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Separation between long- and short-range part of the Coulomb interaction in low dimensional systems: implications for the macroscopic screening length and the collective charge excitations

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.03181 v1 pith:EBD355LD submitted 2025-02-05 cond-mat.mtrl-sci

Pierluigi Cudazzo This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords macroscopicdielectricfunctionlow-dimensionalscreeningCoulombinteractionseparationcollectivechargeexcitations2Dmaterialslengthplasmonsexcitons
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

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$_2$.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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$_2$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

0 major / 6 minor

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.

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.

minor comments (6)
  1. [Sections III and IV, after Eq. (16) and first paragraph of Section IV] 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.
  2. [Eq. (21)] 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.
  3. [Computational details] 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.
  4. [Section III, paragraph after Eq. (21)] 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.
  5. [Fig. 3 and surrounding text] 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.
  6. [Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central 2D result follows from the Dyson decomposition plus an acknowledged ansatz, and the screening length is defined, not fitted.

full rationale

The central derivation is self-contained rather than circular. Starting from the exact Dyson decomposition in Section III (Eqs. 13-15), the paper writes a scalar Dyson equation (Eq. 19) only after introducing the Local Response Approximation (Eqs. 17-18) explicitly as an ansatz based on transverse localization of the response; this is an approximation whose validity is discussed (|q|lambda << 1 in 2D, always approximate in 1D), not a hidden restatement of the target result. The collective-excitation criterion (zeros of Re epsilon_2D_M) is then a standard pole condition on the scalar equation, exactly as in bulk, so it is a consequence rather than an input. The screening length r0 in Eq. 23 is obtained by matching Eq. 22 to the asymptotic form 1 + r0 |q|; the resulting expression r0 = lim_{q->0} -2pi/q^2 bar-chi_2D_M(q,0) is a definition/identity evaluated from the computed response, not a parameter fitted to the quantity being predicted. The self-citations (Refs. 41, 44, 47, 51, 52) are used as external benchmarks and standard asymptotic results (plasmon and exciton dispersions, 2D static dielectric behavior) and are not load-bearing for the derivation. The only caveat is that the statement that the LRA is 'formally exact' in 2D is stronger than what the factorization argument proves, but this is a mathematical-rigor concern about an approximation, not circularity.

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

No numbers are fitted to data. The thickness lambda appears in the model effective interactions of Eq. 21 as a physical input (the transverse wave-function extent), but the 2D numerical applications use the pure 2D limit |q|lambda << 1 where lambda drops out. The screening lengths r0 and q0 are computed from first-principles response functions, not fitted. No new entities are introduced.

assumptions (4)
  • standard math The microscopic charge response function obeys the standard Dyson equation with the Coulomb interaction split into long- and short-range parts.
    Invoked implicitly in Eqs. 3-4 and 13-14; the Dyson equation for chi with bare Coulomb interaction is the standard many-body starting point.
  • domain assumption The low-dimensional macroscopic average is defined by Eq. 6 with Omega_perp -> infinity, which converges because electronic wave functions are confined along the non-periodic directions.
    This definition of the macroscopic limit for low-dimensional systems is taken from Refs. 33, 36, 37 and is used to define chi_lD_M in Eq. 10.
  • domain assumption The appropriate long/short-range separation of the Coulomb interaction in low-dimensional systems is Eq. 12: the long-range part includes all G_perp with G_parallel = 0, and the short-range part excludes G_parallel = 0.
    The central structural choice that makes the modified response bar-chi well defined when G_perp becomes a continuum; sourced from Ref. 39.
  • ad hoc to paper The Local Response Approximation ansatz, Eqs. 17-18, factorizes chi and bar-chi as f(q,omega)|phi(r_perp)|^2 |phi(r'_perp)|^2.
    This is the key closure used to derive the scalar Dyson equation (Eq. 19) and effective interaction (Eq. 20). The paper argues it is exact in 2D for |q|lambda << 1, but it is an approximation for finite thickness and in 1D.

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Pith. "Pith review of Separation between long- and short-range part of the Coulomb interaction in low dimensional systems: implications for the macroscopic screening length and the collective charge excitations." pith.science (2026). https://pith.science/paper/EBD355LD

@misc{pith2026250203181,
  author       = {Pith},
  title        = {Pith review of: Separation between long- and short-range part of the Coulomb interaction in low dimensional systems: implications for the macroscopic screening length and the collective charge excitations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EBD355LD}},
  note         = {Machine review of arXiv:2502.03181}
}
read the original abstract

Collective charge excitations directly probed in electron energy loss and inelastic X rays scattering spectroscopies play a key role in different fields of condensed matter physics. Being induced by the long-range part of the Coulomb interaction between particles, in standard bulk systems they appear as well-defined features in the spectra associated to the zero crossing of the real part of the macroscopic dielectric function. However, this simple criterion cannot be used to identify collective excitations in low dimensional systems where the macroscopic dielectric function is not a well defined concept. In this work, we discuss how this problem can be traced back to the definition of the long-range Coulomb interaction and we show how the appropriate separation between long- and short-range Coulomb interaction allows one to correctly express the low dimensional macroscopic dielectric function in terms of microscopic quantities accessible in first-principles calculations. This allows disentangling collective charge excitations in low dimensional materials in analogy with what one does in standard bulk systems. In addition, we show how important macroscopic quantities, such as the screening length scale, can be extracted from full first principle calculations. As an illustrative example we perform a study of the screening effects and the collective excitations in prototypical 2D materials including metals (NbSe2) as well as semiconductors (BN and MoS2).

Figures

Figures reproduced from arXiv: 2502.03181 by the authors.

Figure 2
Figure 2. FIG. 2. NbSe [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. BN spectra at different [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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