The quantum-mechanical position operator and the polarization problem

The position operator (defined within Schroedinger representation as usual) becomes meaningless when the usual Born-von Karman periodic boundary conditions are adopted: this fact is at the root of the polarization problem. I show how to define the position expectation value by means of rather peculiar many-body (multiplicative) operator acting on the wavefunction of the extended system. This definition can be regarded as the generalization of a precursor work, apparently unrelated to the polarization problem. For uncorrelated electrons, the present finding coincides with the so-called "single-point Berry phase" formula, which can hardly be regarded as the approximation of a continuum integral, and is computationally very useful for disordered systems. Simulations which are based on this concept are being performed by several groups.Comment: 10 pages, 1 embedded figure (in two panels). Presented at the Fifth Williamsburg Workshop on First-Principles Calculations for Ferroelectric

What makes an insulator different from a metal?

The insulating state of matter is characterized by the excitation spectrum, but also by qualitative features of the electronic ground state. The insulating ground wavefunction in fact: (i) displays vanishing dc conductivity; (ii) sustains macroscopic polarization; and (iii) is localized. The idea that the insulating state of matter is a consequence of electron localization was first proposed in 1964 by W. Kohn. I discuss here a novel definition of electron localization, rather different from Kohn's, and deeply rooted in the modern theory of polarization. In fact the present approach links the two features (ii) and (iii) above, by means of essentially the same formalism. In the special case of an uncorrelated crystalline solid, the localization of the many-body insulating wavefunction is measured - according to our definition - by the spread of the Wannier orbitals; this spread diverges in the metallic limit. In the correlated case, the novel approach to localization is demonstrated by means of a two-band Hubbard model in one dimension, undergoing a transition from band insulator to Mott insulator.Comment: 12 pages with 3 figures. Presented at the workshop "Fundamental Physics of Ferroelectrics", Aspen Center for Physics, February 200

Trying Cases in the Media: A Comparative Overview

The essay deals with the problem of media impact on ongoing trials. In particular, it proposes a taxonomy of three comparative models of governance (traditional common law approach; US approach; Continental European approach) and makes a case for the recognition of presumption of innocence as a fundamental rigth vis-à-vis the court of public opinion

Orbital magnetization and Chern number in a supercell framework: Single k-point formula

The key formula for computing the orbital magnetization of a crystalline system has been recently found [D. Ceresoli, T. Thonhauser, D. Vanderbilt, R. Resta, Phys. Rev. B {\bf 74}, 024408 (2006)]: it is given in terms of a Brillouin-zone integral, which is discretized on a reciprocal-space mesh for numerical implementation. We find here the single ${\bf k}$-point limit, useful for large enough supercells, and particularly in the framework of Car-Parrinello simulations for noncrystalline systems. We validate our formula on the test case of a crystalline system, where the supercell is chosen as a large multiple of the elementary cell. We also show that--somewhat counterintuitively--even the Chern number (in 2d) can be evaluated using a single Hamiltonian diagonalization.Comment: 4 pages, 3 figures; appendix adde

Electronic polarization in quasilinear chains

Starting with a finite $k$-mesh version of a well-known equation by Blount, we show how various definitions proposed for the polarization of long chains are related. Expressions used for infinite periodic chains in the 'modern theory of polarization' are thereby obtained along with a new single particle formulation. Separate intracellular and intercellular contributions to the polarization are identified and, in application to infinite chains, the traditional sawtooth definition is found to be missing the latter. For a finite open chain the dipole moment depends upon how the chain is terminated, but the intracellular and intercellular polarization do not. All of these results are illustrated through calculations with a simple H\"uckel-like model.Comment: 5 page

Multifractal analysis of Power Markets. Some empirical evidence

This work is intended to offer a comparative analysis of the statistical properties of hourly prices in the day–ahead electricity markets of several countries. Starting from the intermittent nature of typical price fluctuations in many power markets, we will provide evidence that working into a stochastic multifractal analysis framework can be of help to asses typical features of day–ahead market prices.Multifractals, Hurst Coefficient, Power Markets

Mapping topological order in coordinate space

The organization of the electrons in the ground state is classified by means of topological invariants, defined as global properties of the wavefunction. 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 \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