Anharmonic dephasing in the electron-phonon interaction

Bartomeu Monserrat; Mingran Kong

REVIEW 4 major objections 5 minor 58 references

Anharmonic dephasing in the electron-phonon interaction

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2608.09039 v1 pith:ZILVRMTJ submitted 2026-08-10 cond-mat.mtrl-sci physics.comp-phquant-ph

Mingran Kong , Bartomeu Monserrat This is my paper · ORCID

classification cond-mat.mtrl-sciphysics.comp-phquant-ph

keywords electron-phononcouplinganharmonicphononsfinitephononlifetimesthree-phononscatteringdephasingFan-Migdalself-energyfirst-principlescalculationmany-bodyperturbationtheory

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What'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.

The 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.

Bottom 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[Sec. II E 2, Eqs. (41)-(45)] 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).
[Sec. II E 2, Eqs. (43)-(44)] 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.
[Sec. II D-II E, Fig. 3] 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.
[Sec. II E 1 and Eqs. (43)-(45)] 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.

minor comments (5)

[Eq. (1)] There is a typo: "amd" should be "and" in the definition of the phonon operators.
[Sec. II B] The name "Fan-Midgal" appears in two places; it should be "Fan-Migdal."
[Fig. 7 caption] "restuls" should be "results."
[Sec. IV D] 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.
[Ref. [24]] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central scattering-rate formulas are a standard perturbative derivation from the Hamiltonian in Eq. (31), with no fitted parameter, no load-bearing self-citation, and no quantity defined in terms of the target rate.

full rationale

The central result, Eqs. (42)-(45), is obtained by applying equilibrium Matsubara Green's function techniques to the Hamiltonian in Eq. (31), which contains only first-order electron-phonon vertices and three-phonon vertices. The self-energy is written directly from the Feynman diagram in Fig. 4 via Eq. (32), with a stated symmetry factor S=18 and the internal propagators of Eq. (33). The l-summation is shown explicitly in Eqs. (34)-(40), and the remaining m-summation is formally reduced in Eq. (41). The algebraic finish is deferred to the Supplemental Material with the statement that the summation over m is 'lengthy but straightforward'; that deferral is an auditability concern, not a circular one. No parameter appearing in Eqs. (42)-(45) is fitted to the predicted scattering rates: the electron-phonon coefficients g are DFPT/EPW inputs, the three-phonon coefficients phi come from third-order force constants, and the frequencies and occupations are harmonic inputs. None of these quantities is defined in terms of the final rate Gamma, and the rate itself is not fed back into the derivation. The only self-citations are the companion work on MgB2, Ref. [24], and the reference to the in-house code DaoQuantum, but neither is used to justify the central formulas: the MgB2 discussion is an application of the formalism, not a premise for it. The reviewer-identified sign issue in Eq. (44) and the omission of the real part of the three-phonon self-energy are correctness or completeness concerns, not circular reductions. The derivation is therefore self-contained in the sense relevant to circularity: it reduces to standard many-body perturbation theory applied to a stated Hamiltonian, and the predictions are not equal to their inputs by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The derivation relies on standard many-body perturbation theory, the Migdal approximation, the self-energy relaxation-time approximation, and truncation to the three-phonon bubble diagram. No free parameters are fitted. No new physical entities are introduced.

assumptions (7)

domain assumption Migdal approximation: vertex corrections to the electron-phonon interaction are negligible when the electron mass is much smaller than the ion mass.
Invoked in Sec II B to justify retaining only Fan-Migdal, Debye-Waller, and tadpole diagrams in the standard electron-phonon self-energy.
domain assumption Born-Oppenheimer approximation and cancellation of the electron-phonon tadpole diagram at the DFT equilibrium geometry.
Used in Sec II B to argue the tadpole diagram (Fig. 1c) is zero, following Ref [31].
domain assumption Off-shell approximation with analytic continuation i omega_m -> epsilon + i eta for the electron self-energies.
Used to obtain Eq. (18) and later Eqs. (43)-(45); this on-shell reduction neglects some off-shell self-energy effects.
domain assumption Self-energy relaxation time approximation (SERTA): scattering rates are extracted from the imaginary part of the self-energy using the identity 1/(x+i eta)=P/x - i pi delta(x).
Used in Sec II B and II E to derive Eqs. (19), (29), and (43)-(45).
ad hoc to paper Truncation to the three-phonon bubble diagram: four-phonon diagrams, electron-phonon vertex corrections, and tadpole corrections to the Fan-Migdal self-energy are neglected.
State in Sec II D: 'A systematic study of these effects is beyond the scope of the present work'. The central claim rests on this diagram being the leading anharmonic correction.
ad hoc to paper The real part of the three-phonon self-energy (anharmonic frequency shifts) and off-shell principal-value terms are neglected in the final scattering-rate formulas.
Eqs. (43)-(45) contain only delta functions at harmonic phonon frequencies; no real-part renormalization is included. The authors acknowledge the full spectral-function approach (strategy 2) as more complete but leave it for future work.
domain assumption DFT and DFPT provide accurate electronic and phononic starting points for the perturbative expansion.
All first-principles inputs (band structures, phonon dispersions, coupling coefficients, third-order force constants) are computed within DFT/DFPT using parameters from Refs [52,53,56].

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Cite this review

Pith. "Pith review of Anharmonic dephasing in the electron-phonon interaction." pith.science (2026). https://pith.science/paper/ZILVRMTJ

@misc{pith2026260809039,
  author       = {Pith},
  title        = {Pith review of: Anharmonic dephasing in the electron-phonon interaction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZILVRMTJ}},
  note         = {Machine review of arXiv:2608.09039}
}

read the original abstract

Electron-phonon coupling has been a central topic in condensed matter physics for decades, and firstprinciples methods have demonstrated remarkable success in quantitatively capturing its role in a wide variety of physical phenomena and materials. Conventional calculations of electron-phonon coupling typically assume that phonons have infinite lifetimes, but phonons can exhibit finite lifetimes due to anharmonic phonon-phonon interactions. In this work, we derive an expression for the electron-phonon coupling scattering rates including the effects of anharmonic three-phonon interactions, which lead to phonon dephasing and finite phonon lifetimes. We also describe a first-principles implementation of this anharmonic electron-phonon coupling which can be seamlessly integrated within existing workflows for the evaluation of electron-phonon and phonon-phonon coupling interactions. Finally, we present calculations of electron-phonon scattering rates including phonon dephasing in a range of materials, and discuss the different microscopic mechanisms by which anharmonic phonons influence electron-phonon coupling. This study establishes the importance of finite phonon lifetimes in the evaluation of electron-phonon coupling, and provides a platform to explore these effects in a wide range of materials and phenomena.

Figures

Figures reproduced from arXiv: 2608.09039 by the authors.

**Figure 2.** FIG. 2. Feynman diagrams for phonon-phonon interactions. (a) Second-order bubble diagram driven by three-phonon [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Low-order Feynman diagrams combining electron [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Feynman diagram representing the three-phonon cou [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Workflow for the first principles implementation of the electron-phonon and electron-anharmonic-phonon calculations. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Silicon results, including (a) crystal structure; (b) electron band structure; (c) phonon dispersion; (d) electron [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

**Figure 7.** Figure 7: FIG. 7. Silicon carbide restuls, including (a) crystal structure; (b) electron band structure; (c) phonon dispersion; (d) electron [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Lead telluride results, including (a) crystal structure; (b) electron band structure; (c) phonon dispersion; (d) electron [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9. (a) Electron-phonon scattering rate for PbTe with [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: FIG. 10. (a) Scaling of the harmonic electron-phonon coupling calculation (red), electron–anharmonic-phonon coupling calcu [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]

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General overview The main objective of this work is the incorporation of anharmonic dephasing (or, equivalently, finite phonon lifetimes) into electron-phonon interactions. These ef- fects could be incorporated using different strategies: (i) introducing an energy linewidth to the phonon energy terms in the expression for the electron self-energy; (ii) us...

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Pith tools

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General overview The main objective of this work is the incorporation of anharmonic dephasing (or, equivalently, finite phonon lifetimes) into electron-phonon interactions. These ef- fects could be incorporated using different strategies: (i) introducing an energy linewidth to the phonon energy terms in the expression for the electron self-energy; (ii) us...

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harmonic electron-phonon cou- pling

Derivation of anharmonic dephasing in the electron-phonon interaction In this section, we derive the formula for incorporat- ing anharmonic dephasing in electron-phonon scattering rates. We consider a Hamiltonian incorporating first- order electron-phonon coupling and lowest-order three- phonon coupling: ˆHep+3p = ˆH0 + 1√ N X µ1µ2λ gkq µ2µ1λ ˆAλqˆc† µ2k+...

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