift $\\alpha$ that the numerical exactness relies on."},{"cited_title":"Zollner, M","cited_arxiv_id":null,"evidence_quote":"Supplies the hBN/Co graphene system and its previous exchange-splitting values, the baseline that the downfolded Co resonance must reproduce."}],"review_version":1}