{"id":"25e8aaa6-faa4-4daa-a5cb-8d609abfe281","arxiv_id":"2502.03181","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A controlled separation of the Coulomb interaction yields a scalar macroscopic dielectric function for 2D materials, letting plasmons and excitons be identified by zero crossings of its real part and screening lengths be extracted from first principles.","lead":"This paper shows how to define a proper macroscopic dielectric function for flat 2D materials, which had been missing, by splitting the Coulomb interaction into a long-range and a short-range part. This gives a simple way to identify plasmons and excitons in spectra and to extract the screening length from standard quantum calculations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the LRA ansatz is the fragile step, but the central 2D claim follows from a weaker exact macroscopic closure and is numerically supported.","rationale":"The reader identified the LRA as the weakest assumption, and I partially agree: the factorization of chi00 as f |phi|^2 |phi'|^2 is not generally exact, and the paper's statement that the LRA is formally exact in 2D is stronger than what is proved. However, the central claim only needs the macroscopically integrated response, and at small q the transverse dependence of the long-range potential is negligible, so the scalar Dyson equation follows from the exact equation without the factorization. The 1D case is explicitly approximate, and the numerical tests at small wave vectors match the full calculation. Therefore no load-bearing flaw remains, and the ACCEPT verdict is unchanged.","tokens_in":11215,"tokens_out":20581,"duration_ms":204960,"concrete_test":"Recompute the NbSe2 EEL spectrum at q = 0.02 a.u.^-1 by integrating the exact Eq. 15 over r_perp and r'_perp using the full ab initio chi and bar-chi, and compare the resulting Im chi_M with the LRA scalar result; if the two disagree outside numerical noise, the factorized ansatz is load-bearing, and if they agree, the central claim is validated without invoking LRA.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest step is the LRA factorization (Eqs. 17-18), and I agree that the scalar Dyson equation (19) as derived rests on it. However, this concern does not land against the central claim in the regime the paper actually claims. In 2D with |q|lambda << 1, the long-range potential v0 in Eq. 16 is constant in the transverse coordinates to leading order; integrating the exact equation (15) over r_perp and r'_perp then collapses the convolution to bar-chi_M v0 chi_M without any factorization assumption. Thus the macroscopic scalar Dyson equation and the zero-of-Re-epsilon criterion are exact in that limit for reasons independent of the LRA. The paper's own 1D caveat and the numerical agreement in Figs. 2-3 support the qualified 2D claim. What is not exact is the microscopic factorized ansatz itself; the paper's wording that the 'LRA is formally exact' is stronger than what is proved, but this does not undermine the central conclusion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript addresses the long-standing difficulty of defining a macroscopic dielectric function in low-dimensional (ℓD) systems. The authors propose to separate the Coulomb interaction into a long-range part containing only reciprocal vectors parallel to the periodic directions (G‖ = 0) and a short-range part containing all remaining components. Using this split in the microscopic Dyson equation, they derive a scalar Dyson-like equation for the ℓD macroscopic response function χ_M^ℓD (Eq. 19) with an effective long-range interaction v_eff (Eq. 20), and define an ℓD macroscopic dielectric function ε_M^ℓD. Collective charge excitations are then identified with zeros of Re ε_M^ℓD, in analogy with the 3D case. They also express the 2D screening length r0 and the Thomas-Fermi wavevector q0 in terms of the modified response function (Eqs. 23–24), and validate the approach with RPA and BSE calculations for NbSe2, BN, and MoS2, including comparisons between the full microscopic solution and the local response approximation.","tokens_in":11445,"tokens_out":19950,"duration_ms":174894,"significance":"If the derivation stands, the paper provides a general and parameter-free first-principles route to interpret EELS/IXS spectra of low-dimensional materials and to extract macroscopic screening lengths, replacing phenomenological models. The central formal step—the scalar Dyson equation—is controlled in the regime |q|λ << 1, and the numerical comparisons in Figs. 2 and 3 are convincing. The paper is also transparent about the limitations in 1D and for finite-thickness 2D systems. The main strengths are the explicit connection between microscopically accessible response functions and macroscopic spectroscopic quantities, and the consistency of the resulting collective-mode criterion with the bulk analogy.","major_comments":[],"minor_comments":[{"comment":"The statement that the LRA is 'formally exact' for |q|λ << 1 is stronger than what is proved: the factorized ansatz of Eqs. (17–18) is not exact in general, although the macroscopic scalar equation (Eq. 19) does follow exactly in that regime because v0^2D is constant in the transverse coordinates. Please revise the wording to distinguish the exact macroscopic closure from the approximate microscopic factorization.","section":"Sections III and IV, after Eq. (16) and first paragraph of Section IV"},{"comment":"The Gaussian normalization appears to be incorrect: for ℓ = 2, the normalized transverse wavefunction should be (2/(πλ^2))^{1/4} e^{-z^2/λ^2}, whereas the printed form (2/(πλ))^{1/4} is not normalized, which conflicts with the statement that Eq. (10) implies f = χ_M^ℓD. The same issue affects the ℓ = 1 prefactor. Please correct the prefactors and re-derive the explicit v_eff expressions accordingly.","section":"Eq. (21)"},{"comment":"The paper does not specify the ground-state code, pseudopotentials, k-grids, cutoffs, or the λ values used for the LRA curves in Figs. 2 and 3; for a journal version, please add a computational methods paragraph or a supplementary material section.","section":"Computational details"},{"comment":"The statement that arχ_M^ℓD is proportional to |q|^2 at small wavevectors is only valid at finite frequency; for a metal at ω = 0 the static arχ_M^2D tends to a constant, so the equality lim_{q→0} χ_M = lim_{q→0} arχ_M does not hold in that case. Please add the implied frequency qualification.","section":"Section III, paragraph after Eq. (21)"},{"comment":"The text says the peak at 5.66 eV in -Im[ε^{-1}] and the peak at 5.28 eV in Im[ε] are 'essentially the same'; please clarify that these are the longitudinal and transverse exciton positions related through the long-range interaction, since the 0.38 eV difference is visible in the figure.","section":"Fig. 3 and surrounding text"},{"comment":"The manuscript contains many typos and grammatical errors, including 'Coulom b' in the title, 'ifferent', 'expres sion', 'well behaves', 'at last at small q', and a malformed double bar in Eq. (18). A careful language edit is needed.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid contribution and the central theoretical result is correct in the regime claimed. The main caveats are expository: the 'formally exact' phrasing for the LRA is an overstatement, and the Gaussian normalization in Eq. (21) needs correction. These issues are local and do not undermine the central claim. I recommend minor revision and would be happy to see the revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid paper that clears up a real conceptual gap. The author shows that with the right split of the Coulomb interaction into long- and short-range parts, you can write a scalar Dyson equation for the macroscopic response of a low-dimensional system, and the resulting epsilon_2D_M identifies collective excitations (plasmon in NbSe2, bright exciton in BN) by zero crossings of its real part. The screening-length formula r0 = lim_{q->0} -2π/q^2 \\bar chi_2D_M is clean and immediately usable. The numerical comparisons between full and LRA at small q are convincing, and the 1D caveat is stated honestly.\n\nWhat's actually new? The long/short separation is borrowed from Refs. 33 and 39. The genuinely new pieces are the LRA ansatz (Eqs. 17-18), the explicit scalar equation (Eq. 19) and effective interaction (Eq. 20), and the demonstration that the macroscopic dielectric function is well defined in 2D in the |q| lambda << 1 regime. The BN exciton as a zero of Re epsilon_2D_M is a nice illustrative result.\n\nSoft spots, in proportion: the LRA is presented as 'formally exact in 2D' for |q| lambda << 1. That is stronger than what the factorization argument proves. The factorized ansatz itself is never exact for a real material; what is exact at leading order is the scalar closure obtained by integrating the exact Eq. 15, since v0 is constant in the transverse coordinates to leading order. The paper would be cleaner if it derived that closure directly and then presented the LRA as a convenient way to get finite-thickness corrections. The novelty relative to Refs. 33 and 39 could also be spelled out more explicitly; the reader shouldn't have to guess which equation is new. Minor issues: the notation is dense in places, and the Gaussian effective-interaction examples are illustrative but not pushed.\n\nOverall: the central claim holds up, the numerics back it, and the caveats are mostly in the right places. I'd send this to a referee. The author should be asked to soften the 'formally exact' language and to make the novelty boundaries explicit, but I'd expect acceptance after minor revision.","headline":"Solid and useful: gives a controlled scalar Dyson equation for the 2D macroscopic response with honest validity conditions, though the 'formally exact LRA' claim is a bit stronger than what the factorization strictly establishes.","tokens_in":11956,"tokens_out":2013,"would_cite":true,"duration_ms":17005,"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":"In low-dimensional materials, a redefined split of the Coulomb interaction restores a scalar macroscopic dielectric function whose zero crossings identify collective charge excitations and whose static limit yields the screening length.","keywords":["macroscopic dielectric function","low-dimensional screening","Coulomb interaction separation","collective charge excitations","2D materials","screening length","plasmons","excitons"],"falsifier":"Compute the full microscopic response of a 2D material at wave vectors where $|q|\\lambda$ is not small (for example, a three-layer system at $q\\gtrsim0.1$ a.u.${}^{-1}$) without using the LRA factorization, and compare the loss peaks with the zero crossings of $\\Re[1-v^{2\\mathrm{D}}_{\\mathrm{eff}}\\bar\\chi^{2\\mathrm{D}}_M]$; a mismatch would show the scalar criterion holds only in the strictly long-wavelength regime. In 1D, repeat the LRA-based dielectric calculation for several transverse widths $\\lambda$ and check whether the predicted collective-mode energies drift with the cutoff, as the divergence of $K_0$ implies.","tokens_in":10990,"feed_emoji":"⚡","tokens_out":15217,"duration_ms":114945,"temperature":0.7,"pith_summary":"This paper establishes a way to define a macroscopic dielectric function for low-dimensional materials, which standard bulk theory does not provide. The fix is to split the Coulomb interaction differently, taking the long-range part to be the Fourier components with $\\mathbf{G}_\\parallel=0$ and everything else as short-range. With that split, and with a local-response ansatz for the transverse electronic profile, the macroscopic response satisfies a scalar Dyson equation $\\varepsilon^{\\ell\\mathrm{D}}_M(q,\\omega)=1-v^{\\ell\\mathrm{D}}_{\\mathrm{eff}}(q)\\bar\\chi^{\\ell\\mathrm{D}}_M(q,\\omega)$, so collective charge excitations are the zero crossings of $\\Re\\varepsilon^{\\ell\\mathrm{D}}_M$, as in bulk crystals. These zero crossings let first-principles spectra identify plasmons and collective excitons without phenomenological models, and also yield the screening length $r_0$ directly from the modified response function. The construction is demonstrated on NbSe$_2$, BN, and MoS$_2$.","feed_headline":"Splitting the Coulomb force exposes 2D plasmons and excitons","feed_subtitle":"First-principles spectra now reveal collective modes and screening lengths through one scalar dielectric function.","key_machinery":"The central device is the local response approximation (LRA): the ansatz that the microscopic response functions factor as $\\chi_{00}(q,r_\\perp,r'_\\perp;\\omega)=f(q,\\omega)|\\varphi(r_\\perp)|^2|\\varphi(r'_\\perp)|^2$ and the same form for $\\bar\\chi$, with $\\varphi$ the transverse wave function confined to a thickness $\\lambda$. This factorization collapses the integral Dyson equation for the head of the response into a scalar equation for $\\chi^{\\ell\\mathrm{D}}_M$ in terms of $\\bar\\chi^{\\ell\\mathrm{D}}_M$ and the effective interaction $v^{\\ell\\mathrm{D}}_{\\mathrm{eff}}(q)$. In 2D the ansatz is formally exact in the regime $|q|\\lambda\\ll1$, where the exponential in the long-range potential can be replaced by one; in 1D the divergence of $K_0$ at zero makes the LRA an approximation and leaves a full microscopic treatment as the only exact route.","core_discovery":"In bulk crystals the macroscopic dielectric function is the head of the inverse dielectric matrix, and collective longitudinal modes appear as zeros of its real part. The paper claims that the same structure survives in one and two dimensions once the long-range Coulomb interaction is redefined as the $\\mathbf{G}_\\parallel=0$ part, whose partial Fourier transform is $2\\pi |q|^{-1}e^{-|q||r_\\perp-r'_\\perp|}$ in 2D and $2K_0(|q||r_\\perp-r'_\\perp|)$ in 1D. Under the local response approximation, the low-dimensional macroscopic response $\\chi^{\\ell\\mathrm{D}}_M$ is tied to the modified response $\\bar\\chi^{\\ell\\mathrm{D}}_M$ by a scalar Dyson equation, which introduces $\\varepsilon^{\\ell\\mathrm{D}}_M=1-v^{\\ell\\mathrm{D}}_{\\mathrm{eff}}(q)\\bar\\chi^{\\ell\\mathrm{D}}_M$. On NbSe$_2$, the prominent 0.72 eV loss feature is identified as an intraband plasmon because $\\Re\\varepsilon^{2\\mathrm{D}}_M$ vanishes there; in BN, the lowest bright exciton is also a zero of $\\Re\\varepsilon^{2\\mathrm{D}}_M$, showing its collective character. The paper further derives $r_0=-\\lim_{q\\to0}(2\\pi/q^2)\\bar\\chi^{2\\mathrm{D}}_M(q,0)$ and obtains 11.2 a.u. for BN and 76.0 a.u. for MoS$_2$.","pith_inferences":["If the scalar Dyson equation is used with exchange-correlation kernels beyond RPA/BSE, the zero-crossing criterion could track plasmon softening or exciton-plasmon mixing in 2D materials without phenomenological models; the paper establishes the framework but does not run such calculations.","The $r_0$ and $q_0$ formulas give a common scale for comparing macroscopic screening across 2D materials; one could test whether these lengths correlate with exciton binding energies or with screening by a surrounding dielectric environment.","The 1D analysis implies that effective-interaction models of 1D plasmons and excitons must carry an explicit transverse cutoff; a numerical scan of the $|q|\\ln(\\lambda|q|)$ dispersion as $\\lambda$ varies would quantify how strongly the mode energies depend on that regularization.","For finite-thickness 2D systems the paper's own spectra show small high-frequency residuals, suggesting the LRA can be extended by using a model effective interaction with a finite transverse profile while keeping the scalar Dyson equation; the paper notes this as a regime extension, not a contradiction."],"forward_implications":["For 2D semiconductors, the screening length becomes an ab initio quantity through $r_0=-\\lim_{q\\to0}(2\\pi/q^2)\\bar\\chi^{2\\mathrm{D}}_M(q,0)$, with the paper reporting 11.2 a.u. for BN and 76.0 a.u. for MoS$_2$.","For 2D metals, the metallic screening wave vector follows from $q_0=-\\lim_{q\\to0}2\\pi\\bar\\chi^{2\\mathrm{D}}_M(q,0)$, giving 7.1 a.u.${}^{-1}$ for NbSe$_2$.","A peak in a 2D loss spectrum can be classified as collective by checking whether $\\Re\\varepsilon^{2\\mathrm{D}}_M(q,\\omega)$ vanishes; this identifies the 0.72 eV NbSe$_2$ feature as an intraband plasmon and the lowest bright BN exciton as a collective mode.","The scalar Dyson structure is independent of the approximation used for $\\bar\\chi$, so the same criterion applies at RPA, BSE, or higher levels of theory.","In 1D systems no exact macroscopic dielectric function exists; finite-thickness effects must always be included, and any LRA-based description is only approximate."],"supporting_citations":[{"why":"Supplies the $\\mathbf{G}_\\parallel=0$ split of the Coulomb interaction in low-dimensional systems that defines the short-range and long-range parts.","marker":"[39]"},{"why":"Documents the low-dimensional obstruction: the noninteracting response scales as $1/\\Omega_\\perp$, so the naive macroscopic limit of $\\bar\\chi$ coincides with $\\chi$ and the bulk split fails.","marker":"[31]"},{"why":"Defines the low-dimensional macroscopic average over non-periodic directions, giving the definition of $\\chi^{\\ell\\mathrm{D}}_M$ used throughout.","marker":"[33]"},{"why":"Classical derivation of the macroscopic dielectric function as the head of the inverse dielectric matrix in periodic crystals, the bulk template the paper generalizes.","marker":"[34]"},{"why":"The companion classical derivation of the macroscopic dielectric function and local-field effects, providing the bulk zero-of-$\\Re\\varepsilon$ criterion.","marker":"[35]"},{"why":"Derives the $1+r_0|q|$ static screening law for 2D dielectrics that the paper converts into the explicit formula for $r_0$.","marker":"[52]"},{"why":"Further 2D static screening results supporting the relation between $\\varepsilon^{2\\mathrm{D}}_M(q,0)$ and the screening length $r_0$.","marker":"[51]"},{"why":"Supplies the optical-limit RPA result and the known 2D and 1D plasmon dispersions used to validate the zero-crossing identification.","marker":"[43]"}],"fun_headline_variants":["Redefined Coulomb split enables 2D dielectric and zero-crossing modes","Long-range part as G∥=0 term fixes 2D dielectric","First-principles screening lengths for 2D from modified response","Scalar Dyson equation gives 2D collective mode criterion","First-principles screening lengths: 11.2 a.u. for BN, 76 for MoS2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the local response approximation, which assumes the microscopic charge response can be written as a single function of $q$ and $\\omega$ times the transverse density profile $|\\varphi(r_\\perp)|^2$ for both the full and the modified response; this is exact only in 2D for wavelengths much longer than the material thickness.","fun_headline_variants_meta":{"raw":{"variants":["Redefined Coulomb split enables 2D dielectric and zero-crossing modes","Long-range part as G∥=0 term fixes 2D dielectric","First-principles screening lengths for 2D from modified response","Scalar Dyson equation gives 2D collective mode criterion","First-principles screening lengths: 11.2 a.u. for BN, 76 for MoS2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001109,"raw_usage":{"total_tokens":4702,"prompt_tokens":1108,"completion_tokens":3594,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":724,"completion_tokens_details":{"reasoning_tokens":3494}},"tokens_in":724,"tokens_out":3594,"duration_ms":25789,"temperature":1.0,"reasoning_tokens":3494,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T05:37:36.807733+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full microscopic response of a 2D material at wave vectors where $|q|\\lambda$ is not small (for example, a three-layer system at $q\\gtrsim0.1$ a.u.${}^{-1}$) without using the LRA factorization, and compare the loss peaks with the zero crossings of $\\Re[1-v^{2\\mathrm{D}}_{\\mathrm{eff}}\\bar\\chi^{2\\mathrm{D}}_M]$; a mismatch would show the scalar criterion holds only in the strictly long-wavelength regime. In 1D, repeat the LRA-based dielectric calculation for several transverse widths $\\lambda$ and check whether the predicted collective-mode energies drift with the cutoff, as the divergence of $K_0$ implies.","supporting_citations":[{"cited_title":"Mazzei, and C","cited_arxiv_id":null,"evidence_quote":"Supplies the $\\mathbf{G}_\\parallel=0$ split of the Coulomb interaction in low-dimensional systems that defines the short-range and long-range parts."},{"cited_title":"Sottile, F","cited_arxiv_id":null,"evidence_quote":"Documents the low-dimensional obstruction: the noninteracting response scales as $1/\\Omega_\\perp$, so the naive macroscopic limit of $\\bar\\chi$ coincides with $\\chi$ and the bulk split fails."},{"cited_title":"Mazzei, and C","cited_arxiv_id":null,"evidence_quote":"Defines the low-dimensional macroscopic average over non-periodic directions, giving the definition of $\\chi^{\\ell\\mathrm{D}}_M$ used throughout."},{"cited_title":"Wiser, Phys","cited_arxiv_id":null,"evidence_quote":"Classical derivation of the macroscopic dielectric function as the head of the inverse dielectric matrix in periodic crystals, the bulk template the paper generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The companion classical derivation of the macroscopic dielectric function and local-field effects, providing the bulk zero-of-$\\Re\\varepsilon$ criterion."},{"cited_title":"Cudazzo, C","cited_arxiv_id":null,"evidence_quote":"Derives the $1+r_0|q|$ static screening law for 2D dielectrics that the paper converts into the explicit formula for $r_0$."},{"cited_title":"Cudazzo, I","cited_arxiv_id":null,"evidence_quote":"Further 2D static screening results supporting the relation between $\\varepsilon^{2\\mathrm{D}}_M(q,0)$ and the screening length $r_0$."},{"cited_title":"Giuliani, and G","cited_arxiv_id":null,"evidence_quote":"Supplies the optical-limit RPA result and the known 2D and 1D plasmon dispersions used to validate the zero-crossing identification."}],"review_version":1}