Eliashberg Equations and Retarded Pairing

Eliashberg theory retains the frequency dependence of the normal and anomalous electron self-energies generated by a retarded bosonic interaction. In its conventional isotropic electron–phonon form, the coupled functions $Z_n$ and $\Delta_n$ encode mass renormalization and pairing on Matsubara frequencies. The approximation is controlled only when the vertex hierarchy established by Migdal’s theorem is valid; self-consistency and agreement with one spectrum do not prove the pairing mechanism.

Required background. Migdal validity supplies the vertex-control condition. Nambu–Gor’kov propagators supplies the matrix self-energy.

Helpful background. Matsubara frequencies supplies thermal sums and continuation conventions.

Isotropic imaginary-axis equations

For a particle–hole-symmetric, isotropic, single-band system, write

$$\mathcal G^{-1}(\mathbf k,i\omega_n) =i\omega_n Z_n\tau_0-\xi_{\mathbf k}\tau_3-\phi_n\tau_1, \qquad \Delta_n=\frac{\phi_n}{Z_n}.$$

With fermionic $\omega_n=(2n+1)\pi T$, the standard equations are

$$Z_n=1+\frac{\pi T}{\omega_n}\sum_m \lambda_{n-m} \frac{\omega_m}{\sqrt{\omega_m^2+\Delta_m^2}},$$ $$Z_n\Delta_n=\pi T\sum_m \left[\lambda_{n-m}-\mu^\ast\Theta(\omega_c-\lvert\omega_m\rvert)\right] \frac{\Delta_m}{\sqrt{\omega_m^2+\Delta_m^2}}.$$

The spectral kernel is

$$\lambda_{n-m}=2\int_0^\infty\mathrm d\Omega\, \frac{\Omega\,\alpha^2F(\Omega)} {\Omega^2+(\omega_n-\omega_m)^2}, \qquad \lambda=2\int_0^\infty\frac{\mathrm d\Omega}{\Omega}\alpha^2F(\Omega).$$

These definitions fix the normalization: $Z(0)\simeq1+\lambda$ in the normal low-temperature limit. An implementation should reproduce that check before solving the gap equation.

The Coulomb pseudopotential $\mu^\ast$ summarizes retardation of a short-range repulsion over a specified electronic-to-bosonic scale separation, often schematically

$$\mu^\ast=\frac{\mu}{1+\mu\log(E_{\mathrm{el}}/\omega_{\mathrm{ph}})}.$$

It is not a universal fitted constant: the electronic scale, screening model, double counting, and anisotropy determine whether this reduction is meaningful.

Linearized transition and an Einstein check

At $T_c$, linearize in $\Delta_m$. For an Einstein spectrum $\alpha^2F(\Omega)=(\lambda\Omega_E/2)\delta(\Omega-\Omega_E)$,

$$\lambda_{n-m}=\lambda\frac{\Omega_E^2}{\Omega_E^2+(\omega_n-\omega_m)^2}.$$

In the weak-coupling limit with $Z\simeq1$ and negligible $\mu^\ast$, the kernel acts approximately as a cutoff at $\Omega_E$, recovering $T_c\sim1.13\Omega_Ee^{-1/\lambda}$ and the BCS gap ratio. At stronger coupling, $Z>1$, the ratio $2\Delta_0/T_c$ and thermodynamic quantities change. Numerical convergence requires increasing the Matsubara cutoff and number of frequencies together, checking the $1/\omega_n$ tails and the invariance of $T_c$.

Eliashberg 1960 develops the strong-coupling equations; Allen and Mitrović 1983, §§2–5 provides their conventional normalization and observable consequences.

Continuation and inference ceiling

Real-frequency tunnelling or optical data require analytic continuation of $Z(i\omega_n)$ and $\phi(i\omega_n)$. Direct real-axis equations, controlled rational representations, or statistically regularized continuation must reproduce spectral positivity and high-frequency moments where applicable. Different $\alpha^2F$ can produce similar broadened spectra, so inversion needs covariance, resolution, backgrounds, and sensitivity to $\mu^\ast$. Marsiglio 2020, §§2–4 reviews the imaginary- and real-axis formulations and their numerical interfaces.

An extracted bosonic feature can show that a retarded kernel is compatible with the data. Establishing that it is the dominant pairing cause requires independent isotope, momentum, pressure, impurity, and competing-order tests. For unconventional kernels, vertex structure and sign-changing gaps lie beyond the isotropic phonon equations written here.

The chapter validity map keeps the self-consistent solution downstream of its vertex and inversion assumptions.

Eliashberg self-consistency computes frequency-dependent pairing under a licensed vertex approximation. Kernel normalization, Coulomb retardation, continuation, and nonunique inversion bound the result. Original schematic, not to scale.

See the paired-matter claim test matrix for the mechanism-evidence boundary.

Exercise

Check the Einstein kernel. Evaluate $\lambda_{n-m}$ for the delta-function spectrum above and find its zero-frequency-transfer value.

Solution

Substitution gives $2(\lambda\Omega_E/2)\Omega_E/[\Omega_E^2+(\omega_n-\omega_m)^2]$, hence $\lambda_{n-m}=\lambda\Omega_E^2/[\Omega_E^2+(\omega_n-\omega_m)^2]$. At $n=m$ it equals $\lambda$, which checks both dimensions and normalization.

References

• Allen, P. B., and Mitrović, B. (1983). “Theory of superconducting $T_c$.” Solid State Physics 37, 1–92. doi:10.1016/S0081-1947(08)60665-7.
• Eliashberg, G. M. (1960). “Interactions between electrons and lattice vibrations in a superconductor.” Soviet Physics JETP 11, 696–702. JETP archive.
• Marsiglio, F. (2020). “Eliashberg theory: A short review.” Annals of Physics 417, 168102. doi:10.1016/j.aop.2020.168102.