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.
Abstract. We explain why the conventional argument for deriving the time-dependent Born-Oppenheimer approximation is incomplete and review recent mathematical results, which clarify the situation and at the same time provide a systematic scheme for higher order corrections. We also present a new elementary derivation of the correct second-order time-dependent Born-Oppenheimer approximation and discuss as applications the dynamics near a conical intersection of potential surfaces and reactive scattering.Mathematics Subject Classification. 81Q05, 81Q15, 81Q70.
We investigate the localization properties of independent electrons in a periodic background, possibly including a periodic magnetic field, as e. g. in Chern insulators and in Quantum Hall systems. Since, generically, the spectrum of the Hamiltonian is absolutely continuous, localization is characterized by the decay, as |x| → ∞, of the composite (magnetic) Wannier functions associated to the Bloch bands below the Fermi energy, which is supposed to be in a spectral gap. We prove the validity of a localization dichotomy, in the following sense: either there exist exponentially localized composite Wannier functions, and correspondingly the system is in a trivial topological phase with vanishing Hall conductivity, or the decay of any composite Wannier function is such that the expectation value of the squared position operator, or equivalently of the Marzari-Vanderbilt localization functional, is +∞. In the latter case, the Bloch bundle is topologically non-trivial, and one expects a non-zero Hall conductivity.(2) Throughout this Section, we use Hartree atomic units, and moreover we reabsorb the reciprocal of the speed of light 1/c in the definition of the function A Γ .
Abstract. We consider a real periodic Schrödinger operator and a physically relevant family of m ≥ 1 Bloch bands, separated by a gap from the rest of the spectrum, and we investigate the localization properties of the corresponding composite Wannier functions. To this aim, we show that in dimension d ≤ 3 there exists a global frame consisting of smooth quasi-Bloch functions which are both periodic and time-reversal symmetric. Aiming to applications in computational physics, we provide a constructive algorithm to obtain such a Bloch frame. The construction yields the existence of a basis of composite Wannier functions which are real-valued and almost-exponentially localized.The proof of the main result exploits only the fundamental symmetries of the projector on the relevant bands, allowing applications, beyond the model specified above, to a broad range of gapped periodic quantum systems with a time-reversal symmetry of bosonic type.
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