Decomposing Allen–Dynes error into learnable and irreducible components

Abstract

The Allen–Dynes formula predicts the superconducting transition temperature from three moments ofthe electron–phonon spectral function α2F(ω), and underpins essentially all large-scale computationalsearches for new superconductors. We ask what that compression costs, separating error a better functionof the same three moments could absorb from error no such function can, because the information wasdestroyed when α2F was reduced to three numbers.Solving the linearised isotropic Migdal–Eliashberg equations for 806 DFPT spectral functions, wefind both components governed by the same spectral shape parameter r = ω2/ωlog, and distinguishableby construction: the learnable component is variation predictable from r, an input the formula alreadyreceives, while the irreducible component is the Tc spread among spectra sharing all three moments,which no such function can recover. The irreducible floor rises steeply with r (ρ = +0.92 over ninemoment-matched ensembles); measured at fixed r it is non-monotone in the coupling constant, with aminimum near λ = 0.9 and a maximum near 5.Relating the two requires declaring the recalibration class, the population and the µ∗ convention.Under our declarations Allen–Dynes sits a factor 11.0 above the floor, and between 3.3 and 20.6 acrossevery alternative we tabulate; correcting for the coupling coverage of the moment-matched cells brings itto 5–11, the width of that interval being set by the estimand rather than by the measurement. The orderof magnitude is robust, the leading digit is not. Matched in shape space, headroom falls from 26.7 atr = 1.06 to 2.9 at r = 1.56 (ρ = −0.833), because the floor rises with r far faster than the error does.Where the spectrum is near-Einstein a better closed form has an order of magnitude to give; where it isbroad, three moments are already close to exhausted.Throughout, the reference Tc is the Migdal–Eliashberg solution on DFT input, not the experimentalvalue: this work compares a closed-form approximation with the theory it approximates, on identicalinput.