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arxiv: 2604.14294 · v2 · submitted 2026-04-15 · ❄️ cond-mat.str-el

Recognition: unknown

Breakdown of the Migdal-Eliashberg theory for electron-phonon systems. Role of polarons/bi-polarons

Andrey Chubukov , Ilya Esterlis , Artem Abanov , Nikolay Prokof'ev
Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:08 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Migdal-Eliashberg theorypolaronbipolaronelectron-phonon interactionFermi liquidcharge density wavevariational method
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The pith

Migdal-Eliashberg theory collapses to polaron states at couplings below phonon softening

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Electrons interacting with phonons in the adiabatic limit are conventionally described by Migdal-Eliashberg theory even at strong coupling, with breakdown expected only when the dressed phonon frequency approaches zero. This work demonstrates that at low and near-full electron densities the Fermi liquid instead becomes unstable to a polaron or bipolaron state at substantially smaller values of the coupling constant λ. Variational calculations establish rigorous upper bounds showing this transition occurs before phonon softening in both two and three dimensions, and the bound on λ approaches zero at extreme densities in three dimensions. The instability involves fermions with energies of order the bandwidth and therefore falls outside the Migdal-Eliashberg framework. At densities near half filling the first instability is instead a charge-density-wave state from which the polaronic state develops at still larger λ.

Core claim

Using variational considerations we establish rigorous upper bounds on the coupling λ at which a Fermi liquid state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on λ tends to zero in the limit of small or near-full density. Polaron formation is produced by fermions with energies comparable to the bandwidth and thus lies outside the realm of Migdal-Eliashberg theory.

What carries the argument

Variational upper bounds on the energy that identify the coupling strength where the polaron or bipolaron state becomes energetically favorable over the Fermi liquid.

If this is right

  • At low and high densities the polaron/bipolaron instability precedes phonon softening.
  • In 3D the critical λ bound vanishes at extreme densities.
  • Near half filling a charge-density-wave state appears first.
  • A strong-coupling regime of Migdal-Eliashberg theory exists near the CDW instability before polaron formation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This breakdown mechanism suggests that low-density materials may show polaronic behavior at moderate couplings without phonon softening.
  • The variational approach could be applied to other electron-boson models to find similar early instabilities.
  • Direct simulations at intermediate densities would map the full phase diagram between the variational bounds and exact results.

Load-bearing premise

Variational upper bounds on energy reliably identify the first instability to polaron/bipolaron states for generic densities without a lower-energy competing state existing between the Fermi liquid and polaron regimes.

What would settle it

A numerical calculation of the Holstein model ground state at low density that remains Fermi-liquid-like with finite phonon frequency up to a λ larger than the reported variational upper bound.

Figures

Figures reproduced from arXiv: 2604.14294 by Andrey Chubukov, Artem Abanov, Ilya Esterlis, Nikolay Prokof'ev.

Figure 1
Figure 1. Figure 1: FIG. 1. Density of states [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The phase diagram obtained in variational (left) and analytical (right) analysis for a homogeneous polaron order on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic phase diagram for an electron-phonon [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical results for 3D fermions with tight-binding dispersion, coupled to a phonon with dispersion given by Eq. (11), [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The same plots as in Fig. 4, but now for a 2D system with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The phase diagram for 3D and 2D systems (left and right panels, respectively) for a dispersionless bare phonon, in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The DOS [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) One-loop electron self-energy. (b) Higher-order [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Function [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Function [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of the exact DOS with the one obtained [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The envelope of the DOS, Eq. (50), in the limit of [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Schematic electron DOS in the regime [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. The DOS [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Schematic electron DOS in the regime [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Potential contributions to [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Schematic electron DOS in the regime [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The condensate order parameter [PITH_FULL_IMAGE:figures/full_fig_p027_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. The evolution of the DOS (in units of 2 [PITH_FULL_IMAGE:figures/full_fig_p028_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Ground state energy as a function of [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Analytic phase diagram of the ancilla model for [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. One of the symmetry related momentum points at [PITH_FULL_IMAGE:figures/full_fig_p030_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. (a) Critical value [PITH_FULL_IMAGE:figures/full_fig_p031_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Comparison of the ground state energy [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. (a) Full chemical potential potential of the checker [PITH_FULL_IMAGE:figures/full_fig_p035_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. The evolution of the DOS with decreasing [PITH_FULL_IMAGE:figures/full_fig_p037_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Schematic phase diagram for the checkerboard po [PITH_FULL_IMAGE:figures/full_fig_p037_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. Phonon self-energy diagrams that include particle [PITH_FULL_IMAGE:figures/full_fig_p037_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. (a) Electronic CDW energy [PITH_FULL_IMAGE:figures/full_fig_p038_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. Results for a semi-circular (non-interacting) DOS [PITH_FULL_IMAGE:figures/full_fig_p042_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31. The phase diagram for a 3D system of spin-less fermions for a dispersionless bare phonon spectrum [PITH_FULL_IMAGE:figures/full_fig_p044_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32. (a) One-loop electron self-energy. (b) Two-loop [PITH_FULL_IMAGE:figures/full_fig_p046_32.png] view at source ↗
Figure 34
Figure 34. Figure 34: FIG. 34. Diagrams contributing to the rainbow approxima [PITH_FULL_IMAGE:figures/full_fig_p047_34.png] view at source ↗
Figure 33
Figure 33. Figure 33: FIG. 33. The DOS of a single site Holstein model for [PITH_FULL_IMAGE:figures/full_fig_p047_33.png] view at source ↗
Figure 35
Figure 35. Figure 35: FIG. 35. Rainbow approximation for [PITH_FULL_IMAGE:figures/full_fig_p048_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. The location of the poles in [PITH_FULL_IMAGE:figures/full_fig_p049_36.png] view at source ↗
Figure 38
Figure 38. Figure 38: FIG. 38. The color plot of the argument of 1 [PITH_FULL_IMAGE:figures/full_fig_p050_38.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37. The real (a) and imaginary (b) parts of the Green’s [PITH_FULL_IMAGE:figures/full_fig_p050_37.png] view at source ↗
Figure 39
Figure 39. Figure 39: FIG. 39. The location of the poles of [PITH_FULL_IMAGE:figures/full_fig_p051_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: FIG. 40. Diagrams contributing to the self-consistent one [PITH_FULL_IMAGE:figures/full_fig_p051_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: FIG. 41. Poles of [PITH_FULL_IMAGE:figures/full_fig_p052_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: FIG. 42. Self-consistent one-loop approximation for an Ein [PITH_FULL_IMAGE:figures/full_fig_p053_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: FIG. 43. Self-consistent one-loop approximation for an Ein [PITH_FULL_IMAGE:figures/full_fig_p054_43.png] view at source ↗
Figure 45
Figure 45. Figure 45: FIG. 45. (a) The exact Im [PITH_FULL_IMAGE:figures/full_fig_p055_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: FIG. 46. The color plot of the argument of 1 [PITH_FULL_IMAGE:figures/full_fig_p056_46.png] view at source ↗
Figure 48
Figure 48. Figure 48: FIG. 48. (a) The poles of the partial [PITH_FULL_IMAGE:figures/full_fig_p057_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: FIG. 49. Comparison of the (a) real and (b) imaginary parts [PITH_FULL_IMAGE:figures/full_fig_p058_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: FIG. 50. Evolution of the polaron density of states in a 2D [PITH_FULL_IMAGE:figures/full_fig_p060_50.png] view at source ↗
read the original abstract

The Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option -- collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling $\lambda$, at which a FL state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on $\lambda$ tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth; i.e., polaron formation is outside the realm of MET. Closer to half-filling, the leading instability upon increasing $\lambda$ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger $\lambda$ out of a CDW-ordered state and inherits a CDW order over some range of coupling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that Migdal-Eliashberg theory (MET) for electron-phonon systems breaks down at strong coupling via a transition to a polaronic/bipolaronic ground state before the dressed phonon softens, at least at small and near-maximum densities. Variational considerations are used to derive rigorous upper bounds on the coupling λ for the Fermi-liquid to polaron/bipolaron transformation in both 2D and 3D; these bounds lie below the softening point. Near half-filling the leading instability is instead a charge-density-wave (CDW) state, with the polaron state appearing at larger λ out of the CDW phase. Analytical support is drawn from exact diagrammatic results on the spin-polarized Holstein model.

Significance. If the central claim holds, the work supplies concrete, falsifiable upper bounds on the regime of MET validity and clarifies that polaron formation can preempt phonon softening outside the adiabatic, low-energy window assumed by MET. The variational upper-bound technique and the link to exact Holstein-model diagrammatics are genuine strengths that make the bounds rigorous and reproducible in principle.

major comments (2)
  1. [Variational considerations and low-density analysis] The identification of the polaron/bipolaron state as the first instability (rather than a competing ordered phase) rests on the variational ansatz being globally optimal. While the manuscript correctly notes CDW dominance near half-filling, the low-density regime relies on Holstein-model hints without explicit energy comparisons between the polaron variational wavefunction and alternative states (e.g., modulated CDW or other density-wave ansätze) at the same λ and n. This comparison is load-bearing for the claim that MET breaks down specifically because of polarons.
  2. [3D systems near small or full density] In the 3D small- and near-full-density limits the upper bound on λ is stated to tend to zero. The derivation of this limit and the explicit demonstration that the corresponding phonon-softening λ remains finite should be shown in a dedicated subsection or appendix, including the functional form of the variational energy as density approaches the band edge.
minor comments (2)
  1. [Abstract] The abstract states that the analysis is performed 'numerically and analytically,' yet the quantitative results presented appear to be dominated by variational bounds and Holstein diagrammatics. A brief clarification of the numerical methods (e.g., exact diagonalization or Monte Carlo parameters) and their error bars would improve reproducibility.
  2. [Notation and definitions] Notation for the dimensionless coupling λ, the density n, and the variational parameters should be collected in a single table or early section to aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and constructive suggestions. We address the two major comments point by point below. We will revise the manuscript to incorporate clarifications and additional derivations where needed.

read point-by-point responses

  1. Referee: [Variational considerations and low-density analysis] The identification of the polaron/bipolaron state as the first instability (rather than a competing ordered phase) rests on the variational ansatz being globally optimal. While the manuscript correctly notes CDW dominance near half-filling, the low-density regime relies on Holstein-model hints without explicit energy comparisons between the polaron variational wavefunction and alternative states (e.g., modulated CDW or other density-wave ansätze) at the same λ and n. This comparison is load-bearing for the claim that MET breaks down specifically because of polarons.

    Authors: We agree that an explicit energy comparison to possible competing ordered states would further strengthen the low-density claim. At low densities the Fermi surface lacks the nesting vectors required for a modulated CDW instability (as already indicated by the exact diagrammatic results on the spin-polarized Holstein model that we cite). The variational upper bound we derive is nevertheless rigorous: it shows that the polaron/bipolaron energy lies below the FL energy at a value of λ strictly smaller than the phonon-softening point, independent of whether other instabilities exist. In the revision we will add a short paragraph in the low-density section that explicitly contrasts the variational polaron energy with the expected cost of a modulated CDW at the same n and λ, using the Holstein-model insights already present in the manuscript. revision: partial

  2. Referee: [3D systems near small or full density] In the 3D small- and near-full-density limits the upper bound on λ is stated to tend to zero. The derivation of this limit and the explicit demonstration that the corresponding phonon-softening λ remains finite should be shown in a dedicated subsection or appendix, including the functional form of the variational energy as density approaches the band edge.

    Authors: We accept the suggestion. In the revised manuscript we will add a dedicated appendix that derives the vanishing of the variational upper bound on λ as n → 0 or n → 1 in three dimensions. The appendix will expand the variational energy functional near the band edge, demonstrate that the critical λ for polaron formation approaches zero, and contrast this with the phonon-softening λ of MET, which remains finite because it is controlled by the density of states at the Fermi level (which stays nonzero away from the exact band edge). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard variational bounds and independent diagrammatics.

full rationale

The paper's central result uses variational wavefunctions to obtain rigorous upper bounds on the critical coupling λ where the polaron/bipolaron energy falls below the Fermi-liquid energy. This is a standard, non-circular application of the variational principle: the computed variational energy provides an upper bound on the true ground-state energy without defining the target instability in terms of itself. The comparison to the phonon-softening point from Migdal-Eliashberg theory or diagrammatic calculations is external to the variational step. References to prior exact treatments of the Holstein model supply supporting hints but do not carry the load-bearing argument; the variational bounds stand on their own and are falsifiable by direct energy minimization. No parameters are fitted and then relabeled as predictions, no ansatz is smuggled via self-citation, and no uniqueness theorem is invoked to force the result. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the adiabatic limit assumption of MET and the validity of variational energy bounds to compare Fermi liquid and polaron states; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Adiabatic limit where phonon Debye frequency is much smaller than Fermi energy
    Core premise of Migdal-Eliashberg theory stated in abstract.
  • standard math Variational methods provide rigorous upper bounds on ground-state energies for state comparison
    Used to establish transformation points to polaron/bipolaron states.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Apparent Planckian scattering from local polaron formation

    cond-mat.str-el 2026-04 unverdicted novelty 7.0

    Local polaron formation in the disordered Holstein model generates apparent Planckian scattering Γ_tr = Γ0 + α k_B T / ℏ with α ~ O(1) from quasielastic scattering, as evidenced by Monte Carlo simulations.

Reference graph

Works this paper leans on

94 extracted references · 6 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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    3D case The 3D phase diagram, obtained in the variational study for dispersion-lessω 0 and spin-full fermions, is pre- sented in the middle of the lower panel in Fig. 4. For convenience, in Fig.31 we re-plot this phase diagram in units ofλ p and for spin-less fermions. We note that phase diagrams in 2D and 3D are similar, but there is one key distinction:...

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    Finite T At temperatures above any ordering transition, the evolution from the FL state to the polaron state is smooth. Nevertheless, numerical calculations have shown that this crossover occurs rapidly [27, 68, 70, 72, 74, 75, 78, 79], and can be identified by the onset of a pseudo- gap in the electronic density of states [70]. As detailed in Sec.IV D 2,...

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    andN F = (2/πW). This self- energy can be split into static and dynamic parts: Σ(1)(ωm) = Σ(1)(0) + Σ(1) dyn(ωm) (A3) where Σ(1)(0) =−λ 0ω0 log W ω0 , Σ(1) dyn(ωm) =−λ 0ω0 log ω0 ω0 −iω m ,(A4) The first term, which depends onW, accounts for the renormalization of the chemical potential,µ=µ 0 − Σ(1)(0), and the second, which does not depend on W, is respo...

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    Combining it with the O(λ0) contribution and expanding inω m, we obtain: Σ(1) dyn(ωm) =−iλ 0ωm +iλ 2 0ωm log W ω0 (A5) We now compute the two two-loop self-energies, Σ (2v) and Σ(2r). Both containλ 2 0 in the prefactors, and in the calculations to this order, the renormalization ofµ 0 into µcan be neglected. We start with the vertex correc- tion diagram, ...

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    Rainbow approximation We writeG −1 R (ω) =ω+µ−Σ R(ω), where the subindex indicates that we compute the self-energy by keeping only rainbow diagrams (one diagram at each loop order), see Fig. 34. We measureω,µand Σ R in units ofω 0 and introduce ¯ω=ω/ω 0, ¯µ=µ/ω 0 and ¯ΣR = Σ R/ω0. For the Green’s function, we introduce ¯GR =G R ω0. 48 a. Aδ-functional bos...

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    Self-consistent one-loop approximation We now extend the perturbation series to include all renormalizations of the internal fermionic propagator in the self-energy diagram , e.g., two subsequent rainbow renormalizations. We show the corresponding diagrams in Fig. 40. The sum of the diagrammatic series can be formally represented in a compact form by repl...

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    we again usex= ˜ω+ ˜µ. The perturbation series for the self-energy yield ¯Σ1L(x, β) = β x−1 + β2 (x−1) 2(x−2) + β3 (x−1) 2(x−2) 2 1 (x−3) + 1 x−1 + β4 (x−1) 2(x−2) 2 1 (x−3) 2(x−4) + 1 (x−2)(x−3) 2 + 2 (x−1)(x−2)(x−3) + 1 (x−1) 2(x−2) +...(C29) As before, we introduce a partial sum ofmterms, ¯Σ(m) 1L (x, β) and analyze the poles of ¯G(m) 1L (z, β) in the ...

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