Generalized Eliashberg Approach

• Generalized Eliashberg approach is a framework that extends conventional Migdal–Eliashberg theory by relaxing assumptions like phonon-only pairing, static Coulomb repulsion, and equilibrium conditions.
• It employs a self-consistent treatment of both normal and anomalous self-energies using the Nambu–Gorkov formalism to address multiband, nonadiabatic, and nonequilibrium effects in superconductors.
• The method has been extended to include ab initio Coulomb interactions, spin-fluctuation models, and nonequilibrium Keldysh–Usadel formulations, offering a versatile tool for studying advanced superconducting phenomena.

Generalized Eliashberg approach designates a family of extensions of conventional Migdal–Eliashberg theory in which the self-consistent treatment of normal and anomalous self-energies is preserved while one or more standard assumptions—phonon-only pairing, static Coulomb repulsion, adiabaticity, equilibrium, isotropy, Fermi-liquid normal states, or mean-field-only treatment—are relaxed. In the cited literature, the term covers ab initio Coulomb-inclusive formalisms, nonequilibrium Keldysh–Usadel theories, multiband and full-bandwidth spin-fluctuation frameworks, quantum-critical large-$N$ constructions, external-pair-potential models, and several numerical or functional reformulations (Akashi, 2021, Tikhonov et al., 2019, Schrodi et al., 2020, Esterlis et al., 13 Jun 2025).

1. Formal basis

The common formal starting point is the Nambu–Gorkov Green’s function and Dyson equation,

${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$

together with a decomposition of the superconducting self-energy into diagonal and anomalous components. A representative form is

$${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$

where $Z$ is the renormalization function, $\phi$ the anomalous pairing self-energy, and $\chi$ the diagonal energy-shift or band-renormalization term (Akashi, 2021). In time-reversal-symmetric cases, $Z$ is identified with the odd-in-frequency part of the normal self-energy and $\chi$ with the even-in-frequency part: $$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$

$$\chi(i\omega_j)-V_{\rm eff}=\frac{1}{2}\left[\Sigma(i\omega_j)+\Sigma(-i\omega_j)\right].$$

This separation is not merely notational. Several generalized formulations turn on the fact that ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$0 multiplies frequency while ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$1 shifts the dispersion, so they renormalize observables in inequivalent ways (Akashi, 2021).

A functional-integral formulation makes the same structure explicit from another direction. After integrating out phonons and decoupling the retarded electron-electron interaction, both the anomalous pairing function and the normal self-energy appear as Hubbard–Stratonovich fields. The mean-field Eliashberg equations then arise as saddle-point conditions of a bosonic effective action, with schematic self-consistency relations

${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$2

and Gaussian fluctuations organized around Cooper and density/self-energy channels (Protter et al., 2020). In that sense, generalized Eliashberg theory is often less a single model than a self-consistent architecture for frequency-dependent normal and anomalous sectors.

2. Major directions of generalization

The phrase “generalized Eliashberg approach” is used in several distinct but related ways.

Direction	Characteristic modification	Representative work
External pair potential	${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$3 and ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$4	(Grigorishin, 2017)
Microwave-driven dirty superconductors	Generalized self-consistency ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$5 at arbitrary ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$6 for weak power	(Tikhonov et al., 2019)
Full-bandwidth spin fluctuations	Momentum-, band-, and Matsubara-dependent ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$7 over the full Brillouin zone	(Schrodi et al., 2020)
Quantum-critical and NFL pairing	Coupled ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$8, ${\bm G}={\bm G}_0+{\bm G}_0{\bm \Sigma}{\bm G},$9, and $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$0 equations with critical bosons and large-$${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$1 control	(Esterlis et al., 13 Jun 2025)
Nonadiabatic finite-bandwidth extensions	Separation of pairing $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$2 from mass-renormalization $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$3	(Sadovskii, 2019)
Dynamical Coulomb/plasmonic kernels	Frequency-dependent $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$4 and hybrid Eliashberg–SCDFT treatments	(Davydov et al., 2020)

In the external-pair-potential construction, the parameter $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$5 is not itself a pairing interaction in the usual sense. Rather, it shifts the anomalous self-energy or gap sector directly. The resulting equations remain Eliashberg-like, but the anomalous combination entering the propagators is $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$6 or $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$7, not the standard $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$8 or $${\bm \Sigma}(i\omega_j)=i\omega_j(1-Z(i\omega_j))\sigma_0+\phi(i\omega_j)\sigma_1+(\chi(i\omega_j)-V_{\rm eff})\sigma_3,$$9. Within the model analyzed there, inclusion of the Coulomb pseudopotential restores a finite $Z$0, and the gap vanishes linearly as $Z$1, $Z$2, rather than with the standard mean-field square-root law (Grigorishin, 2017).

In multiorbital unconventional superconductors, the generalization is often toward full momentum and orbital resolution. For FeSe, the interaction is constructed from multiorbital Hubbard–Hund terms, RPA spin and charge susceptibilities, and then transformed from orbital to band space before entering coupled equations for $Z$3, $Z$4, and $Z$5 (Schrodi et al., 2020). In dirty superconductors under microwave irradiation, the same phrase denotes a nonequilibrium weak-power extension in which spectral depairing and quasiparticle redistribution are treated on equal footing (Tikhonov et al., 2019). In quantum-critical settings, it denotes self-consistent pairing out of non-Fermi liquids rather than quasiparticle metals (Esterlis et al., 13 Jun 2025).

3. Diagonal self-energies, Coulomb renormalization, and normalization

A major contemporary use of the generalized Eliashberg approach concerns the explicit treatment of the screened Coulomb interaction. A central claim of Akashi’s homogeneous-electron-gas analysis is that a properly “normed” ab initio Eliashberg theory must reproduce the homogeneous electron gas limit, and in that limit the Coulomb self-energy cannot be treated consistently if one keeps only the usual $Z$6-type renormalization while dropping the diagonal $Z$7 term (Akashi, 2021). The generalized Eliashberg equations already show the asymmetry: $Z$8 In the homogeneous electron gas, both contributions are individually large and largely cancel in the quasiparticle velocity, but they do not cancel in the pairing kernel. The paper therefore concludes that neglect of $Z$9 breaks the normalization of the theory already in the reference electron gas (Akashi, 2021).

That distinction between diagonal and anomalous sectors reappears in a different setting in the pseudogap study based on combined electron-phonon and electron-electron-phonon interactions. There, the density of states is argued to be gapped primarily by the diagonal self-energy components $\phi$0 and $\phi$1, while the direct contribution of the anomalous self-energy $\phi$2 is negligible. In this construction the pseudogap persists above $\phi$3 because $\phi$4 and $\phi$5 remain strongly frequency dependent in the anomalous normal state (Szczesniak et al., 2015). This suggests a broader lesson: generalized Eliashberg theories often cease to identify “the gap” with the anomalous sector alone.

Several recent ab initio extensions reach a similar conclusion from the Coulomb side. A plasmonic extension of Eliashberg theory replaces the static Coulomb kernel by a dynamically screened interaction $\phi$6, represented through an electronic spectral density $\phi$7. This introduces Coulomb contributions both to the anomalous kernel and to the mass renormalization $\phi$8. Numerically, however, fully plasmonic Eliashberg theory substantially overestimates plasmon-mediated pairing and $\phi$9, whereas a hybrid scheme that keeps Eliashberg for the electron-phonon sector and uses an SCDFT-like treatment for the dynamical Coulomb sector gives results close to SCDFT and in excellent agreement with experiment (Davydov et al., 2020). A related beyond-RPA extension replaces the usual test-charge screened Coulomb interaction by the Kukkonen–Overhauser effective interaction. In a set of conventional superconductors, the resulting vertex corrections systematically decrease $\chi$0, from a few percent in bulk lead to more than $\chi$1 in compressed lithium (Pellegrini et al., 2023). Taken together, these works constrain generalized Eliashberg practice: making Coulomb dynamics explicit is necessary, but doing so within a naive $\chi$2-like anomalous self-energy can be quantitatively unreliable.

4. Nonadiabatic, antiadiabatic, and weak-coupling limits

Another major axis of generalization concerns the breakdown of the adiabatic assumption. In the nonadiabatic or antiadiabatic regime, where characteristic phonon frequencies are comparable to or larger than the Fermi energy or bandwidth, the standard identification of the Eliashberg coupling with both pairing strength and mass renormalization ceases to hold. The generalized mass-renormalization coupling becomes

$\chi$3

while the pairing strength remains controlled by $\chi$4 (Sadovskii, 2019). In the antiadiabatic limit, $\chi$5 is parametrically small, and the relevant perturbative parameter becomes

$\chi$6

The transition temperature correspondingly crosses over from a phonon-scale prefactor to an electronic one, with

$\chi$7

in the antiadiabatic regime (Sadovskii, 2019). The general implication is that pairing strength, quasiparticle mass renormalization, and vertex-correction control parameter need not coincide once the phonon energy ceases to be the smallest scale.

The weak-coupling limit of conventional Eliashberg theory also differs from the BCS limit in a precise way. For an Einstein phonon with $\chi$8, the renormalized weak-coupling transition temperature is

$\chi$9

not the usual BCS prefactor. Moreover, the Matsubara-axis gap tends toward

$Z$0

so retardation survives at arbitrarily small coupling and the weak-coupling limit remains non-BCS in functional form (Marsiglio, 2018). This is a useful benchmark for generalized theories: weak coupling does not by itself justify replacing a retarded kernel by an instantaneous one.

5. Nonequilibrium, quantum criticality, and pairing without quasiparticles

Generalized Eliashberg theory has also been extended beyond equilibrium and beyond the Fermi-liquid normal state. In dirty superconductors subject to microwaves, the microscopic weak-power theory is formulated in Keldysh–Usadel language and organized around

$Z$1

where $Z$2 splits into a spectral part and a kinetic part (Tikhonov et al., 2019). The spectral correction encodes direct depairing by the ac field; the kinetic correction encodes nonequilibrium quasiparticle redistribution. The resulting phase diagram contains only a finite region of superconductivity enhancement near $Z$3; at sufficiently high frequencies and low temperatures, direct depairing prevails and superconductivity is suppressed (Tikhonov et al., 2019).

In quantum-critical metals and SYK-like large-$Z$4 models, the generalized Eliashberg approach addresses pairing out of non-Fermi liquids. The common structure is again a coupled set of self-consistent equations, now for fermionic self-energy, anomalous self-energy, and bosonic polarization. In the Yukawa-SYK formulation, the large-$Z$5 saddle-point equations take the Eliashberg form

$Z$6

with the boson itself becoming critical or overdamped (Esterlis et al., 13 Jun 2025). In related analyses of critical Fermi surfaces and SYK-like lattice models, Eliashberg-type equations are asymptotically exact because the dominant diagrams are self-consistent rainbow or melonic contributions while vertex corrections are suppressed by large $Z$7, small $Z$8, or locality (Chowdhury et al., 2019). This is the sense in which the formalism extends to “pairing without quasiparticles.”

The comparative study of phonons and soft nematic fluctuations sharpens the issue of validity. For the electron-phonon problem, the Migdal parameter

$Z$9

controls the neglect of vertex corrections. For the nematic quantum critical point, there is no analogous microscopic fast-boson/slow-electron separation; nevertheless, the two-loop vertex corrections are argued to be not small parametrically but small numerically, while perturbative evaluation of the one-loop self-energy remains rigorously justified when the fermion-boson coupling is small compared to $\chi$0 (Zhang et al., 2024). The paper’s conclusion is unusually strong: Eliashberg theory for the nematic case can hold all the way to a quantum critical point, whereas in the electron-phonon case it breaks down at some distance from where the dressed Debye frequency would vanish (Zhang et al., 2024).

6. Multiband, fluctuation, and computational formulations

A further meaning of “generalized Eliashberg approach” is methodological. In the functional-integral reformulation, the standard Eliashberg saddle point is embedded in a bosonic effective action whose Gaussian fluctuations generate both Cooper-channel and density-channel propagators. The inverse fluctuation kernels take the form

$\chi$1

making explicit that density/self-energy fluctuations are as systematic as pairing fluctuations in the generalized field theory (Protter et al., 2020). This extension was used to analyze strong-coupling fluctuation diamagnetism near $\chi$2 (Protter et al., 2020).

A different reformulation maps nonlinear Migdal–Eliashberg theory onto a classical spin chain in Matsubara-frequency space,

$\chi$3

The mapping yields both a heat-bath Monte Carlo solver for the full nonlinear equations and a high-frequency decimation procedure that reproduces the Morel–Anderson pseudopotential reduction,

$\chi$4

for the Bogoliuov–Tomachov–Morel–Anderson kernel (Chou et al., 2023). Here the generalization lies in turning Eliashberg theory into a classical statistical problem whose renormalization can be carried out directly in Matsubara space.

Material-specific implementations show the same breadth. In $\chi$5, a four-band strong-coupling Eliashberg model was reduced to an effective two-band description to explain superconductivity-induced infrared anomalies. The theory accounts for two gap scales, $\chi$6 and $\chi$7, and attributes optical suppression up to $\chi$8 to spin-fluctuation-assisted processes in the clean limit (Charnukha et al., 2011). In FeSe, a full-bandwidth, multiband, anisotropic Eliashberg theory based on RPA spin and charge fluctuations yields $\chi$9 K and an $$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$0 gap for bulk FeSe, but only a $$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$1-wave solution with $$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$2 K for monolayer FeSe/SrTiO$$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$3, thereby favoring interfacial electron-phonon coupling as the dominant pairing mediator in the monolayer system (Schrodi et al., 2020).

Across these formulations, the invariant core is the self-consistent treatment of frequency-dependent normal and anomalous sectors. The main unresolved issues are equally consistent across the literature: the treatment of the diagonal $$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$4 channel, Coulomb vertex corrections beyond RPA/$$i\omega_j(1-Z(i\omega_j))=\frac{1}{2}\left[\Sigma(i\omega_j)-\Sigma(-i\omega_j)\right],$$5, high-energy incoherent sectors, and the loss of a universal Migdal control parameter outside ordinary phonon problems (Akashi, 2021, Pellegrini et al., 2023, Schrodi et al., 2020). The generalized Eliashberg approach is therefore best understood not as a single extension, but as a broad research program for preserving Eliashberg’s dynamical self-consistency while reworking its assumptions to fit strongly renormalized, multiscale, or nonequilibrium superconducting systems.