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Geometric phases are ubiquitous in chemistry and physics. They can arise when the Hamiltonian of a system depends on a set of parameters. A prototypical example is the electronic Hamiltonian of a molecule in Born-Oppenheimer (BO)approximation, which depends parametrically on the nuclear coordinates. In this case, the geometric phase is obtained as the loop integral over a vector potential, the so-called Berry connection, with the loop enclosing a conical intersection of BO surfaces. The equations of motion of the exact factorization also contain a Berry-connection-like vector potential which gives rise to another geometric phase. How does this exact geometric phase relate to the BO one? We shall discuss a case where the exact Berry phase vanishes although there is a non-trivial Berry phase for the same system in BO approximation [1],while in another example the values of the two phases are similar [2]. In all cases, the BO phase is a topological phase (i.e. it is quantized, with valueseither 0 or π), while the exact phase is geometric (i.e.non-quantized, with values in the interval [0, π]). As a second topic, the ab-initio description of phonon-mediated superconductivity will be discussed.Highly accurate results for the critical temperature will be presented [3] and the relation of the superconducting order parameter in real space to chemical bonding will be highlighted [4]. First steps towards a realistic description of the laser-driven real-time dynamics of superconductors will be discussed.
[1] S.K. Min, A. Abedi, K.S. Kim,E.K.U. Gross, Phys.Rev.Lett. 113, 263004 (2014).
[2] R. Requist, F. Tandetzky, E.K.U. Gross, Phys. Rev. A 93, 042108(2016).
[3] A Sanna, C Pellegrini, EKU Gross, Phys.Rev. Lett. 125, 057001 (2020).
[4] A. Linscheid, A. Sanna, A. Floris, E.K.U. Gross, Phys. Rev.Lett. 115, 097002 (2015).