Searching for high temperature superconductivity: From Mendeleev to Seiberg-Witten via Madelung and beyond

Abstract

Recently, a noticeable progress was achieved in the area of high temperature superconductors. The maximum temperature of 250 K for LaH(10) was reported at megabar pressures. Experimentally, the existing trend -- the highest critical temperature -- is based on uses of hydrides of atoms most of which are Madelung exceptional. In this work we provide a theoretical background describing such atoms. The, thus far empirical, Madelung rule is controlling the Mendeleev periodicity. Although the majority of elements do obey the Madelung rule, there are exceptions. Thus, it is of interest to derive this rule and its exceptions theoretically and to relate the results to superconductors. This task is achieved by employing the extended Bertrand theorem. The standard Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials: Kepler, Coulomb and harmonic oscillator. Volker Perlick extended this theorem by designing new static spherically symmetric (Bertrand) spacetimes. In this work we use these spacetimes to solve the quantum many-body problem for any atom of periodic system exactly. The obtained solution is not universal though since the Madelung rule has exceptions. Nevertheless, it is demonstrated that the newly obtained results are capable of describing analytically the exceptions as well as the rule. This is achieved by some uses of the Seiberg-Witten and Hodge-de Rham theories. With their help we describe the topological transition: from the standard Madelung atoms to the Madelung anomalous atoms. As a byproduct, the second quantized many-body theory is reformulated in the gauge-theoretic form.