Increased performance of Matsubara space calculations: A case study within Eliashberg theory

We present a method to considerably improve the numerical performance for solving Eliashberg-type coupled equations on the imaginary axis. Instead of the standard practice of introducing a hard numerical cutoff for treating the infinite summations involved, our scheme allows for the efficient calculation of such sums extended formally up to infinity. The method is first benchmarked with isotropic Migdal-Eliashberg theory calculations and subsequently applied to the solution of the full-bandwidth, multiband, and anisotropic equations focusing on the $\mathrm{FeSe}/{\mathrm{SrTiO}}_3$ interface as a case study. Compared to the standard procedure, we reach similarly well converged results with less than one fifth of the number of frequencies for the anisotropic case, while for the isotropic set of equations we spare approximately ninety percent of the complexity. Since our proposed approximations are very general, our numerical scheme opens the possibility of studying the superconducting properties of a wide range of materials at ultralow temperatures.

Figure 1

Convergence study with the number of Matsubara frequencies used for the self-consistent solution of isotropic Eliashberg equations. Purple colored lines correspond to solutions of the Eqs. (1) and (2), red colored curves to our method for $Z_m^{\left(\mathcal{A}\right)}$ and ${\mathrm{\Delta }}_m^{\left(\mathcal{A}\right)}$. Dashed: $\lambda =0.3, T=0.1\mathrm{K}$. Dot-dashed: $\lambda =0.5, T=10\mathrm{K}$. [(a) and (b)] Double logarithmic plot of the average errors ${\varepsilon }_Z$ and ${\varepsilon }_{\mathrm{\Delta }}$, as defined in the main text, for the mass renormalization and the gap function, respectively. The reference results, calculated for $\mathcal{M}=400000$, are considered in an energy window of 200 Matsubara frequencies, centered around the zero. Solid lines correspond to $\lambda =0.5$ and $T=0.01\mathrm{K}$. (c) Maximum gap as a function of temperature, normalized with respect to $T_c$. The inset shows the logarithmic deviation of ${\mathrm{\Delta }}_{\max }$, calculated with $\mathcal{M}=200$, with respect to the cyan solid reference curve obtained for 4 000 frequencies.

Figure 2

Thermodynamic properties obtained from the solution of the isotropic Eliashberg equations for a coupling strength $\lambda =0.3$. Beneath each panel we show the logarithmic error with respect to the reference curve. The color code and the number of Matsubara frequencies is chosen precisely as in Fig. 1. (a) Free energy difference Eq. (11) between normal and superconducting state. (b) Difference in entropy, Eq. (12). (c) Specific heat difference [Eq. (13)] that shows the characteristic jump at $T_c$.

Figure 3

Convergence study of the full-bandwidth, multiband, and anisotropic Eliashberg equations for the FeSe/STO system at $T=10\mathrm{K}$. (a) Computed ARPES spectrum at the $M$ point of the first Brillouin zone; the solid and cyan curves correspond to the analytically continued results for $Z_{\mathbf{k},m}, {\varphi }_{\mathbf{k},m}$, and ${\chi }_{\mathbf{k},m}$ for 12 000 Matsubara frequencies (both lines are similar, but shifted relative to each other for visibility purpose). Purple and dotted (red and dashed) curves: results for 100 Matsubara frequencies taking (not taking) into account the infinite tail correction. The inset shows the corresponding logarithmic error with respect to the reference line. (b) Logarithmic average error of the real-frequency dependent functions as defined in the main text, using the same color code. Dashed: analytically continued solutions for $Z_{\mathbf{k},m}^{\left(\mathcal{A}\right)}, {\chi }_{\mathbf{k},m}^{\left(\mathcal{A}\right)}$, and ${\varphi }_{\mathbf{k},m}^{\left(\mathcal{A}\right)}$ compared to the reference solutions obtained from Eqs. (15, 16, 17).

Figure 4

Self-consistently obtained superconducting gap of monolayer FeSe/STO projected on the electronic Fermi surface pockets. The gap is determined at $T=10\mathrm{K}$ and using a coupling as described in the main text. The size of the band and momentum dependent gap is depicted by the color code.