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The theory of adiabatic electron transport in a correlated condensed-matter system is rooted in a seminal paper by Niu and Thouless [J. Phys. A: Math. Gen. 17, 2453 (1984)]; I adopt here an analogous logic in order to retrieve the known expression for the adiabatic electronic flux in a molecular system [L. A. Nafie, J. Chem. Phys. 79, 4950 (1983)]. Its derivation here is considerably simpler than those available in the current quantum-chemistry literature; it also explicitly identifies the adiabaticity parameter, in terms of which the adiabatic flux and the electron density are both exact to first order. It is shown that the continuity equation is conserved to the same order. For the sake of completeness, I also briefly outline the relevance of the macroscopic electronic flux to the physics of solids and liquids.

Faraday's experiment measures-within a modern view-the charge adiabatically transported over a macroscopic distance by a given nuclear species in insulating liquids: the reason why it is an integer is deeply rooted in topology. Whole numbers enter chemistry in a different form: atomic oxidation states. They are not directly measurable and are determined instead from an agreed set of rules. Insulating liquids are a remarkable exception; Faraday's experiment indeed measures the oxidation numbers of each dissociated component in the liquid phase, whose topological values are unambiguous. Ionic conductivity in insulating liquids is expressed in terms of the autocorrelation function of the fluctuating charge current at a given temperature in a zero electric field; topology plays a major role in this important observable as well. The existing literature deals with the above issues by adopting the independent-electron framework; here, I provide the many-body generalization of all the above findings, which, furthermore, allows for compact and very transparent notations and formulas.

Adiabatic transport in a many-electron system is expressed in terms of the appropriate Berry curvature, owing to the Niu-Thouless theory [J. Phys A {\bf 17}, 2453 (1984)]; the main equation is very compact and very general. I address here three paradigmatic adiabatic response tensors -- -the atomic polar tensor, the atomic axial tensor, and the rotational $g$ factor -- and I show that, for all of them, the known formulas do not need an independent proof. They are just case studies of the general expression, for different choices the curvature's two arguments., Comment: 3 1/2 pages, no figures

The Berry phase plays an outstanding role in condensed-matter physics, where it provides the main entry for defining and computing a key observable: macroscopic electrical polarization. This phase obtains as a loop integral in reciprocal space, which has the topology of a torus. I show here that in some circumstances the loop may be contracted to a point, and polarization can be computed in practice from a single Hamiltonian diagonalization.

The quantum-mechanical expression for the polarization of a crystalline solid does not bear any resemblance to the (trivial) expression for the dipole of a bounded crystallite; in fact, it has been proved via a conceptually different path. Here, I show how to alternatively define the dipole of a bounded sample in a somewhat unconventional way; from such a formula, the crystalline polarization formula-as routinely implemented in electronic-structure codes-follows almost seamlessly.

A many-electron conducting system undergoes free acceleration in response to a macroscopic field. The Drude weight $D$---also called charge stiffness---measures the adiabatic (inverse) inertia of the electrons; the $D$ formal expression requires periodic boundary conditions. When instead a bounded sample is addressed within open boundary conditions, no current flows and a constant (external) field only polarizes the sample: the Faraday cage effect. Nonetheless a low-frequency field induces forced oscillations: we show here that the low-frequency linear response of the bounded system is dominated by the adiabatic inertia and allows an alternative evaluation of $D$. Simulations on model one-dimensional systems demonstrate our main message., Comment: 8 pages, 6 figures | Minor corrections w.r.t. v2 (mostly typos) | Accepted for pubblication in Phys. Rev. B

The modern theory of orbital magnetization addresses crystalline materials at the noninteracting level: Therein the observable is the $\mathbf{k}$-space integral of a geometrical integrand. Alternatively, magnetization admits a local representation in $\mathbf{r}$ space, i.e., a ``density,'' which may address noncrystalline and/or inhomogeneous materials as well; the Chern number admits an analogous density. Here, we provide the formulation for ribbons, polymers, nanowires, and single-layer materials, where both $\mathbf{k}$-space and $\mathbf{r}$-space integrations enter the definition of the two observables.

An insulator differs from a metal because of a different organization of the electrons in their ground state. In recent years this feature has been probed by means of a geometrical property, the quantum metric tensor, which addresses the system as a whole, and is therefore limited to macroscopically homogenous samples. Here we show that an analogous approach leads to a localization marker, which can detect the metallic versus insulating character of a given sample region using as the sole ingredient the ground state electron distribution, even in the Anderson case (where the spectrum is gapless). When applied to an insulator with a nonzero Chern invariant, our marker is capable of discriminating the insulating nature of the bulk from the conducting nature of the boundary. Simulations (both model Hamiltonian and first principles) on several test cases validate our theory.

The Drude weight D (also called charge stiffness) measures the effective electron density contributing to dc conductivity; it vanishes in insulators. It is a general concept, which applies to any metal, including cases with disorder and electron-electron interaction. We provide a thorough analysis of D and of its meaning, both in the general case and in the special case of band metals, where we also show that D has a close relationship to orbital magnetization. The superconducting weight D s measures instead the superfluid density accounting for the Meissner effect. The two quantities D and D s are the main criteria to discriminate between insulators, metals, and superconductors.

The modern theory of polarization, based on a Berry phase, is currently implemented in most first-principle electronic structure codes. Many KS-DFT calculations have addressed various phenomena (ferroelectricity, piezoelectricity, lattice dynamics, infrared spectra of liquid and amorphous systems) in several materials. Notwithstanding, the KS polarization does not coincide with the exact one, even when ideally implemented with the exact KS crystalline potential. This is at odds with the fact that the KS electrical dipole of a bounded crystallite coincides by definition with the exact one: we analyze this issue from several viewpoints. According to the modern theory, the polarization of a centrosymmetric crystal does not vanish in general; we show that the polarization of a centrosymmetric quasi-1d systems (stereoregular linear polimer) is a topological invariant: ergo in this case the KS polarization coincides with the exact one. © 2018, EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature.

The geometrical intrinsic contribution to the anomalous Hall conductivity (AHC) of a metal is commonly expressed as a reciprocal-space integral: as such, it only addresses unbounded and macroscopically homogeneous samples. Here we show that the geometrical AHC has an equivalent expression as a local property. We define a "geometrical marker" which actually probes the AHC in inhomogeneous systems (e.g. heterojunctions), as well as in bounded samples. The marker may even include extrinsic contributions of geometrical nature., Comment: 4 pages, 4 figures

The orbital magnetic moment of a finite piece of matter is expressed in terms of the one-body density matrix as a simple trace. We address a macroscopic system, insulating in the bulk, and we show that its orbital moment is the sum of a bulk term and a surface term, both extensive. The latter only occurs when the transverse conductivity is nonzero and it is due to conducting surface states. Simulations on a model Hamiltonian validate our theory.

The insulating state of matter can be probed by means of a ground state geometrical marker, which is closely related to the modern theory of polarization (based on a Berry phase). In the present work we show that this marker can be applied to determine the metal-insulator transition in disordered systems. In particular, for noninteracting systems the geometrical marker can be obtained from the configurational average of the norm-squared one-body density matrix, which can be calculated within open as well as periodic boundary conditions. This is in sharp contrast to a classification based on the static conductivity, which is only sensible within periodic boundary conditions. We exemplify the method by considering a simple lattice model, known to have a metal-insulator transition as a function of the disorder strength, and demonstrate that the transition point can be obtained accurately from the one-body density matrix. The approach has a general ab initio formulation and could in principle be applied to realistic disordered materials by standard electronic structure methods., T.O. acknowledges support from the Danish Council for Independent Research, Sapere Aude Program; R.R. acknowledges support from the ONR (USA) Grant No. N00014-12-1-1041; I.S. acknowledges support from Ministerio de Economía y Competitividad (Spain) Grant No. FIS2016-77188-P, and from the European Commission Grant No. CIG-303602.

The concept of macroscopic polarization is the basic one in the electrostatics of dielectric materials: but for many years this concept has evaded even a precise microscopic definition, and has severely challenged quantum-mechanical calculations. This concept has undergone a genuine revolution in recent years (1992 onwards). It is now pretty clear that (contrary to a widespread incorrect belief) macroscopic polarization has nothing to do with the periodic charge distribution of the polarized crystal: the former is essentially a property of the phase of the electronic wavefunction, while the latter is a property of its modulus. An outline of the modern theory, based on a Berry-phase, is presented. Experiments invariably address polarization derivatives (permittivity, piezoelectricity, pyroelectricity, etc.) or polarization differences (ferroelectricity), and these differences are measured as an integrated electrical current. The modern theory addresses this same current, which is cast in terms of the phase of the electronic wavefunction. Ab initio implementations have concerned various phenomena (ferroelectricity, piezoelectricity, and lattice dynamics) in several materials.

Ferroelectric (FE) materials display very peculiar structural, electrical, and electromechanical features. Their properties have been addressed, by means of ab initio simulations, from 1990. Since then, a great number of different contributions has greatly expanded our understanding of this important class of materials; quite often, these contributions were based on novel theories, methods, and algorithms. Nowadays, many properties can be accurately computed and predicted from first principles for a given FE material. Even more important, the work of the last decade has revolutionized our qualitative view of the phenomenon of ferroelectricity at large. We review the major advances concerning structural and lattice-dynamical properties, spontaneous and induced polarization, dielectric and piezoelectric properties, and structural phase transitions at finite temperature. Special attention is devoted to illustrating the novel methods which made such advances possible.

Under the terms of the Creative Commons Attribution License 3.0 (CC-BY)., The anomalous Hall conductivity of “dirty” ferromagnetic metals is dominated by a Berry-phase contribution which is usually interpreted as an intrinsic property of the Bloch electrons in the pristine crystal. In this work we evaluate the geometric Hall current directly from the electronic ground state with disorder and then recast it as an integral over the crystalline Brillouin zone. The integrand is an effective k-space Berry curvature, obtained by unfolding the Berry curvature from the small Brillouin zone of a large supercell. Therein, disorder yields a net extrinsic Hall contribution, which we argue is related to the elusive side-jump effect. As an example, we unfold the first-principles Berry curvature of an ordered Fe3Co alloy from the original fcc-lattice Brillouin zone onto a bcc-lattice zone with four times the volume. Comparison with the virtual-crystal Berry curvature clearly reveals the symmetry-breaking effects of the substitutional Co atoms., This work was supported by Grants No. MAT2012-33720 from the Ministerio de Economía y Competitividad (Spain), No. CIG-303602 from the European Commission, and ONR Grant No. N00014-12-1-1041 (USA).

Since the appearance of Berry's seminal paper in 1984, geometric phases have been discovered in virtually all fields of physics. Here we address molecules and solids, and we limit our scope to the Berry's phases of the many-electron wavefunction. Many advances have occurred in very recent years relating to the theory of such phases and their observable consequences. After discussing the basic features of Berry's phases in a generic quantum system, we specialize to selected examples taken from molecular physics and condensed matter physics; in each of these cases, a Berry's phase of the electronic wavefunction leads to measurable effects.

The dynamical charge of a given ion measures the dipole induced in the material when this ion is displaced. In many oxides these charges are much different from the formal ionic charges: for instance, the oxygen dynamical charge can be of the order of −8 unit charges in ferroelectrics. The mechanisms responsible for these “anomalous” values have been elucidated in the past few years. I present here some most recent results where a given oxide is investigated using different, and complementary, theoretical schemes, in order to provide lower and upper bounds for the theoretical prediction. As case studies, I address the ferroelectric oxide KNbO3 and the antiferromagnetic oxide MnO.

Recently, a great deal of attention has been devoted to the near-exact Kohn-Sham (KS) atomic potentials, mainly obtained from quantum Monte Carlo (QMC) calculations for a few significant test cases. We show that the potential of the highest occupied orbital in the self-interaction corrected local-density approximation (SIC-LDA) approach provides an excellent approximation for the QMC potentials, far better than the commonly used ones. Moreover, near-exact KS potentials of the same quality as the QMC ones can be obtained from the SIC-LDA potentials of the highest occupied and lowest empty orbitals, by a simple two-parameter fitting procedure to reproduce the experimental values of the ionization and first excitation energies. This has been demonstrated by a quantum-defect analysis of the empty states of Ne and Be atoms, for which QMC potentials are available to us. The introduced method is used to construct a near-exact KS potential for Ar.

We perform first-principles calculations on the ferroelectric perovskite KNbO 3 , using both Hartree–Fock (HF) and density functional theory in the local density approximation (LDA). We compute longitudinal and transverse dynamical charges, electronic dielectric constant, and the spontaneous polarization. Longitudinal dynamical charges are computed with a supercell technique, while transverse dynamical charges are computed through the Berry phase. Combining the two types of dynamical charges, the electronic dielectric constant is determined. The spontaneous polarization is also computed as a Berry phase. Band–by–band decomposition of the dynamical charges is performed and analyzed. The differences between the HF and LDA results are pointed out and their origin is scrutinized. The results show that in this material, having an intermediate ionic/covalent character, HF underestimates the covalence mechanism, while LDA overestimates it. Both methods provide results of comparable accuracy, but on opposite sides of the experimental values.

The bonding pattern of a covalent semiconductor is disrupted when a surface is cut while keeping a rigid (truncated bulk) geometry. The covalent bonds are partly reformed (with a sizeable energy gain) when reconstruction is allowed. We show that the ``electron localization function'' (ELF)---applied within a first--principles pseudopotential framework---provides un unprecedented insight into the bonding mechanisms. In the unreconstructed surface one detects a partly metallic character, which disappears upon reconstruction. In the surface reformed bonds, the ELF sharply visualizes strongly paired electrons, similar in character to those of the bulk bonds., 6 pages, 8 included figures; revtex file. To appear on Solid State Communications

We study several surface properties of ${\mathrm{BaTiO}}_{3}$: namely, the surface charge, surface energy, and surface dynamical charges, within an ab initio Hartree-Fock scheme. We use both genuinely isolated slabs and periodically repeated slabs, with different terminations. We report these properties as functions of the thickness and termination of the slabs. The ability to deal with genuinely isolated slabs is a virtue of the localized-basis implementation that we adopt. In particular, the slab surface charge provides a value for the spontaneous polarization in excellent agreement with the calculation of the latter as a Berry phase, and with experiment as well. On the other hand, we find that when periodically repeated slabs are used, the interactions among slabs and the fictitious field imposed by the periodic boundary conditions can significantly affect the accuracy of the calculations, even at quite large vacuum separations.

We give evidence that some nonmagnetic (i.e., spin-integrated) properties of the antiferromagnetic late transition-metal monoxides (MnO, FeO, CoO, NiO) are substantially noncubic below the N\'eel temperature even when assuming the ideal rocksalt structure for the ions. Our findings, which are at variance with the currently accepted picture, are based on ab initio and model calculations for the case study of MnO. The calculated zone-center optic phonon frequencies and Born effective charge tensor of this material show a significant magnetic-induced anisotropy.

The dipole moment of any finite and neutral system, having a square-integrable wave function, is a well-defined quantity. The same quantity is ill-defined for an extended system, whose wave function invariably obeys periodic (Born–von Karman) boundary conditions. Despite this fact, macroscopic polarization is a theoretically accessible quantity, for either uncorrelated or correlated many-electron systems: in both cases, polarization is a rather “exotic” observable. For an uncorrelated—either Hartree–Fock or Kohn–Sham—crystalline solid, polarization has been expressed and computed as a Berry phase of the Bloch orbitals (since 1993). The case of a correlated and/or disordered system received a definitive solution only very recently (1998): This latest development allows us to present here the whole theory from a novel, and very general, viewpoint. The modern theory of polarization is even relevant to the foundations of density functional theory in extended systems. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 599–606, 1999

When describing a crystalline system using periodic boundary conditions, the Bloch vector assumes discrete values on a regular mesh in the first Brillouin zone. This mesh is used here in a fundamental way to express the dielectric polarization as a discrete Berry phase of the single-determinant wave function. The present discrete formulation can be used to recover the results of the well-established continuum theory [R. Resta, Rev. Mod. Phys. 66, 899 (1994)], but also provides some other findings. In particular, when the crystal cell is taken as very large (supercell), the Brillouin zone is very small and a single reciprocal point is enough to provide the Berry phase, which measures the macroscopic polarization. The properties of this highly unusual single-point discrete Berry phase are thoroughly investigated. As a simple test case, we provide the calculation of the Hartree-Fock dynamical charge of MgO: the result is compared to experiment and to previous calculations within density-functional theory.

In practical implementations of density-functional theory, the only term where an orbital description is needed is the kinetic one. Even this term in principle depends on the density only, but its explicit form is unknown. We provide a novel solution of the N-representability problem for an extended system, which implies an explicit form for the Kohn-Sham kinetic energy in terms of the density. Our approach is based on a periodic coordinate mapping, uniquely defined by the Fourier coefficients of the metric. The density functional is thus expressed as an explicit functional of the metric tensor: since N-representability is enforced, our constructive recipe provides a variational approximation. Furthermore, we show that our geometric viewpoint is quite naturally related to the electron localization function (ELF), which provides a very informative analysis of the electron distribution. Studies of ELF, as obtained from accurate Kohn-Sham orbitals in real materials, allow an appraisal of the variational approximate density functional. We show that the value of an approximate functional-either the present geometric-based one or some previous ones based on different constructive recipes-strongly depends on the nature of the chemical bonding in the material., Comment: 6 pages, 2 figures

Within the Hartree-Fock (HF) scheme, we investigate some basic polarization features of the paradigmatic ferroelectric perovskite ${\mathrm{KNbO}}_{3}$, using a supercell technique. With regard to the spontaneous polarization and the transverse dynamical charge, the HF results are in good agreement with both experiment and density-functional calculations, despite the large difference in the one-electron energy levels calculated with the two methods. On the other hand, the HF value of the electronic dielectric constant ${\ensuremath{\varepsilon}}_{\ensuremath{\infty}}$ is substantially lower than experiment, while density-functional values are known to be substantially higher than that.

We have investigated the ferroelectric polarization of the perovskite oxide ${\mathrm{KNbO}}_{3}$ within the self-consistent Hartree-Fock (HF) method, where the crystalline orbitals are expanded over a set of localized functions. According to the modern theory, macroscopic polarization is a geometric quantum phase: here we show that---within the HF framework---polarization can be cast as a Berry phase of Slater determinants. We calculate this observable for ${\mathrm{KNbO}}_{3}$ in its tetragonal phase. Besides polarization, we investigate several other properties of the electronic ground state, including the broken-symmetry instability of the tetragonal structure. We therefore assess the reliability and the predictive power of the HF approach when dealing with this material, which is a paradigmatic case of intermediate ionic/covalent crystal.

The dynamical charges measure the macroscopic polarization induced by a relative sublattice displacement in a polar insulator. When the material has a mixed ionic/covalent character such charges present nontrivial features. Apparently similar oxides have strikingly different behaviors: some (like the ferroelectric perovskites) have giant dynamical charges, while others (like ZnO) are quite normal and display dynamical charges close to the nominal static ones. The rationale for such differences is found, upon closely scrutinizing the covalence mechanism. Furthermore, the effects of electron-electron interaction on the polarization of a model highly correlated dielectric are discussed.

The modern expressions for polarization $��$ and orbital magnetization $\M$ are $\k$-space integrals. But a genuine bulk property should also be expressible in $\r$-space, as unambiguous function of the ground-state density matrix, "nearsighted" in insulators, independently of the boundary conditions---either periodic or open. While $��$---owing to its "quantum" indeterminacy---is {\it not} a bulk property in this sense, $\M$ is. We provide its $\r$-space expression for any insulator, even with nonzero Chern invariant. Simulations on a model Hamiltonian validate our theory., Phys. Rev. Lett, in press

We calculate ab initio the electronic states, spontaneous polarization P, and Born dynamical charge ${\mathit{Z}}^{\mathrm{*}}$ of ZnO, using the local-density approximation (LDA), the Hartree-Fock (HF), and a model GW scheme. Upon going from HF to GW and to LDA, the d bands raise substantially in energy, the model GW providing the best overall agreement with experiment. By contrast, the three schemes give the concordant values P=-0.047 C/${\mathrm{m}}^{2}$ and ${\mathit{Z}}^{\mathrm{*}}$=\ifmmode\pm\else\textpm\fi{}2.1, in agreement with available experimental data. While the value of ${\mathit{Z}}^{\mathrm{*}}$ suggests a rigid-ion-like behavior of ZnO, its band-by-band decomposition reveals anomalous contributions from O 2s, Zn 3d, and O 2p bands, indicating a substantial interaction between the corresponding occupied atomic orbitals.

In a mixed ionic/covalent insulator the dynamical ionic charges are much larger than the static ones. The phenomenon is particularly relevant in ferroelectric perovskites, and is well understood at the mean-field level. Here we use a model Hamiltonian to investigate polarization in a strongly correlated, and partially ionic, insulator. Charge transport is evaluated as a Berry phase. At a critical value of the electron-electron interaction, the model has a transition from a band insulator to a Mott insulator: the static ionic charge is continuous across the transition, whereas the polarization is discontinuous. Above the transition, the anion transports a positive charge.

Recently, the Berry's phase approach has made the macroscopic polarization of ferroelectric perovskites accessible to full quantum-mechanical treatment. First-principle calculations for KNbO3 have demonstrated—amongst other things—the overwhelming importance of asymmetric Nb-O bond length variations: such variations drag a giant macroscopic current across the sample during the ferroelectric distortion. Here we investigate the microscopic mechanism mostly responsible for such giant currents. It is shown that only the highest (so-called O2p) electronic bands significantly contribute to the polarization; furthermore evidence is given that a sizeable hydridization between the O2p oxygen orbitals and the Nb 4d orbitals occurs, this hybridization being the sole responsible for the giant currents.

In ferroelectric perovskites, the strong spontaneous polarization is due to giant values of the Born effective charges, which are much larger than predicted on the basis of the nominal ionicity: we investigate the physical origin of such charges, which by definition measure the current traversing the sample during polarization reversal. Our first-principles calculations for ${\mathrm{KNbO}}_{3}$ identify unambiguously the dominant mechanism with the presence of covalent bonding. The giant charges are reduced to their nominal values in a computational experiment performed on a fake material, where covalence has been artificially suppressed.

The macroscopic electric polarization of a crystal is often defined as the dipole of a unit cell. In fact, such a dipole moment is ill defined, and the above definition is incorrect. Looking more closely, the quantity generally measured is differential polarization, defined with respect to a "reference state" of the same material. Such differential polarizations include either derivatives of the polarization (dielectric permittivity, Born effective charges, piezoelectricity, pyroelectricity) or finite differences (ferroelectricity). On the theoretical side, the differential concept is basic as well. Owing to continuity, a polarization difference is equivalent to a macroscopic current, which is directly accessible to the theory as a bulk property. Polarization is a quantum phenomenon and cannot be treated with a classical model, particularly whenever delocalized valence electrons are present in the dielectric. In a quantum picture, the current is basically a property of the phase of the wave functions, as opposed to the charge, which is a property of their modulus. An elegant and complete theory has recently been developed by King-Smith and Vanderbilt, in which the polarization difference between any two crystal states---in a null electric field---takes the form of a geometric quantum phase. The author gives a comprehensive account of this theory, which is relevant for dealing with transverse-optic phonons, piezoelectricity, and ferroelectricity. Its relation to the established concepts of linear-response theory is also discussed. Within the geometric phase approach, the relevant polarization difference occurs as the circuit integral of a Berry connection (or "vector potential"), while the corresponding curvature (or "magnetic field") provides the macroscopic linear response.

We study the screening of a macroscopic electric field in a crystalline dielectric. Density-functional perturbation theory provides the static dielectric constant (or tensor) as a bulk property; we give a formulation which extends the local-density approximation, and specifically we discuss its implementation within gradient-corrected schemes. We briefly consider the relevance (if any) of the so-called ``gap problem'' to static linear response. As a case study, we perform an an initio calculation of the dielectric constant in silicon within a popular gradient-corrected local-density scheme. We find that the gradient corrections reduce the discrepancy found so far between local-density predictions and experiments in covalently bonded materials. The amount of this reduction is sizable if the calculations are performed at the experimental equilibrium lattice constant of the crystal, while, however, it is only marginal when the calculations are carried out, at the calculated lattice constants, consistently within each given theoretical scheme.

A modern theory of the macroscopic electric polarization has been recently founded. Polarization itself is not an observable, while the polarization difference ΔP between any two crystal states can be measured, defined, and calculated as a bulk material property. An hysteresis cycle measures ΔP as the integrated current flowing through the sample: the present theory provides an expression for exactly this quantity. Being a current, ΔP is a property of the phases of the crystal wavefunctions, which can be cast as a Berry's phase, i.e. as a gauge-invariant phase feature of the valence Bloch orbitals. I stress that the periodic charge density of the polarized crystal-where any phase information is deleted-is irrelevant to macroscopic polarization. The present viewpoint elucidates the fundamental quantum nature of polarization: no Clausius-Mossotti-like model applies whenever the valence electrons are delocalized, as is the case in perovskites. Besides its conceptual importance, the Berry's phase app...

The organization of the electrons in the ground state is classified by means of topological invariants, defined as global properties of the wave function. Here we address the Chern number of a two-dimensional insulator and we show that the corresponding topological order can be mapped by means of a ``topological marker,'' defined in $\mathbf{r}$ space, and which may vary in different regions of the same sample. Notably, this applies equally well to periodic and open boundary conditions. Simulations over a model Hamiltonian validate our theory.

Nonlinear piezoelectricity has recently been detected in strained-layer superlattices where CdTe is pseudomorphically grown along (111) on different materials. We investigate this problem from first principles using density-functional theory. As a byproduct, we provide the ab initio calculation of several properties of this material, including the equilibrium lattice constant, Born effective charges, internal strain parameter, zone-center optical-phonon frequencies, dielectric constant, elastic constants, and the «clamped-ions» piezoelectric constant. Nonlinear effects on all of them are considered. Our major result is that piezoelectricity is accurately linear over a wide range of volume-conserving strains, while it displays strong nonlinearity whenever the strain is not volume conserving