Draft:Eliashberg theory

Eliashberg theory, also known as Nambu–Migdal–Eliashberg theory, is a theoretical framework in condensed matter physics that describes superconductivity beyond the approximations of the BCS theory. Developed in the 1960s by Soviet physicist Gerasim M. Eliashberg, the theory allows for a quantitative description of superconductors with strong electron–phonon coupling, including corrections due to the finite timescale (retardation) of phonon-mediated interactions. Eliashberg theory has become the standard for describing conventional superconductors, especially when experimental precision is required.

Eliashberg theory builds on Migdal's theorem, formulated by Arkady Migdal, which demonstrates that vertex corrections to electron–phonon interactions can typically be neglected because the characteristic phonon energies are much smaller than electronic energies. Eliashberg's major contribution was to derive a set of self-consistent equations for the superconducting gap and electronic self-energy that fully incorporate the dynamical nature of the electron–phonon interaction.

In addition to phonon-mediated attraction, Eliashberg theory includes the repulsive Coulomb interaction between electrons, which tends to suppress superconductivity. This repulsion is captured through the Coulomb pseudopotential $\mu ^{*}$, an effective parameter that accounts for the renormalized electron–electron interaction after high-energy processes have been integrated out.

Eliashberg theory successfully predicts the tunneling spectrum observed in superconductors. Unlike BCS theory, which predicts a symmetric and sharp energy gap in the tunneling spectrum, Eliashberg theory captures features that reflect the density of states. of the phonons.

History

The Bardeen–Cooper–Schrieffer (BCS) theory, formulated in 1957, successfully described superconductivity as the formation of Cooper pairs due to an effective phonon-mediated attraction between the electrons. Although the BCS theory provided a qualitative understanding of superconductivity, it relied on a highly simplified model of the phonon-mediated interaction.

In 1958, Lev Gor'kov reformulated BCS theory in the language of Green's functions, facilitating its extension to diagrammatic methods. He showed that Ginzburg–Landau theory, a phenomenological theory which predated the BCS theory, could be derived near the critical temperature from his reformulated BCS theory, both for clean superconductors and for superconducting alloys. In 1958, Arkady Migdal introduced the result now known as Migdal's theorem, which showed that vertex corrections to the electron–phonon interaction can be neglected when the typical phonon energy is much smaller than the electronic energy scale. The difference between the two energy scales is linked to the mass difference between the light electrons and the heavy ions. Due to their lightness, the electrons are quick to respond. This result provided a theoretical basis for treating electron–phonon interactions in a perturbative way.

In 1960, Gerasim M. Eliashberg built upon these developments to formulate what became known as Eliashberg theory. He introduced a self-consistent set of equations that account for the frequency dependence of both the electron self-energy and the superconducting gap function, enabling a quantitative description of superconductivity in the strong-coupling regime. Independently, Yoichiro Nambu developed a gauge-invariant formalism for superconductivity using what are now called Nambu spinors, providing a complementary foundation for the theory..

This framework was extended in 1962 by Pierre Morel and Philip W. Anderson, who incorporated the effects of Coulomb repulsion between electrons. They introduced the concept of the Coulomb pseudopotential, which accounts for the dynamic screening of the repulsive interaction due to retardation effects. Within the BCS theory, a similar description of the Coulomb interaction had previously been given by ...?

The electron-phonon interaction had been considered in the context of superconductivity since the early 1950s and lies in the heart of the BCS theory, albeit in an approximate way. Russian physicist Arkady Migdal's was the first to successfully apply the diagrammatic technique to this problem. According to the Migdal's theorem most of the diagrams can be neglected and the theory should be accurate to the order of (m/M)^1/2, where m is the mass of an electron, and M is the mass of the ion.

Coulomb pseudopotential

In Eliashberg theory, the superconducting state is described by a set of coupled equations for three functions that depend on the fermionic Matsubara frequencies

• Z(iωₙ): the mass renormalization function, describing how electron motion is slowed by interactions.
• φ(iωₙ): the anomalous self-energy, related to the formation of Cooper pairs.
• χ(iωₙ): a correction to the band structure energy, often negligible under particle-hole symmetry.

These functions appear in the self-energy matrix of the Nambu-Gor'kov Green function. The full isotropic Eliashberg equations on the imaginary axis are:

Z(iωₙ) = 1 + (πT / ωₙ) ∑ₘ λ(iωₙ - iωₘ) ωₘ / √(ωₘ² + Δ²(iωₘ))

Z(iωₙ)Δ(iωₙ) = πT ∑ₘ [λ(iωₙ - iωₘ) - μ* θ(ω_c - |ωₘ|)] Δ(iωₘ) / √(ωₘ² + Δ²(iωₘ))

Here:

• Δ(iωₙ) = φ(iωₙ) / Z(iωₙ) is the superconducting gap function.
• λ(iωₙ - iωₘ) is the electron–phonon pairing kernel, derived from the Eliashberg spectral function α²F(ω).
• μ* is the Coulomb pseudopotential, describing screened electron–electron repulsion.
• θ is the Heaviside function, and ω_c is a high-energy cutoff.

The function χ(iωₙ) is often omitted in practice when particle-hole symmetry is assumed and the density of states is approximately constant near the Fermi surface.

The spectral function α²F(ω) encapsulates the strength and frequency distribution of the electron–phonon interaction. It enters the pairing kernel via the relation:

λ(iωₙ - iωₘ) = ∫₀^∞ dω [2ω α²F(ω)] / [(ωₙ - ωₘ)² + ω²]

This frequency dependence is what distinguishes Eliashberg theory from the simpler BCS model.

Mass renormalization and the Z function

The function Z(iωₙ) describes how much the quasiparticle energies are renormalized by interactions. It modifies the dispersion of electronic excitations and reflects the influence of the lattice on electron motion. At low frequencies, Z is approximately:

Z(0) ≈ 1 + λ

where λ is the dimensionless electron–phonon coupling constant:

λ = 2 ∫₀^∞ [α²F(ω) / ω] dω

A larger value of Z corresponds to stronger mass enhancement and indicates that the effective electron mass is increased due to dressing by phonons.

Role of χ(iωₙ)

The function χ(iωₙ) describes a shift in the electronic band structure due to interactions. In most cases where the electron band structure is symmetric about the Fermi level, χ can be set to zero. However, it plays a role in systems with strong asymmetry or where accurate modeling of normal-state properties is required.

Coulomb pseudopotential

In addition to the attractive interaction mediated by phonons, electrons also repel each other via the Coulomb interaction. Eliashberg theory incorporates this repulsion through a dynamically screened effective interaction called the Coulomb pseudopotential, μ*. The typical form is:

μ* = μ / [1 + μ ln(E_F / ω_D)]

where:

• μ is the bare Coulomb parameter,
• E_F is the Fermi energy,
• ω_D is a characteristic phonon frequency.

The logarithmic denominator reflects the retardation effect: phonons act on a slower timescale than electrons, which reduces the effective repulsion. This screening allows superconductivity to emerge even in the presence of electron–electron repulsion.

Predictions and experimental comparison

Eliashberg theory has been successfully applied to predict and explain a wide range of experimental properties of conventional superconductors, particularly in the strong-coupling regime. One of its most powerful features is the ability to compute the tunneling density of states and compare it to spectroscopic measurements.

In particular, the theory predicts the tunneling conductance spectrum of a superconductor–insulator–normal-metal (SIN) junction as:

$\left({\frac {dI}{dV}}\right)_{S}{\Big /}\left({\frac {dI}{dV}}\right)_{N}\propto \Re \left\{{\frac {|V|}{\sqrt {V^{2}-\Delta ^{2}(V)}}}\right\}$

where Δ(V) is the complex-valued gap function analytically continued to the real axis. This expression generalizes the BCS result by allowing for an energy-dependent gap and mass renormalization.

Eliashberg theory also predicts deviations from BCS "universal" ratios in strong-coupling materials. For example, the BCS value of the zero-temperature energy gap is:

2Δ₀ / k_B T_c = 3.53 (BCS)

In strongly coupled superconductors, this ratio can be significantly larger. Other predicted quantities include:

• The temperature dependence of the gap function,
• Specific heat jump at T_c,
• Critical magnetic fields (H_c and H_c2),
• London penetration depth and optical conductivity.

Because the electron–phonon spectral function α²F(ω) can be extracted from tunneling or optical data, Eliashberg theory also enables a kind of “inversion”: experimental spectra can be used to reconstruct the microscopic interaction that drives superconductivity.

Extensions

Although Eliashberg theory was originally formulated for phonon-mediated pairing, the formalism can be extended to other bosonic interactions, such as spin fluctuations. In this case, the phonon propagator is replaced by the spin susceptibility. However, these generalizations lack a counterpart to Migdal's theorem, and therefore may not be controlled approximations in the strong-coupling regime.

Some theories of unconventional superconductivity, such as the resonating valence bond theory, posit that even the normal state is not a Fermi liquid. In such cases, Eliashberg theory is not applicable.

Despite these limitations, Eliashberg theory remains the most accurate and widely used framework for describing conventional superconductors. It continues to serve as a benchmark for first-principles predictions and for testing extensions to more exotic systems.

Modern first-principles approaches

A key limitation of traditional Eliashberg theory is the treatment of Coulomb repulsion using the Coulomb pseudopotential μ∗, which is introduced phenomenologically and typically treated as a parameter (e.g. μ∗≈0.1). While this approach works well for fitting experimental data, it is not predictive and cannot be derived directly from the microscopic Hamiltonian.

To overcome this, modern ab initio approaches have been developed that aim to compute superconducting properties without relying on adjustable parameters like μ∗. These methods typically follow one of two main strategies:

1. Ab initio Eliashberg-type methods In these approaches, the screened Coulomb interaction is calculated explicitly, often using the constrained random phase approximation (cRPA) or GW methods. The electron–phonon coupling and Coulomb terms are treated on equal footing, and the pairing kernel is constructed entirely from first principles. This allows for direct computation of critical temperature Tc​, gap functions, and tunneling spectra without empirical inputs. However, these methods are numerically demanding and require detailed knowledge of screening and frequency-dependent interactions.
2. Superconducting Density Functional Theory (SCDFT) SCDFT is an alternative, parameter-free approach that reformulates superconductivity within the framework of density functional theory. It introduces an exchange-correlation functional that depends on both the normal and anomalous (pair) densities. The pairing interaction in SCDFT includes both the electron–phonon coupling and the screened Coulomb interaction, and can be evaluated using linear-response and perturbation theory techniques. SCDFT avoids the need for a μ∗ parameter by including the Coulomb interaction explicitly in the exchange-correlation kernel. Moreover, the theory is computationally efficient and can be used to study real materials at reasonable cost. It has been successfully applied to elemental superconductors, hydrides under pressure, and other systems where the standard Eliashberg approach becomes unwieldy.

While SCDFT lacks the full dynamical structure of Eliashberg theory, it offers a practical alternative for materials discovery and large-scale computations. Conversely, advanced Eliashberg calculations with ab initio input remain the method of choice when detailed spectral functions or frequency-resolved observables are required.

References

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