The Localization Dichotomy in Solid State Physics

Gianluca Panati (University of Rome Sapienza)

16-Mar-2021, 14:00-15:00 (3 years ago)

Abstract: Solid state systems exhibit a subtle intertwining between the topology of the "manifold" of occupied states and the localization of electrons, if the latter is appropriately defined.
This correspondence has been first noticed and proved in the case of periodic gapped systems: for non interacting electrons, localization is measured via the localization of the Composite Wannier Bases (CWBs) spanning the range of the Fermi projector $P_F$, while the topological information is encoded in the (first) Chern class of the Fermi projector. It has been proved, for dimension $d \leq 3$ that only two regimes are possible:
i) either there exists an exponentially localized CWB, and correspondingly the Chern class of $P_F$ vanishes (ordinary insulator);
ii) or any possible choice of a CWB is delocalized, in the sense that it yields an infinite expectation value for the square of the position operator, and the Chern class is non zero (Chern insulator).
More recently, the previous dichotomy has been reformulated in such a way that it applies also to non-periodic systems, and a corresponding "Localization Dichotomy conjecture" for non-periodic systems has been stated.
In my talk, I will gently introduce the subject, starting from the periodic case, and summarize the recent attempts, by several groups, to prove the Localization Dichotomy for non-periodic systems.
The talk is based on work in collaboration with D. Monaco, M. Marcelli, M. Moscolari, A. Pisante, and S. Teufel, and on discussions with several colleagues.

mathematical physics

Audience: researchers in the discipline

( video )


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