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The Localization-Topology Correspondence: periodic systems and beyond

Banff International Research Station for Mathematical Innovation and Discovery

As realized by TKNN in 1982, a relevant Transport-Topology Correspondence holds true for gapped periodic 2D systems, in the sense that a non-vanishing Hall conductivity corresponds to a non-trivial topology of the space of occupied states, decomposed with respect to the crystal momentum (the Bloch bundle). More recently, a related Localization-Topology Correspondence has been noticed and mathematically proved for 2D and 3D gapped periodic quantum system. The result states that the Bloch bundle is (Chern) trivial if and only if there exists a system of composite Wannier functions on which the expectation value of the squared position operator is finite. In other words, whenever the system is in a Chern-non-trivial phase, the composite Wannier functions are very delocalized, while in the Chern trivial phase they can be chosen exponentially localized (joint work with D. Monaco, A. Pisante and S. Teufel). During my talk, I will report on this result and the essential ideas of its proof, as well as on the ongoing attempt to generalize this correspondence to non-periodic gapped quantum systems (work in progress with G. Marcelli and M. Moscolari).

Mathematics; Quantum theory; Mathematical physics

video/mp4

Author affiliation: Sapienza Universita di Roma

2019-12-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72540

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/