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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: Math. Gen. 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 of the curvature's two arguments.
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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.
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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.
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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.
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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.
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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.
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The macroscopic current density responsible for the mean magnetization M of a uniformly magnetized bounded sample is localized near its surface. In order to evaluate M one needs the current distribution in the whole sample: bulk and boundary. In recent years it has been shown that the boundary has no effect on M in insulators: therein, M admits an alternative expression not based on currents. M can be expressed in terms of the bulk electron distribution only, which is "nearsighted" (exponentially localized); this virtue is not shared by metals, having a qualitatively different electron distribution. We show, by means of simulations on paradigmatic model systems, that even in metals the M value can be retrieved in terms of the bulk electron distribution only.
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The modern expressions for polarization P and orbital magnetization M are k-space integrals. But a genuine bulk property should also be expressible in r space, as an unambiguous function of the ground-state density matrix, "nearsighted" in insulators, independently of the boundary conditions--either periodic or open. While P--owing to its "quantum" indeterminacy--is not a bulk property in this sense, M is. We provide its r-space expression for any insulator, even with a nonzero Chern invariant. Simulations on a model Hamiltonian validate our theory.
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In a linear magnetoelectric the lattice is coupled to electric and magnetic fields; both affect the longitudinal-transverse splitting of zone-center optical phonons on equal footing. A response matrix relates the macroscopic fields (D, B) to (E, H) at infrared frequencies. It is shown that the response matrices at frequencies 0 and ∞ fulfill a generalized Lyddane-Sachs-Teller relationship. The right-hand side member of such relationship is expressed in terms of weighted averages over the longitudinal and transverse excitations of the medium, and assumes a simple form for a harmonic crystal.
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Flexoelectricity is the linear response of polarization to a strain gradient. Here we address the simplest class of dielectrics, namely, elemental cubic crystals, and we prove that therein there is no extrinsic (i.e., surface) contribution to flexoelectricity in the thermodynamic limit. The flexoelectric tensor is expressed as a bulk response of the solid, manifestly independent of surface configurations. Furthermore, we prove that the flexoelectric responses induced by a long-wavelength phonon and by a uniform strain gradient are identical.
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The qualitative difference between insulators and metals stems from the nature of the low-lying excitations, but also--according to Kohn's theory [W. Kohn, Phys. Rev. 133, A171 (1964)]--from a different organization of the electrons in their ground state: electrons are localized in insulators and delocalized in metals. We adopt a quantitative measure of such localization, by means of a "localization length" lambda, finite in insulators and divergent in metals. We perform simulations over a one-dimensional binary alloy model, in a tight-binding scheme. In the ordered case the model is either a band insulator or a band metal, whereas in the disordered case it is an Anderson insulator. The results show indeed a localized/delocalized ground state in the insulating/metallic cases, as expected. More interestingly, we find a significant difference between the two insulating cases: band versus Anderson. The insulating behavior is due to two very different scattering mechanisms; we show that the corresponding values of lambda differ by a large factor for the same alloy composition. We also investigate the organization of the electrons in the many body ground state from the viewpoint of the density matrices and of Boys' theory of localization.
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Macroscopic polarization P and magnetization M are the most fundamental concepts in any phenomenological description of condensed media. They are intensive vector quantities that intuitively carry the meaning of dipole per unit volume. But for many years both P and the orbital term in M evaded even a precise microscopic definition, and severely challenged quantum-mechanical calculations. If one reasons in terms of a finite sample, the electric (magnetic) dipole is affected in an extensive way by charges (currents) at the sample boundary, due to the presence of the unbounded position operator in the dipole definitions. Therefore P and the orbital term in M--phenomenologically known as bulk properties--apparently behave as surface properties; only spin magnetization is problemless. The field has undergone a genuine revolution since the early 1990s. Contrary to a widespread incorrect belief, P 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. Analogously, the orbital term in M has nothing to do with the periodic current distribution in the magnetized crystal. The modern theory of polarization, based on a Berry phase, started in the early 1990s and is now implemented in most first-principle electronic structure codes. The analogous theory for orbital magnetization started in 2005 and is partly work in progress. In the electrical case, calculations have concerned various phenomena (ferroelectricity, piezoelectricity, and lattice dynamics) in several materials, and are in spectacular agreement with experiments; they have provided thorough understanding of the behaviour of ferroelectric and piezoelectric materials. In the magnetic case the very first calculations are appearing at the time of writing (2010). Here I review both theories on a uniform ground in a density functional theory (DFT) framework, pointing out analogies and differences. Both theories are deeply rooted in geometrical concepts, elucidated in this work. The main formulae for crystalline systems express P and M in terms of Brillouin-zone integrals, discretized for numerical implementation. I also provide the corresponding formulae for disordered systems in a single k-point supercell framework. In the case of P the single-point formula has been widely used in the Car-Parrinello community to evaluate IR spectra.
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A theorem for end-charge quantization in quasi-one-dimensional stereoregular chains is formulated and proved. It is a direct analog of the well-known theorem for surface charges in physics. The theorem states the following: (1) Regardless of the end groups, in stereoregular oligomers with a centrosymmetric bulk, the end charges can only be a multiple of 12 and the longitudinal dipole moment per monomer p can only be a multiple of 12 times the unit length a in the limit of long chains. (2) In oligomers with a noncentrosymmetric bulk, the end charges can assume any value set by the nature of the bulk. Nonetheless, by modifying the end groups, one can only change the end charge by an integer and the dipole moment p by an integer multiple of the unit length a. (3) When the entire bulk part of the system is modified, the end charges may change in an arbitrary way; however, if upon such a modification the system remains centrosymmetric, the end charges can only change by multiples of 12 as a direct consequence of (1). The above statements imply that-in all cases-the end charges are uniquely determined, modulo an integer, by a property of the bulk alone. The theorem's origin is a robust topological phenomenon related to the Berry phase. The effects of the quantization are first demonstrated in toy LiF chains and then in a series of trans-polyacetylene oligomers with neutral and charge-transfer end groups.
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The static dielectric properties of liquid and solid water are investigated within linear response theory in the context of ab initio molecular dynamics. Using maximally localized Wannier functions to treat the macroscopic polarization we formulate a first-principles, parameter-free, generalization of Kirkwood's phenomenological theory. Our calculated static permittivity is in good agreement with experiment. Two effects of the hydrogen bonds, i.e., a significant increase of the average local moment and a local alignment of the molecular dipoles, contribute in almost equal measure to the unusually large dielectric constant of water.
Asunto(s)
Agua/química , Enlace de Hidrógeno , Modelos Químicos , Electricidad EstáticaRESUMEN
The authors provide a reformulation of the modern theory of polarization for one-dimensional stereoregular polymers, at the level of the single determinant Hartree-Fock and Kohn-Sham methods within a basis set of local orbitals. By starting with localization of one-electron orbitals, their approach naturally arrives to the Berry phases of Bloch orbitals. Then they describe a novel numerical algorithm for evaluation of longitudinal dipole moments, computationally more convenient than those presently implemented within the local basis periodic codes. This method is based on the straightforward evaluation of the usual direct space dipole matrix elements between local orbitals, as well as overlap matrices between wave functions at two neighboring k points of the reciprocal space mesh. The practical behavior of the algorithm and its convergence properties with respect to the k-point mesh density are illustrated in benchmark calculations for water chains and fluorinated trans-polyacetylene.
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The ground-state fluctuation of polarization P is finite in insulators and divergent in metals, owing to the SWM sum rule [I. Souza, T. Wilkens, and R. M. Martin, Phys. Rev. B 62, 1666 (2000)]. This is a virtue of periodic (i.e., transverse) boundary conditions. I show that within any other boundary conditions the P fluctuation is finite even in metals, and a generalized sum rule applies. The boundary-condition dependence is a pure correlation effect, not present at the independent-particle level. In the longitudinal case inverted triangle x P = -rho, and one equivalently addresses charge fluctuations: the generalized sum rule reduces then to a well-known result of the many-body theory.
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The qualitative difference between insulators and conductors not only manifests itself in the excitation spectra but also--according to Kohn's theory [Phys. Rev. 133, A171 (1964)]--in a different organization of the electrons in their ground state: the wave function is localized in insulators and delocalized in conductors. Such localization, however, is hidden in a rather subtle way in the many-body wave function. The theory has been substantially revisited and extended in modern times, invariably within a periodic-boundary-condition framework, i.e., ideally addressing an infinite condensed system. Here we show how the localization/delocalization of the many-body wave function shows up when considering either three-dimensional clusters of increasing size or quasi-one-dimensional systems (linear polymers, nanotubes, and nanowires) of increasing length, within the ordinary "open" boundary conditions adopted for finite systems. We also show that the theory, when specialized to uncorrelated wave functions, has a very close relationship with Boy's theory of localization [Rev. Mod. Phys. 32, 296 (1960)]: the Boys orbitals in the bulk of the sample behave in a qualitatively different way in insulating versus conducting cases.
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We report a simulation of deuterated water using a Car-Parrinello approach based on maximally localized Wannier functions. This provides local information on the dynamics of the hydrogen-bond network and on the origin of the low-frequency infrared activity. The oscillator strength of the translational modes, peaked around approximately 200 cm-1, is anisotropic and originates from intermolecular--not intramolecular--charge fluctuations. These fluctuations are a signature of a tetrahedral hydrogen-bonding environment.
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The theory of the insulating state discriminates between insulators and metals by means of a localization tensor, which is finite in insulators and divergent in metals. In absence of time-reversal symmetry, this same tensor acquires an off-diagonal imaginary part, proportional to the dc transverse conductivity, leading to quantization of the latter in two-dimensional systems. I provide evidence that electron localization--in the above sense--is the common cause for both vanishing of the dc longitudinal conductivity and quantization of the transverse one in quantum Hall fluids.