{"id":"4e8eefeb-a508-4b55-a17e-57632e3a4bfe","arxiv_id":"2608.09039","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive an electron-phonon scattering rate that includes anharmonic three-phonon dephasing and implement it from first principles, finding small corrections in Si, SiC, and PbTe.","lead":"This paper derives and implements a first-principles formula for electron-phonon scattering rates that includes finite phonon lifetimes from three-phonon interactions. It applies the method to silicon, silicon carbide, and lead telluride, finding small corrections in these materials and pointing to a companion study that reports a large effect in MgB2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eqs. (42)-(45) are effectively unauditable: the derivation is deferred to the Supplemental Material, and the visible occupation-factor structure of Eq. (44) does not transparently reduce to the standard two-phonon golden-rule limit, so the central formula needs an independent check.","rationale":"The reader already placed the paper in CONDITIONAL status, and I agree that the central formulas need verification before the anharmonic-dephasing claim can be accepted. My concern differs somewhat from the reader's weakest assumption: the reader focuses on the omission of the real part of the three-phonon self-energy and off-shell principal-value contributions in anharmonic materials, whereas I find the more immediate issue to be that the derivation itself is not present in the paper and that the one structurally testable feature, the occupation-factor structure of Eq. (44), does not transparently match the standard golden-rule form. The missing real-part/off-shell issue is real but is a scope limitation the authors explicitly acknowledge in Sec. II E.1, where they choose the Dyson-diagram strategy over the full spectral-function approach. The occupation-factor concern, if correct, would directly invalidate Eq. (42) rather than merely limit its range of applicability. I am not asserting the formula is wrong; the sign convention in the self-energy formalism could account for the apparent discrepancy, but that convention is not explained in the main text. The concrete toy-model re-derivation would settle the matter cheaply, because Eq. (41) contains all the information needed to reproduce or falsify Eqs. (43)-(45). I therefore keep the reader's CONDITIONAL verdict unchanged rather than upgrading to REJECT, since the concern is a demand for verification rather than a demonstrated error.","tokens_in":21517,"tokens_out":26797,"duration_ms":288331,"concrete_test":"Independently re-evaluate the Matsubara sums in Eq. (41) for a minimal toy model: one electronic band with a constant density of states and two Einstein phonon modes coupled through one nonzero three-phonon vertex, then compare the analytic result term by term with Eqs. (43)-(45). In particular, verify the T = 0, f_μ1 = 0 limit of the two-phonon-emission term: if the coefficient is not the golden-rule value (1 + N_2)(1 + N_3)(1 - f_μ1) up to an explicitly stated sign convention, Eq. (44) is incorrect and the central result fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that Eq. (42), with the partial rates in Eqs. (43)-(45), is the leading finite-phonon-lifetime correction to the Fan-Migdal scattering rate. The paper does not actually present the derivation: after Eq. (41), the text states that the remaining Matsubara summation over m is lengthy, straightforward, and detailed in the Supplemental Material, with Ref. [45] pointing to a URL to be inserted by the publisher. The final formulas are therefore the only checkable content, and they contain a structural feature that is at minimum nonstandard: Eq. (44), the two-phonon-emission term, has an occupation factor [N_λ2 + N_λ3 + 1](f_μ1 - 1) + N_λ2 N_λ3. In the zero-temperature, empty-final-state limit N_λ2 = N_λ3 = 0 and f_μ1 = 0, this equals -1, whereas the golden-rule rate for spontaneous emission of two phonons into an empty final electron state is positive and proportional to (1 + N_λ2)(1 + N_λ3)(1 - f_μ1) = 1. If Γ^(2e) is a partial scattering rate, the sign appears inverted; if it is instead a correction to the imaginary part of the self-energy with a nonstandard sign convention, that convention is not stated and the relation to the physical rate is not given. More broadly, Eqs. (43)-(45) are written as |principal-value amplitude|^2 times an on-shell delta, with occupation factors that do not transparently follow from the standard (1-f)(1+N)... + fN... structure of electron-two-phonon scattering (cf. Ref. [15]). Because the algebraic core of the derivation is hidden in the SM, this cannot be checked from the manuscript.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a first-principles expression for the anharmonic-dephasing correction to the electron-phonon scattering rate. Starting from a Hamiltonian with linear electron-phonon coupling and cubic three-phonon coupling, the authors evaluate the three-phonon bubble insertion into the Fan-Migdal self-energy (Fig. 4) and obtain the central result Eq. (42) with partial rates in Eqs. (43)-(45). They describe an implementation in their in-house code DaoQuantum and apply it to silicon, silicon carbide, and lead telluride, reporting small anharmonic corrections at 300 K and a larger correction for PbTe at 500 K. They also mention a companion study of MgB2 where the effect is claimed to be large.","tokens_in":21971,"tokens_out":15590,"duration_ms":180601,"significance":"If the central formulas are correct, the paper addresses a real gap in the standard electron-phonon framework: the traditional assumption of infinite phonon lifetimes is relaxed, and a concrete first-principles workflow is provided. The authors give no fitted parameters in the central derivation, use only DFT/DFPT and third-order force constants as inputs, and present calculations in three different materials, which is a useful proof of concept. The potential payoff is significant because the same formalism could be applied to transport, thermoelectric, and superconducting materials where phonon lifetimes are short. However, the significance is currently conditional: the core algebraic derivation is not present in the manuscript, and the occupation-factor structure of the final formulas raises questions that must be resolved before the result can be accepted.","major_comments":[{"comment":"The central result of the paper is not auditable from the main text. After Eq. (41), the remaining Matsubara sum over m is stated to be \"lengthy but straightforward\" and deferred to the Supplemental Material, which is absent from the preprint and whose URL in Ref. [45] is a placeholder. Since Eqs. (42)-(45) are the entire physical content of the work, this is a load-bearing omission. Please provide the complete derivation in the main text or in an accessible Supplemental Material, including the analytic continuation iωn → ε + iη, the identification of the poles, and the explicit step in which the imaginary part is taken to reach Eqs. (43)-(45).","section":"Sec. II E 2, Eqs. (41)-(45)"},{"comment":"The occupation factors in the final formulas do not transparently reduce to known golden-rule limits, and the sign convention needs to be stated explicitly. In the zero-temperature, empty-final-state limit (Nλ2 = Nλ3 = 0, fμ1 = 0), Eq. (44) equals -1, so Γ^(2e) is negative in this limit; this is consistent with the later statement that two-phonon emission gives a negative contribution in Si and SiC, but it means Γ^(2e) is not itself a partial scattering rate in the usual positive-rate sense. Additionally, the 1e1a factor in Eq. (43), Nλ2Nλ3 + Nλ2 = Nλ2(Nλ3+1), appears to correspond to absorbing λ2 and emitting λ3, whereas the delta function δ(ωn - ωλ2 + ωλ3 - εμ1) appears to correspond to emitting λ2 and absorbing λ3. Please clarify the labeling and state explicitly whether Eqs. (43)-(45) are corrections to the Fan-Migdal rate rather than standalone positive rates, and show how the sign structure follows from the derivation.","section":"Sec. II E 2, Eqs. (43)-(44)"},{"comment":"The manuscript does not justify why diagram 3(a), evaluated in Sec. II E, is the only leading-order anharmonic-dephasing contribution from the Hamiltonian in Eq. (31). Figure 3 also shows vertex corrections [panels (e) and (f)] and phonon-loop corrections [(c) and (d)] that can enter at comparable orders in the coupling constants once the interaction vertices are counted. The text says these are \"beyond the scope\" of the present work, but a power-counting argument, or a numerical estimate of at least one omitted diagram, is needed to support the claim that Eq. (42) is the leading finite-phonon-lifetime correction. Without this, the central claim is incomplete.","section":"Sec. II D-II E, Fig. 3"},{"comment":"The final scattering-rate formulas contain only on-shell delta functions at the harmonic phonon frequencies, with no real part of the three-phonon self-energy and no off-shell principal-value contributions. The text acknowledges this in Sec. II E 1, but for strongly anharmonic materials such as PbTe (Ref. [55]), the neglected real frequency shifts can be comparable to the included imaginary-part effects. Please provide a quantitative argument, or an explicit numerical check, that these omissions are small for the materials studied, or state more cautiously that the result captures only the on-shell dephasing contribution and not the full leading-order anharmonic correction.","section":"Sec. II E 1 and Eqs. (43)-(45)"}],"minor_comments":[{"comment":"There is a typo: \"amd\" should be \"and\" in the definition of the phonon operators.","section":"Eq. (1)"},{"comment":"The name \"Fan-Midgal\" appears in two places; it should be \"Fan-Migdal.\"","section":"Sec. II B"},{"comment":"\"restuls\" should be \"results.\"","section":"Fig. 7 caption"},{"comment":"The discussion says the two-phonon absorption corrections are \"always negative,\" which follows from Eq. (45) only if the occupation factor is nonnegative; this is true for fμ1 ≥ 0, but the point would be clearer if the factorization of the bracket in Eq. (45) were shown explicitly.","section":"Sec. IV D"},{"comment":"The companion work is cited only as \"accompanying manuscript\" with no arXiv or journal reference; since the MgB2 result is used to motivate the significance of the present work, a reference or at least a preprint identifier should be provided.","section":"Ref. [24]"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central formula is deferred to a placeholder Supplemental Material, so the referee cannot verify the main result. The sign of Eq. (44) in the zero-temperature limit, together with the apparent mismatch between the occupation factor and the energy delta in Eq. (43), suggests that the derivation may contain a subtle error, or at least a sign-convention ambiguity that must be resolved. I would not recommend rejection at this stage because the research direction is sound and the issue may be fixable with a complete derivation and a careful statement of the sign conventions. However, if the derivation, when supplied, confirms that Eq. (44) has the opposite sign to the standard electron-two-phonon golden rule in the zero-temperature limit, the central claim would need substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious extension of electron-phonon theory with real first-principles results. The specific diagram and the scattering-rate formulas are new. But the central derivation lives in a Supplemental Material that isn't in the preprint, and the occupation-factor structure of Eq. (44) looks odd enough that I would not trust it without seeing the steps.\n\nWhat's genuinely new: the three-phonon bubble inserted into the Fan-Migdal self-energy, leading to Eqs. (42)–(45), is not in the cited literature. The paper also gives a clean diagrammatic taxonomy of anharmonic corrections and makes a pragmatic choice to compute the leading bubble correction instead of the full spectral-function approach, which they correctly note is not yet practical. The implementation is real: EPW and ShengBTE-style third-order force constants, convergence tests, and a careful comparison of processes for Si, SiC, and PbTe. That part is solid and reproducible from the text.\n\nThe soft spots, in order. First, the main derivation is deferred to an unavailable SM. After Eq. (41) the paper says the remaining Matsubara sum is lengthy but straightforward and hands it to the SM. That makes Eqs. (43)–(45) unauditable from the manuscript. The stress-test note is right that in the T=0 empty-final-state limit, the occupation factor in Eq. (44) equals −1, while a golden-rule two-phonon emission rate would be positive. It's possible Γ^(2e) is a correction to the imaginary part of the self-energy rather than a full golden-rule partial rate, and the sign is fine. But the paper never says that, and calling it a scattering rate invites the wrong reading. Second, the final formulas keep only on-shell delta functions and drop the real part of the three-phonon self-energy and principal-value terms. They acknowledge this in Sec. II E 1, but for PbTe, where anharmonicity is large, the neglected pieces might be more than a small correction. Third, the abstract and conclusions claim the work establishes the importance of finite phonon lifetimes, yet the three materials here show small corrections. The big effect is in the companion MgB2 paper, which isn't part of this preprint. That claim is too strong for what this paper contains.\n\nBottom line: I'd send this to peer review. It's a legitimate new contribution and the implementation is real. But I'd insist the derivation be moved into the main text or an accessible supplement, and I'd tell the referee to spend time on the analytic continuation and the sign of Eq. (44). If that survives contact, it's a useful paper. If not, it's fixable.","headline":"Serious new derivation of anharmonic dephasing corrections to electron-phonon scattering, but the central formula is unauditable as-is because the derivation is in a missing supplement.","tokens_in":22417,"tokens_out":4094,"would_cite":true,"duration_ms":37780,"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":"This paper derives the anharmonic-dephasing contribution to the electron-phonon scattering rate as a sum of three process terms and shows it is computable from first principles.","keywords":["electron-phonon coupling","anharmonic phonons","finite phonon lifetimes","three-phonon scattering","phonon dephasing","Fan-Migdal self-energy","first-principles calculation","many-body perturbation theory"],"falsifier":"Compute the same anharmonic electron-phonon scattering rate in lead telluride at 300 K and 500 K using the full phonon spectral function, including the real part of the three-phonon self-energy, and compare it with the delta-function rate of Eqs. (43)-(45); a sizeable difference would show that the on-shell approximation misses part of the anharmonic dephasing.","tokens_in":21321,"feed_emoji":"⚫️","tokens_out":9642,"duration_ms":90712,"temperature":0.7,"pith_summary":"This paper sets out to establish that phonon lifetimes, which conventional electron-phonon calculations treat as infinite, contribute a computable correction to electron-phonon scattering rates. The correction comes from three-phonon interactions that damp a phonon while it is exchanged between electrons, and the paper derives the lowest-order scattering-rate formula $\\Gamma^{\\mathrm{el\\text{-}ah\\text{-}ph}}_{\\mu k} = 2\\Gamma^{(1e1a)}_{\\mu k}+\\Gamma^{(2e)}_{\\mu k}+\\Gamma^{(2a)}_{\\mu k}$ from many-body perturbation theory. A first-principles implementation is then used to show that, in silicon, silicon carbide, and lead telluride at 300 K, the correction is small but material-dependent in sign, and that it grows with temperature in lead telluride. If the derivation is right, standard electron-phonon workflows can include anharmonic dephasing using quantities they mostly already compute.","feed_headline":"Anharmonic phonon dephasing alters electron-phonon scattering rates","feed_subtitle":"A derivation adds three-phonon dephasing to electron-phonon scattering; tests show small, sign-dependent corrections.","key_machinery":"The load-bearing mechanism is the second-order Feynman diagram in Fig. 4, in which the phonon line of the Fan-Migdal self-energy is dressed by the three-phonon bubble: an electron emits or absorbs a phonon that then splits into or absorbs another phonon via three-phonon coupling, before recombining with the electron. The derivation evaluates two nested bosonic frequency sums by Cauchy residue techniques, and the final rates in Eqs. (43)-(45) factor into the square of a combination of electron-phonon and three-phonon matrix elements, times an energy-conserving $\\delta$-function, times Bose-Einstein and Fermi occupation factors. This factorization is what makes the correction computable with existing first-principles inputs.","core_discovery":"On its own terms, the paper's central claim is that the leading correction to the electron-phonon scattering rate from finite phonon lifetimes is the sum of three three-phonon-mediated terms in Eq. (42), obtained by inserting the three-phonon bubble into the Fan-Migdal self-energy and taking the imaginary part of the resulting electron self-energy. Each term describes a distinct real anharmonic process: one phonon emitted and one absorbed ($\\Gamma^{(1e1a)}$, always positive), two phonons emitted ($\\Gamma^{(2e)}$, sign depends on occupation factors), and two phonons absorbed ($\\Gamma^{(2a)}$, always negative). The paper reports first-principles evaluations in Si, SiC, and PbTe at 300 K, where the correction is small, negative in Si and SiC, positive in PbTe, and increasing with temperature in PbTe.","pith_inferences":["One consequence the paper leaves implicit is that existing Fan-Migdal calculations in materials with short phonon lifetimes should be revisited, because the required inputs, electron-phonon and three-phonon matrix elements, are already produced by standard first-principles workflows.","The sign pattern in Eqs. (43)-(45) suggests a practical classification: materials whose phonon spectra support two-phonon emission will tend to show reduced electron-phonon scattering, while materials with resonant one-emission-one-absorption channels, such as lead telluride, will show enhanced scattering; this could matter for thermoelectric design.","A natural extension not computed here is the four-phonon bubble and the vertex corrections the paper lists, which may become important in strongly anharmonic crystals and could change the magnitude or sign of the correction.","A direct comparison of this on-shell delta-function rate with a full spectral-function calculation that includes anharmonic frequency shifts would test whether the present formula is quantitatively sufficient in strongly anharmonic materials like lead telluride."],"forward_implications":["The standard Fan-Migdal electron-phonon scattering rate should be augmented by $\\Gamma^{\\mathrm{el\\text{-}ah\\text{-}ph}}$ of Eq. (42), with the three process terms given by Eqs. (43)-(45).","In silicon and silicon carbide at 300 K, the anharmonic correction reduces the scattering rate slightly, with a negative two-phonon emission process dominating.","In lead telluride at 300 K, the correction increases the scattering rate, driven by a positive one-phonon emission and one-phonon absorption process linked to the resonant LA+LO→TO phonon decay.","Raising lead telluride to 500 K increases both the harmonic scattering rate and the anharmonic correction, consistent with larger phonon populations.","The implementation fits into existing electron-phonon and phonon-phonon first-principles workflows and scales like the harmonic electron-phonon calculation with a larger prefactor, about $O(N^{5.7})$ as both momentum grids grow."],"supporting_citations":[{"why":"Supplies the many-body Green's function formalism, Feynman diagram rules, and bosonic frequency summation techniques on which the entire derivation is built.","marker":"[25]"},{"why":"Provides the Feynman rules for anharmonic phonon diagrams used to construct the three-phonon bubble self-energy.","marker":"[41]"},{"why":"Defines the density-functional perturbation theory that produces the harmonic phonons and electron-phonon matrix elements used as inputs.","marker":"[8]"},{"why":"Sets out the standard first-principles electron-phonon framework, including the Fan-Migdal scattering rate, that this work extends.","marker":"[9]"},{"why":"Contains the detailed frequency-summation steps leading to the final scattering-rate formulas in Eqs. (43)-(45).","marker":"[45]"},{"why":"Documents the strong anharmonic phonon scattering in lead telluride, the strongly anharmonic case against which the correction is tested.","marker":"[55]"},{"why":"Reports the companion application to magnesium diboride showing that the anharmonic dephasing correction can strongly affect electron conductivity.","marker":"[24]"},{"why":"Supplies the finite-displacement third-order force constants from which the three-phonon coupling coefficients are obtained.","marker":"[51]"},{"why":"Provides the localized-orbital interpolation machinery used to obtain dense electron-phonon matrix elements across the Brillouin zone.","marker":"[47]"}],"fun_headline_variants":["Anharmonic phonons alter electron-phonon scattering","Electron-phonon rates get an anharmonic correction","Phonon lifetimes matter for electron-phonon coupling","Three-phonon effects enter electron-phonon rates","Sign-changing anharmonic term in electron-phonon scattering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formulas replace the damped phonon line by a sharp energy-conserving $\\delta$-function at the harmonic phonon frequency, keeping only the imaginary part of the three-phonon self-energy; if the anharmonic frequency shift or off-shell virtual processes are large, as they can be in strongly anharmonic crystals such as lead telluride, the correction is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Anharmonic phonons alter electron-phonon scattering","Electron-phonon rates get an anharmonic correction","Phonon lifetimes matter for electron-phonon coupling","Three-phonon effects enter electron-phonon rates","Sign-changing anharmonic term in electron-phonon scattering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000381,"raw_usage":{"total_tokens":2019,"prompt_tokens":942,"completion_tokens":1077,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":997}},"tokens_in":558,"tokens_out":1077,"duration_ms":9383,"temperature":1.0,"reasoning_tokens":997,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:17:50.125023+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the same anharmonic electron-phonon scattering rate in lead telluride at 300 K and 500 K using the full phonon spectral function, including the real part of the three-phonon self-energy, and compare it with the delta-function rate of Eqs. (43)-(45); a sizeable difference would show that the on-shell approximation misses part of the anharmonic dephasing.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the many-body Green's function formalism, Feynman diagram rules, and bosonic frequency summation techniques on which the entire derivation is built."},{"cited_title":"Tadano and S","cited_arxiv_id":null,"evidence_quote":"Provides the Feynman rules for anharmonic phonon diagrams used to construct the three-phonon bubble self-energy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets out the standard first-principles electron-phonon framework, including the Fan-Migdal scattering rate, that this work extends."},{"cited_title":"Noffsinger, F","cited_arxiv_id":null,"evidence_quote":"Documents the strong anharmonic phonon scattering in lead telluride, the strongly anharmonic case against which the correction is tested."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the companion application to magnesium diboride showing that the anharmonic dephasing correction can strongly affect electron conductivity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the localized-orbital interpolation machinery used to obtain dense electron-phonon matrix elements across the Brillouin zone."}],"review_version":1}