Fast Real-Axis Eliashberg Calculations: Full-bandwidth solutions beyond the constant density of states approximation

Abstract

Experimentally relevant signatures of superconductivity require access to real-frequency quantities, such as the spectral functions, optical response, and transport properties, yet Migdal-Eliashberg calculations are commonly performed on the imaginary axis and then analytically continued, a step that is numerically delicate and can obscure physically relevant spectral features. Here we present a practical route to solving the finite-temperature Migdal-Eliashberg equations directly on the real-frequency axis, while retaining the effects from the full-bandwidth electronic structure. Our formulation accounts for particle-hole asymmetry through an energy-dependent electronic density of states, avoiding the constant density of states approximation often used in real-axis calculations, and includes a static screened Coulomb contribution. We introduce an efficient numerical technique to solve the Migdal-Eliashberg integrals whose computational cost scales linearly with the real-frequency grid, making high-resolution, full-bandwidth real-axis calculations feasible and providing direct access to the interacting Green's function and derived observables without analytic continuation. As an illustration, we apply the method to H$_{3}$S, where a van-Hove singularity near the Fermi level produces strong particle-hole asymmetry. The full-bandwidth solution yields noticeably different spectra than the constant density of states approximation and brings the superconducting gap and lineshapes into closer agreement with experiment, highlighting when band-structure details are essential. Furthermore, the methods presented here open the door to time-dependent, nonequilibrium simulations within Eliashberg theory.

Figures (8)

• Figure 1: The superconducting and electronic properties of H$_3$S derived from its crystal structure. (a) The Eliashberg spectral function $\alpha^2F(\omega)$ obtained from density functional perturbation theory. (b) The electronic density of states $N(\varepsilon)$ for H$_3$S computed using density functional theory. The van-Hove singularity near the Fermi level ($\varepsilon = 0$) is highlighted in red. (c) The effective screened Coulomb potential $W(\varepsilon,\varepsilon')$ as a function of electron energy $\varepsilon$ calculated from the GW approximation. These quantities are used as input for numerical solutions to the Migdal-Eliashberg equations. The corresponding superconducting gap edge (d) $\Delta(\omega)$, (e) renormalization parameter $Z(\omega)$, and (f) root-mean square deviation between iterations of $\Delta(\omega)$ (convergence) in the constant density of states and $\mu^*$ approximation at $T=1\,$mK. (g) With access to the real-axis solutions, one can compute the transport properties of a material and use it to model the response in nonequilibrium perturbations, such as in pump-probe experiments.
• Figure 2: Solutions to the Migdal-Eliashberg equations for H$_3$S at $T=1\,$mK. The top panel corresponds to solutions considering the full variable electronic density of states (vDOS) for (a) the superconducting order parameter at the Fermi level $\phi(\omega,\varepsilon_k=0)$, (b) the wavefunction renormalization parameter $Z(\omega)$, (c) the shift in the chemical potential due to superconducting correlations $\chi(\omega)$, and (f) superconducting order parameter at the gap-edge as a function of $\varepsilon$. The bottom panel corresponds to the solution to the Migdal-Eliashberg equations assuming a constant electronic density of states at the Fermi level (cDOS) for (d) $\phi(\omega)$ and (e) $Z(\omega)$. In each panel, the real part corresponds to the solid black line, and the grey dashed line indicates the imaginary part. The solid red bars indicate the location of the van-Hove singularity in H$_3$S. The vDOS solution provides results closer in agreement with experiment.
• Figure 3: A comparison between solutions of the Migdal-Eliashberg equations for (a,d) $\phi(\omega)$, (b,e) $Z(\omega)$, and (c,f) $\chi(\omega)$ at (a-c) $T=100\,$K and (d-f) $T=1\,$K obtained with the direct real-frequency axis solver shown in black and the analytic continuation done with the Padé approximation shown in red. Numerical instabilities are clearly present in the analytically continued solutions, with large unphysical gradients notably appearing in (a), (b), (d), and (e).
• Figure 4: (a) The spectral function $A(\varepsilon,\omega)$ for H$_3$S at (1) $T=1\,$K, (b) $100\,$K, and (c) $200\,$K. The dashed red line indicates the Bogoliubov dispersion relation. Bright yellow regions indicate high spectral weight, whereas dark purple regions indicate low or zero spectral weight.
• Figure 5: (a) The spectral function $A(\varepsilon=0,\omega)$ at $T=1\,$mK, $1\,$K, $30\,$K, $100\,$K, and $T=200\,$K. (b) The corresponding quasiparticle density of states $N(\omega)$ is computed from $A(\varepsilon, \omega)$. In (a) and (b), each temperature is offset on the y-axis for clarity. (c) The interacting quasiparticle occupancy $\langle n(\varepsilon) \rangle$ at $T=1\,$mK plotted alongside the BCS and non-interacting (Fermi-Dirac) distributions for comparison. The solid red bar indicates the location of the van-Hove singularity in H$_3$S.