Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

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Berry phase effects on electronic properties

https://cfm.ehu.es/ivo/publications/mostofi-cpc08.pdf

wannier90: A tool for obtaining maximally-localised Wannier functions ✩

The electronic ground state of a periodic system is usually described in terms of extended Bloch orbitals, but an alternative representation in terms of localized "Wannier functions" was introduced by Gregory Wannier in 1937. The connection between the Bloch and Wannier representations is realized by families of transformations in a continuous space of unitary matrices, carrying a large degree of arbitrariness. Since 1997, methods have been developed that allow one to iteratively transform the extended Bloch orbitals of a first-principles calculation into a unique set of maximally localized Wannier functions, accomplishing the solid-state equivalent of constructing localized molecular orbitals, or "Boys orbitals" as previously known from the chemistry literature. These developments are reviewed here, and a survey of the applications of these methods is presented. This latter includes a description of their use in analyzing the nature of chemical bonding, or as a local probe of phenomena related to electric polarization and orbital magnetization. Wannier interpolation schemes are also reviewed, by which quantities computed on a coarse reciprocal-space mesh can be used to interpolate onto much finer meshes at low cost, and applications in which Wannier functions are used as efficient basis functions are discussed. Finally the construction and use of Wannier functions outside the context of electronic-structure theory is presented, for cases that include phonon excitations, photonic crystals, and cold-atom optical lattices.

Maximally-localized Wannier Functions: Theory and Applications

Real and Complex Analysis

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a or a topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via 'dimensional reduction' by compactifying one or more spatial dimensions (in 'Kaluza–Klein'-like fashion). For -topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The -topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent -topological insulators in the same class, from which they inherit their topological properties. The eightfold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle–hole symmetries) is a reflection of the eightfold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). Furthermore, we derive for general spatial dimensions a relation between the topological invariant that characterizes topological insulators and superconductors with chiral symmetry (i.e., the winding number) and the Chern–Simons invariant. For lower-dimensional cases, this formula relates the winding number to the electric polarization (d=1 spatial dimensions) or to the magnetoelectric polarizability (d=3 spatial dimensions). Finally, we also discuss topological field theories describing the spacetime theory of linear responses in topological insulators (superconductors) and study how the presence of inversion symmetry modifies the classification of topological insulators (superconductors).

https://iopscience.iop.org/article/10.1088/1367-2630/12/6/065010/pdf

Topological insulators and superconductors: Tenfold way and dimensional hierarchy

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

Exponential localization of Wannier functions in insulators.

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, φ(ɛx), and vector potential A(ɛx), with xℝ
 d
 and ɛ≪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 ɛ. 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.

Effective dynamics for bloch electrons: Peierls substitution and beyond

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.

/pdf/triviality-of-bloch-and-bloch-dirac-bundles-4wehaqdx6q.pdf

Triviality of Bloch and Bloch–Dirac Bundles

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\phi(\epsi x)$, and vector potential $A(\epsi x)$, with $x \in \R^d$ and $\epsi \ll 1$. 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 $\epsi$. 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.

https://arxiv.org/pdf/math-ph/0212041.pdf

Effective dynamics for Bloch electrons: Peierls substitution and beyond

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.

Gianluca Panati

Papers

Exponential localization of Wannier functions in insulators.

Effective dynamics for bloch electrons: Peierls substitution and beyond

Triviality of Bloch and Bloch–Dirac Bundles

Effective dynamics for Bloch electrons: Peierls substitution and beyond

Triviality of Bloch and Bloch-Dirac bundles