[CvGmt News] seminario Pisante 12/11

Daniele Castorina castorin at mat.uniroma2.it
Thu Nov 7 11:09:31 CET 2013


Cari colleghi,

vi segnalo il prossimo seminario di ED a Roma Tor Vergata:

*Martedi' 12 Novembre 2013, h 14:15, Aula Dal Passo*

*Adriano Pisante (Sapienza Università di Roma) *

*"Wannier functions, Bloch bundles and Marzari-Vanderbilt functional" 
(joint with Gianluca Panati)

**Many properties of electrons in crystalline solids are described by 
Schroedinger operators with potentials whose periodicity reflects 
the periodic structure of the crystal. The construction of generalized 
eigenfunctions with exponential decay  is crucial, both from a 
theoretical and from a numerical viewpoint. Through the Bloch-Floquet 
transform a band structure for the spectrum of the Hamiltonian naturally 
appears (Bloch bands) and in presence of a spectral gap, typical of 
insulators and semiconductors, to the relevant family of bands there is 
an associated Bloch bundle, whose rank is the number of relevant bands.*

*A natural basis for simple or multiband systems is given by composite 
Wannier functions, whose existence together with their exponential 
localization can be obtained using geometric obstruction theory whenever 
the Hamiltonian satisfies  the time reversal symmetry (which ensures the 
triviality of the associated Bloch bundle and the existence of a smooth 
trivializing frame). This result largely extends partial results in case 
of a simple band or in one dimension. *

*In the talk we describe how to construct exponentially localized 
composite Wannier functions by variational methods, minimizing the 
localization functional introduced by Marzari and Vanderbilt. This 
functional can be transformed in one of calculus of variations type for 
maps from a torus to the unitary group. To study both existence and 
regularity of minimizers (and the corresponding localization property 
for the Wannier functions), it is very useful to exploit the connection 
with the theory of harmonic maps between Riemannian manifolds and the 
regularity theory for the corresponding elliptic systems. Here we aim to 
show existence and exponential localization for maximally localized 
Wannier functions (minimizers), in presence of the time reversal symmetry.*


-- 
Daniele Castorina
Dipartimento di Matematica - Studio 1221
Università di Roma "Tor Vergata"
Via della Ricerca Scientifica 00133 Roma
email: castorin at mat.uniroma2.it
tel: +390672594653

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