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REVIEW 2 major objections 4 minor 61 references

First-Principles Wannier Representation of Proximity Effects

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Pith's one-line read A dynamical proximity operator from DFT reproduces what static fits miss.

desk verdict 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. read the letter →

arxiv 2607.25690 v1 pith:LL2UI43N submitted 2026-07-28 cond-mat.mes-hall cond-mat.mtrl-sciphysics.comp-ph

Yaroslav Zhumagulov , Johan Félisaz , Stepan S. Tsirkin , Denis Kochan , Oleg V. Yazyev This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sciphysics.comp-ph
keywords proximityeffectWannierdownfoldingvirtualhybridizationdynamicaleffectiveHamiltoniangrapheneheterostructuresspin-orbitcouplingLöwdinpartitioningexchangeinteraction
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Method, Eq. (2) and following paragraph] 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.
  2. [Method, paragraph after Eq. (1)] 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.
minor comments (4)
  1. [Abstract] 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.
  2. [Method, Eq. (2)] 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.
  3. [Applications, Figs. 1(c), 2(c), 3(a)-(b)] 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.
  4. [Graphene/hBN/Co, Fig. 1(d)] 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).

Circularity Check

1 steps flagged · score 4.0 of 10

Exact P-projected spectrum reproduction is a Löwdin identity by construction, but the substantive proximity parameters are DFT-derived and externally consistent, so circularity is partial only.

  1. self definitional [Abstract; Method, Eq. (1) and the G_P equation following Eq. (3)]
    "Löwdin partitioning [42] gives the energy-dependent effective Hamiltonian V(z)=PVP+PVQ 1/(z−QHQ) QVP, (1)... Because the P subspace is invariant under H0 (PH0Q=0), the coupling between the P and Q sectors is carried entirely by V... The proximity-modified low-energy spectrum follows from the retarded P-projected Green's function G_P(ω)=[ω+i0+−PH0P−V(ω)]−1=P(ω+i0+−H)−1P."

    The displayed identity is the algebraic Löwdin/Feshbach formula, not an empirical result. Given PH0Q=0 and V=H−H0, the equality [z−PH0P−V(z)]^{-1}=P(z−H)^{-1}P holds exactly by construction, so the abstract's claim of 'reproducing its spectrum exactly within that subspace' is guaranteed by the definition of V(z) and cannot serve as an independent test. The paper nevertheless uses this exactness as a headline. The substantive outputs (exchange resonance, intervalley λ_K, bond-resolved λ_R) are computed from V matrix elements of the DFT Hamiltonian rather than fitted, and the k·p values agree in order with independent prior fits [8,27], so the central physical content does not reduce to this identity.

full rationale

The derivation is mostly self-contained. Eq. (1) is standard Löwdin partitioning; its use requires PH0Q=0, and for graphene the p_z Wannier manifold is an exact invariant subspace of the isolated-layer Hamiltonian by mirror symmetry, so no unsupported external theorem is imported. The α-shifted resolvent and Padé continuation introduce numerical approximation; the main text defers the value of α, leakage control, and convergence tests to its own Supplemental Material [45]. That is a numerical-correctness risk for the quantitative contrasts (0.24 meV versus below 1 μeV, and the >99% hybridization share), but it is not circularity: the quantities are still computed from the DFT Hamiltonian rather than fitted or reinserted. The k·p parameters for PtSe2 and WSe2 are obtained by projection of V onto a known operator basis, and their agreement in order with independent band-structure fits [8,27] provides external grounding. The only by-construction element is the exact reproduction of the P-projected spectrum, which follows identically from the Löwdin formula; that is a self-definitional consistency property rather than a falsifiable prediction. Factoring in that partial self-definitional element while recognizing the independent content of the computed couplings yields a score of 4.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim is parameter-free in the physics sense: λ_K, λ_R, and exchange splittings are outputs, not fits. The non-trivial inputs are the DFT Hamiltonian, the projected p_z Wannier subspace P, the invariance assumption PH0Q=0, and numerical parameters α and η. The choice of commensurate supercells and DFT functional are inherited modeling decisions.

free parameters (1)
  • Energy shift α in Eq. (2) = not reported (convergence study deferred to Supplemental Material [45])
    Lifts the P sector in Eq. (2) so that the shifted linear system mimics the QHQ resolvent. The result depends on α through a leakage that the paper says is controlled by α; convergence details are deferred to the Supplemental Material.
assumptions (4)
  • domain assumption The projected subspace P is invariant under H0: PH0Q=0.
    Introduced after Eq. (1); the p_z Wannier manifold is not exactly invariant under the full bare-layer Hamiltonian, so this is an approximation that the paper does not justify.
  • standard math Löwdin partitioning (Schur complement) gives the exact block-effective Hamiltonian for any Hermitian H.
    Standard resolvent identity used in Eq. (1).
  • domain assumption The DFT Kohn-Sham Hamiltonian of the heterostructure is an accurate reference for low-energy proximity effects.
    All conclusions inherit the exchange-correlation functional and pseudopotential accuracy; no beyond-DFT checks are made.
  • domain assumption The α-shifted linear system solved on a sparse Matsubara grid plus Padé continuation recovers the real-frequency resolvent.
    The leakage is stated to be controlled by α and convergence is delegated to Supplemental Material [45]; no convergence data appear in the main text.

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Pith. "Pith review of First-Principles Wannier Representation of Proximity Effects." pith.science (2026). https://pith.science/paper/LL2UI43N

@misc{pith2026260725690,
  author       = {Pith},
  title        = {Pith review of: First-Principles Wannier Representation of Proximity Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LL2UI43N}},
  note         = {Machine review of arXiv:2607.25690}
}
abstract

Proximity effects in layered heterostructures are usually represented by static parameters fitted to first-principles bands, which discards the energy dependence of the virtual hybridization, the momentum transfer, and the spatial structure. We overcome this limitation by deriving a dynamical proximity operator $\mathcal{V}(\mathbf{k},\mathbf{k}';\omega)$ directly from density functional theory, downfolding the Kohn-Sham Hamiltonian of the heterostructure onto a fixed low-energy target Wannier subspace and reproducing its spectrum exactly within that subspace. The construction separates direct matrix elements from virtual hybridization through all remaining states. In graphene on hBN/Co(0001), virtual hybridization generates more than $99\%$ of the proximity exchange and gives it a resonant frequency dependence set by the Co $d$ states. In graphene/PtSe$_2$ it resolves a sublattice-selective intervalley coupling with a $\sqrt{3}\times\sqrt{3}$ charge modulation, and in graphene/WSe$_2$ a bond-resolved Rashba coupling of $0.24$~meV, against below $1$~$\mu$eV for the direct projection alone. Our results expose the limitations of static projections and establish a fitting-free microscopic foundation for low-energy modeling, spin-relaxation theory, and transport calculations.

Figures

Figures reproduced from arXiv: 2607.25690 by the authors.

Figure 1
Figure 1. FIG. 1. Graphene/hBN/Co(0001) proximity results. (a) Atomic structure: graphene on a single hBN layer on an hcp Co(0001) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Graphene/PtSe [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Graphene/WSe [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reviewed August 15, 2026 · model on record in the stance chip above.