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Wannier functions for periodic Schrödinger operators and harmonic maps into the unitary group

Friday 27th March 2015 - 11:30 to 12:30
INI Seminar Room 1
Co-author: Adriano Pisante ("La Sapienza" University of Rome)

The localization of electrons in crystalline solids is often expressed in terms of the Wannier functions, which provide an orthonormal basis of L2(Rd) canonically associated to a given periodic Schrödinger operator.

A very popular tool in theoretical and computational solid-state physics are the maximally localized Wannier functions, which are defined as the minimizers (in a suitable space of Wannier functions) of a localization functional introduced by Marzari and Vanderbilt in 1997. While early confirmed by numerical evidence, the exponential localization of such minimizers has remained an open question until recently.

In the talk, the concept of Wannier basis will be reviewed in detail, with emphasis on its geometric counterpart (Bloch frame). Then a recent result proving the existence and the exponential localization of the minimizers, under suitable assumptions, will be presented (joint work with A. Pisante). The proof exploits methods and techniques from the regularity theory of harmonic maps into the unitary group and the so-called "decomposition into unitons" of such maps.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons