Electron-phonon coupling in correlated materials: insights from the Hubbard-Holstein model

Andrew J. Millis; Jennifer Coulter

REVIEW 4 major objections 3 minor 2 cited by

Electron-phonon coupling in correlated materials: insights from the Hubbard-Holstein model

T0 review · 4 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Electron correlations suppress electron-phonon coupling by a factor of two to four, leaving low-energy Fermi liquid physics intact.

desk verdict Qualitative suppression of electron-phonon coupling by correlations is likely right; the quantitative factor of four rests on shaky analytic-continuation differences and should be treated as provisional. read the letter →

arxiv 2505.08081 v2 pith:QMLI5CFO submitted 2025-05-12 cond-mat.str-el

Jennifer Coulter , Andrew J. Millis This is my paper · ORCID

classification cond-mat.str-el

keywords Hubbard-Holsteinmodelelectron-phononcouplingdynamicalmean-fieldtheorycorrelation-inducedrenormalizationMigdal-EliashbergFermiliquidself-energyanalyticcontinuation

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 asks whether the standard picture of electron-phonon coupling, inherited from density functional theory, survives in a correlated metal. It studies the single-band Hubbard-Holstein model with dynamical mean-field theory and argues that repulsive correlations at $U=2t$ suppress the effective electron-phonon coupling by roughly a factor of two to four while modestly softening the phonon. With these renormalized parameters, the phonon part of the electron self energy is additive to the electron-electron part and follows the same low-frequency formulas as the uncorrelated case. The paper also argues that phonons barely change the low-energy Fermi liquid properties at moderate coupling, but do alter the self energy at frequencies of order the electronic bandwidth. A sympathetic reader would care because this is evidence that practical beyond-DFT calculations can keep the Migdal-Eliashberg structure and absorb correlations into a renormalized coupling.

What carries the argument

The load-bearing object is the phonon-induced self-energy difference $\Sigma^{\mathrm{ph}}_{U,\lambda}(\omega)=\Sigma_{U,\lambda}(\omega)-\Sigma_{U,\lambda=0}(\omega)$, computed from separately continued real-frequency self energies. This difference isolates phonon effects from electron-electron effects. The calculations are done for the Hubbard-Holstein Hamiltonian with on-site $U$, a dispersionless Einstein phonon of frequency $\omega_0$, and coupling $g$, solved by single-site dynamical mean-field theory using a canonical transformation that removes the explicit electron-phonon coupling, so phonon dynamics are treated without a Migdal approximation. Fitting this difference to the perturbative forms yields the renormalized $\tilde{\lambda}$ and $\tilde{\omega}_0$ that carry the paper's main conclusions.

What would settle it

A direct check is to solve the same impurity model on the real frequency axis, for example by exact diagonalization with many phonon levels, at $U/t=2$, $\omega_0/t=0.2$, $\lambda\approx 0.5$, and $\beta t=100$; if the extracted $\tilde{\lambda}$ is not roughly $1/2$ to $1/4$ of the bare value, or if the high-frequency self-energy change above $\omega\approx 2t$ disappears, the central claim is wrong.

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Extended reading notes

Core claim

The paper's central claim is that in the Hubbard-Holstein model at $U/t=2$, electronic correlations reduce the effective electron-phonon coupling $\lambda$ to a renormalized $\tilde{\lambda}$ roughly one quarter to one half of the bare value, while also shifting the effective phonon frequency $\tilde{\omega}_0$. Once these renormalized parameters are used, the phonon contribution to the low-frequency self energy is additive to the electron-electron contribution and matches the perturbative form. Conversely, low-frequency correlation signatures, such as the mass enhancement and the $T^2$ scattering coefficient, are almost independent of $\lambda$ for moderate coupling, with phonon-induced changes concentrated at frequencies of order the bandwidth. On this evidence the paper proposes a renormalized Migdal-Eliashberg description: correlations enter mainly by rescaling the electron-phonon coupling, not by invalidating the phonon-perturbation framework.

Load-bearing premise

The quantitative extraction of $\tilde{\lambda}$ and the high-frequency effects rests on subtracting two separately continued real-frequency self energies, and the authors warn that this amplifies systematic errors of rational-approximant analytic continuation, especially at frequencies above about $t$ and at small $\lambda$.

Editorial extensions

If this is right

Renormalized Migdal-Eliashberg calculations should use $\tilde{\lambda}$ and $\tilde{\omega}_0$ extracted from a correlated calculation rather than bare density-functional values.
At moderate $\lambda$, low-frequency Fermi liquid quantities such as the mass enhancement and the $T^2$ scattering coefficient can be taken from the Hubbard model alone, with phonons added later.
Phonon-induced changes to the self energy at frequencies above about $2t$ survive even when low-energy behavior is additive, so band-edge spectral features carry phonon signatures.
Beyond-DFT electron-phonon methods that omit dynamical correlations are likely to overestimate phonon scattering rates and coupling strengths in moderately correlated metals.

Reading between the lines

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

If the suppression pattern persists beyond this single-band, single-phonon model, first-principles electron-phonon calculations for correlated metals should multiply the bare coupling by a correlation-dependent factor before computing transport or superconducting transition temperatures.
A direct experimental test would compare the phonon contribution to the quasiparticle scattering rate, measured by optical conductivity or photoemission, with the DFT value; the predicted ratio at $U\approx 2t$ is about $1/2$ to $1/4$.
The high-frequency, bandwidth-scale phonon-induced self-energy changes suggest that phonons may renormalize Hubbard bands even when they leave low-energy Fermi liquid properties intact; this could be probed by photoemission line-shape studies above the quasiparticle peak.

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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

The headline is simple: this is a careful DMFT study that does what it says, and the central physical picture is likely correct—correlations renormalize the effective electron-phonon coupling downward, while phonons have little effect on low-frequency Fermi-liquid coefficients. The paper is worth taking seriously, but the quantitative suppression factor (about 2–4 at U=2t) is softer than the abstract implies.

The genuinely new part is the systematic extraction of renormalized coupling λ̃ and phonon frequency ω̃0 from the phonon-induced self-energy difference Σ_ph = Σ_{U,λ} − Σ_{U,0}, obtained from nonperturbative CTSEG DMFT. That goes beyond the earlier Huang-Hanke-Arrigoni-Scalapino vertex-correction work, which was limited to leading order and high temperature. The paper also shows that the C coefficient in ImΣ = C(ω² + (πT)²) is nearly λ-independent, which is a concrete and useful statement. The honest limitation paragraphs are a strength, not a weakness: the authors explicitly flag that subtracting two Padé continuations amplifies errors.

Where it gets shaky: the factor-of-4 suppression is reported for small λ, where the phonon-induced difference is tiny and the continuation error is largest. The authors even say they believe larger λ results are more reliable, yet Figure 3 presents the small-λ factor-of-4 as a headline. The high-frequency self-energy changes in Sec. V.A are also flagged as less reliable, so they should be treated as suggestive. There is also an internal inconsistency in Table I: C=0.3552 at U/t=1, λ=0.3183, while neighboring entries are ~1.0–1.04. That is likely a typo, but it contradicts the text's claim that C is λ-independent and needs fixing. No code or data are provided, which makes independent checking hard.

The math and citation pattern are fine. The comparison to Huang et al. is appropriate, and the broader DFT+DMFT context is well framed. This is not a polemic or a toy model; it is a solid numerical study with a clear claim.

Who is this for? Anyone building beyond-DFT electron-phonon schemes, and people interpreting transport in correlated metals. The qualitative suppression finding will be cited. The paper deserves a serious referee: the methods are appropriate, the question is important, and the flaws are addressable. I would send it to peer review with a request to either provide the raw Matsubara data or explain the Table I anomaly and to soften the small-λ quantitative claims. My verdict would be minor revision, not rejection.

Referee Report

4 major / 3 minor

Summary. This paper uses single-site DMFT with the CTSEG continuous-time solver to compute electron self energies of the Hubbard-Holstein model at U/t = 0, 1, 2, and 3, for phonon frequencies ω0/t = 0.02 and 0.2, and for several electron-phonon coupling strengths λ. By comparing the self energy with and without phonons, Σ_{U,λ} − Σ_{U,0}, the authors argue that phonon effects are additive at low frequencies once λ and the phonon frequency are renormalized, that electronic correlations suppress the effective electron-phonon coupling by roughly a factor of 2 to 4 at U=2t, and that low-frequency Fermi-liquid correlation effects are only weakly modified by phonons while high-frequency self energies are noticeably changed. The paper concludes by advocating a renormalized Migdal-Eliashberg picture for correlated materials.

Significance. If the conclusions hold, this is a valuable benchmark for beyond-DFT electron-phonon treatments: it goes beyond Migdal perturbation theory with a nonperturbative DMFT solution, provides explicit suppression factors, and makes a falsifiable prediction about the effective coupling in correlated metals. A particular strength is that the qualitative suppression of phonon-induced structure is directly visible by comparing the U=0 and U=2t self energies in Fig. 1, so the central qualitative claim does not rest solely on the subtracted continuations. The authors also deserve credit for explicitly flagging the analytic-continuation limitations of their own analysis. The remaining issue is that the quantitative suppression factors and the renormalized-Migdal-Eliashberg conclusion are currently tied to fitted parameters extracted from differences of Padé continuations, and one table entry appears inconsistent with the stated λ-independence of the C coefficient.

major comments (4)

[§IV, Fig. 3] The quantitative suppression factors in Fig. 3(b,c) are obtained by fitting tilde-λ and tilde-ω0 to Σ_ph = Σ_{U,λ} − Σ_{U,λ=0}, a difference of two independently Padé-continued self energies. The authors state in §II that continuations may be less reliable for frequencies above about t and in §IV that analytic continuation errors may be magnified in the difference, especially for small λ where Σλ is very close to Σλ=0; they then say they believe the larger λ results to be more reliable. However, Fig. 3(b,c) reports the largest suppression ratios for the smaller λ values. Because the abstract's 'strongly suppressed' claim and the factor-of-roughly-2-to-4 statement rest on these fitted values, the quantitative conclusion is currently under-supported. Please add an uncertainty estimate or a continuation-independent cross-check, for example fitting the Matsubara-axis differences directly, and if necessary restrict the quantitative claim to the large-λ regime where it is robust.
[§V, Table I] The entry for U/t=1, λ=0.3183 gives C=0.3552, while the neighboring entries for λ=0 and λ=0.1791 give 0.9814 and 1.0420, respectively. This is inconsistent with the text's statement that the C coefficient is essentially λ-independent and with the conclusion that phonons only weakly affect the T^2 coefficient. Either the table contains a typographical error or the λ dependence is non-monotonic in a way the text does not acknowledge. This must be resolved before the 'weak modification of correlations' conclusion is accepted.
[§V.A, Fig. 4(b)] The claim that phonons produce significant changes in the self energy for ω ≳ 2t is load-bearing for the abstract's statement that phonon-induced modifications are most evident at high frequencies. Yet §II states that the continuation methods may be less reliable above about t, and §VI concedes that this effect should be further investigated using methods not subject to the limitations of analytic continuation. The high-frequency claim is therefore not established by the presented evidence alone. Please provide a quantitative estimate of the continuation error in this regime or an independent cross-check, or soften the corresponding abstract and conclusion statements.
[§IV, Fig. 2] The conclusion states that the phonon contribution to the low-frequency self energy is additive once λ and ω0 are renormalized, but §IV itself notes that the small-λ subtracted self energies are not fully consistent with the renormalized-coupling picture: for ω0=0.02t there is no visible phonon onset, and for ω0=0.2t the imaginary part shows a peak and a sign change whose origin 'remains to be determined.' The additive/renormalized-Migdal-Eliashberg statement should either be restricted to the larger-λ, larger-ω0 regime where the evidence is cleaner, or supported by additional evidence that the small-λ discrepancies are continuation artifacts.

minor comments (3)

[§VI] The sentence 'T^2 and ω^2 contributions to the scattering rate and the are only weakly impacted' is grammatically incomplete, and 'coupling strengths However' needs a punctuation or paragraph break before 'However.'
[Fig. 2 caption] The caption notation '(c,d) 2x' is cryptic; the text explains that panel (d) is the real part of the subtracted self energy magnified by 2×, so the caption should say explicitly which panels are scaled and by what factor.
[Fig. 3 caption] The dotted line described as 'λ = λ/4' should read 'tilde-λ = λ/4' to match the notation used in the text and axes; as written in the plain-text version it is self-referential and confusing.

Circularity Check

1 steps flagged · score 3.0 of 10

The suppression itself is in the raw DMFT self energies, but the 'renormalized Migdal-Eliashberg' effective coupling is fit to the same data, making the quantitative factor-4 conclusion partly self-referential.

fitted input called prediction [Section IV, Fig. 3 and the paragraph beginning 'We now use characteristic features of Σ_ph...']
"We now use characteristic features of Σ_ph to define renormalized couplings λ̃ and phonon frequencies ω̃0 quantitatively. For the larger phonon frequency, we defined ω̃0 from the maximum in ReΣ (equivalently from the midpoint of the rise in ImΣ) and then we fit the peak in ImΣ to the perturbative form for Σ as in Equations 2 and 4 using the base density of states N(μ). ... In the correlated (U/t = 2) case, we see that at weak coupling the effective electron-phonon coupling is reduced by approximately a factor of 4 for smaller λ values."

The claim that electron-phonon coupling is suppressed by a factor of about four at U/t=2 is a report of λ̃, and λ̃ is obtained by requiring the peak of the computed ImΣ_ph to match the Migdal expression of Eq. 2. The peak height therefore agrees with the 'renormalized Migdal-Eliashberg' form by construction; only the frequency dependence away from the fitted point provides independent evidence. Because the same fitted λ̃ is later used to split the low-frequency slope into phonon and electron-electron parts in Sec. V.A, the conclusion that the electron-electron part is weakly modified by phonons is partly a rearrangement of the fit, although the raw DMFT comparison in Fig. 1 does independently show a smaller phonon-induced feature at U=2t.

full rationale

This paper's central observation—that at U=2t the phonon-induced self-energy features are much smaller than at U=0—is read directly from the DMFT self energies in Fig. 1 and does not depend on any fitted parameter; that part of the claim is not circular. No load-bearing uniqueness theorem or self-citation chain is used: the cited Werner-Millis solver is an external method, and the comparison to Huang et al. is a consistency check rather than the basis of the conclusion. The only circular element is in the renormalized-Migdal section: λ̃ and ω̃0 are 'defined' and the peak of ImΣ_ph is 'fit' to the perturbative form, so the agreement at the fitted peak is guaranteed. The paper is explicit that continuation errors are amplified by subtraction and that the small-λ results are less reliable; that is a correctness caveat, not additional circularity. Overall the central suppression claim has independent content, but the quantitative factor-four and the 'renormalized Migdal-Eliashberg' justification are partially built from the same self-energy data they are used to explain.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The model has bare inputs U, g, omega0, bandwidth, and temperature, but the load-bearing quantitative claims depend on fit-extracted lambda-tilde, omega-tilde, and C coefficients. The main assumptions are the adequacy of single-site DMFT, the reliability of Padé continuation and subtraction, the separability of phonon and electron-electron contributions, and the use of perturbative Migdal formulas as the fitting template. No new physical entities are postulated.

free parameters (3)

Renormalized electron-phonon coupling lambda-tilde (U/t=2) = Approximately lambda/4 at weak coupling; about 0.28 for bare lambda=0.6446 at omega0/t=0.2
Extracted in Section IV by fitting the phonon contribution to the self energy to the perturbative form. The central suppression claim is quantified through this fitted quantity.
Renormalized phonon frequency omega-tilde_0 (U/t=2) = Reduced from 0.2t to about 0.1t at the largest studied lambda
Obtained from the peak in Re Sigma_ph or the midpoint of the step in Im Sigma_ph, as described in Section IV and shown in Figure 3(a).
C coefficient in Im Sigma(0) = alpha T + C T^2 = C about 5.0 for U/t=2, about 16 to 17 for U/t=3, about 1.0 for U/t=1 except C=0.355 at lambda=0.3183
Fit coefficients in Table I are used to support the claim that phonons barely change the T^2 electron-electron scattering rate. The U/t=1 outlier is unexplained.

assumptions (6)

domain assumption Single-site DMFT with a local self energy is an adequate approximation for the Hubbard-Holstein model at the studied parameters.
Introduced in Section II when the lattice model is mapped to a single-site impurity model. Nonlocal and momentum-dependent correlations are neglected.
domain assumption Padé analytic continuation of Matsubara self energies gives reliable real-frequency results at least below frequencies of order t, and the difference of two continuations is meaningful.
Stated in Section II: continuations must reproduce imaginary-axis data to 0.1 percent error, but the authors note that higher frequencies may be less reliable and that subtraction amplifies errors. This is load-bearing for lambda-tilde extraction and high-frequency results.
ad hoc to paper The phonon contribution is cleanly isolated by the difference Sigma_{U,lambda} minus Sigma_{U,0}.
Used throughout Sections IV and V to define Sigma_ph and to separate phonon and electron-electron effects. The additivity it implies is one of the conclusions, making this assumption partly circular.
ad hoc to paper The perturbative Migdal-type formulas (Eqs. 2 to 4) are the correct functional form for the phonon self energy once lambda and omega_0 are renormalized.
Invoked in Section IV to define and fit lambda-tilde and omega-tilde, for example the dashed curve in Figure 2(c). The good fit is then cited as evidence for a renormalized Migdal-Eliashberg picture.
domain assumption The semicircular density of states and a dispersionless Einstein phonon capture the qualitatively relevant physics of correlated materials.
The model is introduced in Section II as a projection of band theory and DFT electron-phonon matrix elements onto a single band. Real materials have multiple orbitals and phonon dispersions.
domain assumption Beta t = 100 is effectively zero temperature for omega0/t=0.2, and the temperature range with T greater than omega0 permits a classical phonon analysis.
Used in Sections II and V.B to interpret the temperature dependence of decay rates and to justify the alpha T + C T^2 fit.

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Pith. "Pith review of Electron-phonon coupling in correlated materials: insights from the Hubbard-Holstein model." pith.science (2026). https://pith.science/paper/QMLI5CFO

@misc{pith2026250508081,
  author       = {Pith},
  title        = {Pith review of: Electron-phonon coupling in correlated materials: insights from the Hubbard-Holstein model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QMLI5CFO}},
  note         = {Machine review of arXiv:2505.08081}
}

read the original abstract

Dynamical mean-field theory computations of the electron self energy of the Hubbard-Holstein model as a function of electron-phonon and electron-electron interactions are analyzed to gain insight into the dependence of electron-phonon couplings on correlation strength in quantum materials. We find that the electron-phonon interaction is strongly suppressed by electronic correlations, while electron-electron correlation effects at Fermi liquid scales are only weakly modified by coupling to phonons, with phonon-induced modifications most evident at high frequencies on the order of the electronic bandwidth. Implications for beyond-density functional theories of the electron-phonon interaction are discussed.

Figures

Figures reproduced from arXiv: 2505.08081 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: plots the imaginary parts of the calculated self energies against ω 2 over low-moderate frequency regime (left panel) and a broad frequency regime extending to energies greater than the half bandwidth 2t (right panel). Panel (a) shows that the C coefficient is essentia…

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Pith tools

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

[1] [1]

renormalized Migdal- Eliashberg

for ˜g = g/1.5 and ˜ω0 = 0.1, plotted as a grey dashed line. Note that ( d) shows the real part of the subtracted self energy magnified by 2x. energy computed with λ̸= 0 and with λ = 0. As this quantity is the difference between two independently con- tinued quantities, analytic continuation errors may be magnified in the difference, especially for small ...

work page

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J. M. Ziman, Electrons and phonons: the theory of transport phenomena in solids, Oxford university press (2001)

work page 2001

[3] [3]

Giustino, Electron-phonon interactions from first prin- ciples, Reviews of Modern Physics 89, 015003 (2017)

F. Giustino, Electron-phonon interactions from first prin- ciples, Reviews of Modern Physics 89, 015003 (2017)

2017

[4] [4]

Mandal, R

S. Mandal, R. E. Cohen, and K. Haule, Strong pressure- dependent electron-phonon coupling in fese, Physical Re- view B 89, 220502 (2014)

work page 2014

[5] [5]

Gerber, S.-L

S. Gerber, S.-L. Yang, D. Zhu, H. Soifer, J. Sobota, S. Re- bec, J. Lee, T. Jia, B. Moritz, C. Jia, et al., Femtosecond electron-phonon lock-in by photoemission and x-ray free- electron laser, Science 357, 71 (2017)

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