TIME DEPENDENT DENSITY FUNCTIONAL THEORY ... - TDDFT.org

Localization and polarization in the insulating
state of matter
Raffaele Resta
INFM DEMOCRITOS National Simulation Center, Trieste, and Dipartimento di Fisica Teorica,
University of Trieste
At variance with what happens in metals, the electronic charge in insulators cannot flow freely
under an applied dc field, and undergoes instead static polarization. Both these features owe to the
different nature of the excitation spectrum, but also to a different organization of the electrons in
their ground state: electrons are localized in insulators and delocalized in metals. Such localization
feature, however, is hidden in a rather subtle way into the many-body wavefunction. There have
been recent advances in the theory of the insulating state, which in turn are deeply rooted into
the modern theory of polarization, based on a Berry phase. Localization and polarization can be
regarded as two aspects of the same phenomenon, and stem from essentially the same formalism.
The theory is very general, and applies on the same footing to either correlated wavefunctions
or independent electrons, and to either disordered or crystalline systems. Its implications for a
Kohn-Sham crystal can be stated as follows. In insulators, a set of well localized orbitals (the
Wannier functions) spans the same Hilbert space as do the Bloch orbitals of the occupied bands.
The first moment (dipole) of these orbitals defines macroscopic polarization, while their spherical
second moment (spread) can be made finite. In metals macroscopic polarization is ill-defined, while
it is impossible to span the Hilbert space of the occupied Bloch states using localized orbitals whose
spread is finite.
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