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In his book Condensed Matter in a Nutshell, the late Gerald D. Mahan wrote that The calculation of a crystal's superconducting properties is not yet possible (p. 5). The book was published in 2010.

However, in 2013, Margine and Giustino published a paper titled Anisotropic Migdal-Eliashberg theory using Wannier functions where they provided a way to calculate superconducting properties such as superconducting gap and critical temperature. Also, in 2023, there was a virtual school on many-body calculations using EPW and BerkleyGW where they had session on calculating superconducting properties. Moreover, google search reveals many such papers on calculating critical temperature using first-principle such as this one (2016) or this one with Roeser–Huber Formalism (2022).

Then, in light of these new developments, can we say Mahan's comment is obsolete and nowadays we can indeed study superconductors using first-principle?

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    $\begingroup$ +1 but how accurate are those Tc predictions? $\endgroup$ Commented Dec 21, 2023 at 14:13
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    $\begingroup$ Is the question about conventional, phonon-mediated superconductivity or also unconventional SC? While the latter has alluded simple solutions as of now, conventional SC has had some development in recent years as you have pointed out (additional sources: here and this paper, as well as the special edition it appeared in) $\endgroup$
    – manju9
    Commented Dec 21, 2023 at 14:14
  • $\begingroup$ @NikeDattani I am not sure about that and couldn't find any study on this but the preliminary results I am seeing from the above-mentioned virtual school tutorial, it seems promising. They found Tc to be 6.0K for FCC Pb where the experimental result is 7.2K. $\endgroup$ Commented Dec 21, 2023 at 14:32
  • $\begingroup$ @manju9 that's a good point but I didn't consider it while asking the question. I'll leave the question as it is so that people can answer for both cases. NikeDattani, forgot to mention, for MgB2, they found 50K in contrast to the experimental 39K. $\endgroup$ Commented Dec 21, 2023 at 14:43
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    $\begingroup$ Ilya Esterlis has some recent interesting papers on when Migdal-Elishberg theory is valid, and when it breaks down. $\endgroup$
    – Anyon
    Commented Dec 21, 2023 at 17:36

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