Triviality of Bloch and Bloch-Dirac bundles

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 for any d < 4, 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.Comment: 20 pages, no figure

Symmetry and localization in periodic crystals: triviality of Bloch bundles with a fermionic time-reversal symmetry

We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, called the Bloch bundle. The role of additional $\mathbb{Z}_2$-symmetries, as time-reversal and space-reflection symmetry, is discussed, showing how time-reversal symmetry implies the triviality of the Bloch bundle, both in the bosonic and in the fermionic case. Moreover, the same $\mathbb{Z}_2$-symmetry allows to define a finer notion of isomorphism and, consequently, to define new topological invariants, which agree with the indices introduced by Fu, Kane and Mele in the context of topological insulators.Comment: Contribution to the proceedings of the conference "SPT2014 - Symmetry and Perturbation Theory", Cala Gonone, Italy (2014). Keywords: Periodic Schr\"{o}dinger operators, composite Wannier functions, Bloch bundle, Bloch frames, time-reversal symmetry, space-reflection symmetry, invariants of topological insulator

Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene

We investigate the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the multilayer case. Since the decrease of the Wannier functions is characterised by the structure of the Bloch eigenspaces around the Dirac points, we introduce a geometric invariant of the family of eigenspaces, baptised eigenspace vorticity. We compare it with the pseudospin winding number. For every value $n \in Z$ of the eigenspace vorticity, we exhibit a canonical model for the local topology of the eigenspaces. With the help of these canonical models, we show that the single band Wannier function $w$ satisfies $|w(x)| \leq \mathrm{const} |x|^{- 2}$ as $|x| \rightarrow \infty$, both in monolayer and bilayer graphene.Comment: 54 pages, 4 figures. Version 2: Section 1.0 added; improved results on the decay rate of Wannier functions in graphene (Th. 4.3 and Prop. 4.6). Version 3: final version, to appear in JSP. New in V3: previous Sections 3.1 and 3.2 are now Section 2.2; Lemma 2.4 modified (previous statement was not correct); major modifications to Section 2.3; Assumption 4.1(v) on the Hamiltonian change

The topological Bloch-Floquet transform and some applications

We investigate the relation between the symmetries of a Schr\"odinger operator and the related topological quantum numbers. We show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch-Floquet transform induces a direct integral decomposition of the algebra of observables. More relevantly, we prove that the generalized transform selects uniquely the set of "continuous sections" in the direct integral decomposition, thus yielding a Hilbert bundle. The proof is constructive and provides an explicit description of the fibers. The emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere. Two running examples provide an Ariadne's thread through the paper. For the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.Comment: 34 pages, 1 figure. Key words: topological quantum numbers, spectral decomposition, Bloch-Floquet transform, Hilbert bundle. V3: a subsection has been added; V4: some proofs have been simplified; V5: final version to be published (with a new title

Space-Adiabatic Perturbation Theory

We study approximate solutions to the Schr\"odinger equation i\epsi\partial\psi_t(x)/\partial t = H(x,-i\epsi\nabla_x) \psi_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 \Hi_{\rm f} of fast ``internal'' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \cite{NS} we prove that interband transitions are suppressed to any order in \epsi. As a consequence, associated to that energy band there exists a subspace of L^2(\mathbb{R}^d,\Hi _{\rm f}) almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern 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 time-adiabatic theory.Comment: 49 page

The Haldane model and its localization dichotomy

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of |x| 2 diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane [10]. We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in [19]