The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, φ(εx), and vector potential A(εx), with x ∈ R d and ε ≪ 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L 2 (R d ) and an effective Hamiltonian governing the evolution inside this subspace to all orders in ε. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.

In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer in any dimension d ≤ 3, thus generalizing a previous result by G. Nenciu. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.

We study approximate solutions to the time-dependent S c hr odinger equation i"@ t t (x)=@t = H(x ;i"r x ) t (x) with the Hamiltonian given as the Weyl quantization of the symbol H(q p ) taking values in the space of bounded operators on the Hilbert space H f of fast \internal" degrees of freedom. By assumption H(q p ) has an isolated energy band. Using a method of Nenciu and Sordoni NeSo] we p r o ve that interband transitions are suppressed to any order in ". As a consequence, associated to that energy band there exists a subspace of L 2 (R d H f ) almost invariant under the unitary time evolution. We d e v elop a systematic perturbation scheme for the computation of e ective Hamiltonians which g o vern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the timeadiabatic theory.

We consider a periodic Schrödinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced in Marzari and Vanderbilt (Phys Rev B 56:12847-12865, 1997) and we prove some results about the existence and exponential localization of its minimizers, in dimension d le; 3. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds. © 2013 Springer-Verlag Berlin Heidelberg