IsoME v2.0.0 released

Eliashberg equations solved directly on the real-frequency axis

• Christoph Heil

Image by Alejandro Simon

We are happy to announce the release of IsoME v2.0.0, the largest update to our Julia package for high-precision Eliashberg calculations since its first release. The headline addition is a second solver, RealAxisSolver(), which solves the finite-temperature isotropic Migdal-Eliashberg equations directly on the real-frequency axis—no analytic continuation required. Get it from GitHub, or simply Pkg.add("IsoME").

Why the real axis?

Almost every quantity an experiment actually measures on a superconductor—tunneling spectra, optical conductivity, quasiparticle lifetimes—is a real-frequency quantity. The Migdal-Eliashberg equations, however, are conventionally solved on the imaginary (Matsubara) axis and then analytically continued. That continuation is an ill-conditioned inverse problem: it amplifies numerical noise, smears out spectral structure, and becomes increasingly unstable at low temperatures—exactly where the physics is most interesting.

RealAxisSolver() removes that step. It returns the complex gap function Δ(ω) and renormalization function Z(ω) on a real-frequency grid, retaining the fine spectral structure inherited from α²F(ω) that analytic continuation irretrievably loses, and it remains numerically stable from millikelvin temperatures up to above Tc.

Crucially, the real-axis solver keeps the full energy dependence of the electronic density of states—something most real-axis implementations give up in favour of a constant density of states. All three approximation levels familiar from the Matsubara solver are available on the real axis:

  • cDOS+μ — constant density of states with the Morel-Anderson pseudopotential
  • vDOS+μ — full-bandwidth variable density of states with the Morel-Anderson pseudopotential
  • vDOS+W — full-bandwidth with the ab initio screened Coulomb interaction W(ε,ε′)

Keeping the full bandwidth matters. For H3S, whose van Hove singularity at the Fermi level makes the electronic structure strongly particle-hole asymmetric, the full-bandwidth real-axis solution gives a zero-temperature gap of 2Δ ≈ 60 meV, close to recent tunneling measurements, while the constant density of states approximation overshoots at 75 meV.

Fast enough to use in a loop

A direct real-axis solution has historically been considered expensive. The key algorithmic ingredient in IsoME v2.0.0 is a reformulation of the integral kernel K(ω,ω′) that reduces the cost from the conventional O(N²) to O(N). High-resolution solutions therefore converge in milliseconds in the cDOS case and in minutes with the full variable density of states—on a standard laptop.

That speed is what turns the real-axis solver from a curiosity into a building block. Updating the full Migdal-Eliashberg solution at every time step of a kinetic-equation simulation would be hopeless with imaginary-axis methods and analytic continuation; with a linear-scaling real-axis solver it becomes routine. This is precisely what underpins our recent work on the non-equilibrium response of driven superconductors: the ab initio framework for ultrafast dynamics and light-induced superconductivity, including our account of photo-induced superconductivity in K3C60, and the modeling of single-photon detection in superconducting nanowire detectors and qubits carried out with the group of Prof. Karl Berggren at MIT.

The methodology behind the solver is described in detail in arXiv:2603.18199.

Using it

Both solvers share the same arguments interface, so switching between them is a one-line change:

using IsoME

inp = arguments(
  a2f_file = "path-to-your-α2F-data",
  outdir   = "path-to-output-directory"
)

EliashbergSolver(inp)   # Matsubara axis: fast and robust, ideal for Tc searches
RealAxisSolver(inp)     # real axis: spectral quantities without analytic continuation

The Matsubara solver is unchanged in spirit and remains the right tool for critical-temperature searches and high-throughput screening, where its automatic Tc search mode does the work for you. Reach for the real-axis solver when you need real-frequency self-energy components—and be prepared to treat grids and cutoffs with a little more care. Runnable examples for both are included in test/Nb/examples.jl.

A note for existing users: v2.0.0 is a major version and contains breaking changes. If you are upgrading from the 1.x series, please have a look at the documentation before rerunning old input scripts.

Get the code

If you use IsoME, please cite the package paper, IsoME: Streamlining High-Precision Eliashberg Calculations, Comput. Phys. Commun. 315, 109720 (2025), and, when using the real-axis solver, arXiv:2603.18199. BibTeX entries for both are in CITATION.bib.

As always, we would be glad to hear how it works for you. Bug reports, feature requests, and contributions are very welcome on the issue tracker.

Happy computing!

Christoph Heil