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We investigate how similar heatmaps of soccer players are when constructed from (i) event datasets and (ii) tracking datasets. When using event datasets, we show that the scale at which the events are grouped strongly influences the correlation with the tracking heatmaps. Furthermore, there is an optimal scale at which the correlation between event and tracking heatmaps is the highest. However, even at the optimal scale, correlations between both approaches are moderate. Furthermore, there is high heterogeneity in the players' correlation, ranging from negative values to correlations close to the unity. We show that the number of events performed by a player does not crucially determine the level of correlation between both heatmaps. Finally, we analyzed the influence of the player position, showing that defenders are the players with the highest correlations while forwards have the lowest., Comment: 6 pages, 5 figures

We propose using Network Science as a complementary tool to analyze player and team behavior during a football match. Specifically, we introduce four kinds of networks based on different ways of interaction between players. Our approach's main novelty is to use tracking datasets to create football tracking networks, instead of constructing and analyzing the traditional networks based on events. In this way, we are able to capture player interactions that go beyond passes and introduce the concepts of (a) Ball Flow Networks, (b) Marking Networks, (c) Signed Proximity Networks and (d) Functional Coordination Networks. After defining the methodology for creating each kind of network, we show some examples using tracking datasets from four different matches of LaLiga Santander. Finally, we discuss some of the applications, limitations, and further improvements of football tracking networks., Comment: 13 pages, 5 figures

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., Comment: 5 pages, 1 figure (2 panels)

We introduce an alternative approach to the first-principles calculation of NMR shielding tensors. These are obtained from the derivative of the orbital magnetization with respect to the application of a microscopic, localized magnetic dipole. The approach is simple, general, and can be applied to either isolated or periodic systems. Calculated results for simple hydrocarbons, crystalline diamond, and liquid water show very good agreement with established methods and experimental results., Comment: submitted to Phys. Rev. Lett; second half completely rewritten to address referee's comments

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 added

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) BCs. 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 div P = -rho, and one equivalently addresses charge fluctuations: the generalized sum rule reduces then to a well known result of many-body theory., Comment: 4 pages, no figure

We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator (i.e., one in which the Chern invariant, or in 2d the Chern number, vanishes). Following the approach used recently to develop the single-band formalism [T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett. {\bf 95}, 137205 (2005)], we work in the Wannier representation and find that the magnetization is comprised of two contributions, an obvious one associated with the internal circulation of bulk-like Wannier functions in the interior and an unexpected one arising from net currents carried by Wannier functions near the surface. Unlike the single-band case, where each of these contributions is separately gauge-invariant, in the multi-band formulation only the \emph{sum} of both terms is gauge-invariant. Our final expression for the orbital magnetization can be rewritten as a bulk property in terms of Bloch functions, making it simple to implement in modern code packages. The reciprocal-space expression is evaluated for 2d model systems and the results are verified by comparing to the magnetization computed for finite samples cut from the bulk. Finally, while our formal proof is limited to normal insulators, we also present a heuristic extension to Chern insulators (having nonzero Chern invariant) and to metals. The validity of this extension is again tested by comparing to the magnetization of finite samples cut from the bulk for 2d model systems. We find excellent agreement, thus providing strong empirical evidence in favor of the validity of the heuristic formula., Comment: 14 pages, 8 figures. Fixed a typo in appendix A

Working in the Wannier representation, we derive an expression for the orbital magnetization of a periodic insulator. The magnetization is shown to be comprised of two contributions, an obvious one associated with the internal circulation of bulk-like Wannier functions in the interior, and an unexpected one arising from net currents carried by Wannier functions near the surface. Each contribution can be expressed as a bulk property in terms of Bloch functions in a gauge-invariant way. Our expression is verified by comparing numerical tight-binding calculations for finite and periodic samples., Comment: submitted to PRL; signs corrected in Eqs. (11), (12), (19), and (20)

We investigate some surfaces of a paradigmatic sp bonded metal--namely, Al(110), Al(100), and Al(111)--by means of the "electron localization function" (ELF), implemented in a first-principle pseudopotential framework. ELF is a ground-state property which discriminates in a very sharp, quantitative, way between different kinds of bonding. ELF shows that in the bulk of Al the electron distribution is essentially jelliumlike, while what happens at the surface strongly depends on packing. At the least packed surface, Al(110), ELF indicates a free-atom nature of the electron distribution in the outer region. The most packed surface, Al(111), is instead at the opposite end, and can be regarded as a jellium surface weakly perturbed by the presence of the ionic cores., Comment: 5 pages with 6 figures. Surface Science (in press)

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., Comment: 6 pages, 8 included figures; revtex file. To appear on Solid State Communications

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) sustains macroscopic polarization, and (ii) is localized. We give a sharp definition of the latter concept, and we show how the two basic features stem from essentially the same formalism. Our approach to localization is exemplified by means of a two--band Hubbard model in one dimension. In the noninteracting limit the wavefunction localization is measured by the spread of the Wannier orbitals., Comment: 5 pages including 3 figures, submitted to PRL

Within an ab-initio HF scheme, we use both Berry-phase calculations and supercell calculations in order to compute the dynamical charges for lattice dynamics and the electronic dielectric constant for KNbO_3 and BaTiO_3. Comparison with experimental data indicates that HF provides a description of the electronic properties of this material whose accuracy is of the same order as the LDA one. There are however significant differences between the two sets of results, whose origin is scrutinized. Motivated by the study of surface and domain-boundary properties, we also present some results for BaTiO_3 slabs, including both genuinely isolated and periodically repeated slabs with different terminations. The capability of dealing with a genuinely isolated slab is a virtue of the localized-basis implementation adopted here. We demonstrate, amongst other things, the nontrivial dynamical-charge neutrality of BaTiO_3 [001] surfaces., Comment: 11 pages, 6 figures. Presented at the Fifth Williamsburg Workshop on First-Principles Calculations for Ferroelectrics

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 Ferroelectrics

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

The position operator (defined within the Schroedinger representation in the standard way) becomes meaningless when periodic boundary conditions are adopted for the wavefunction, as usual in condensed matter physics. We show how to define the position expectation value by means of a simple many-body operator acting on the wavefunction of the extended system. The relationships of the present findings to the Berry-phase theory of polarization are discussed., Comment: Four pages in RevTeX

We calculate ab-initio the electronic states and the Born dynamical charge Z* of the alkaline-earth oxides in the local-density approximation. We investigate the trend of increasing Z* values through the series, using band-by-band decompositions and computational experiments performed on fake materials with artificially-modified covalence. The deviations of Z* from the nominal value 2 are due to the increasing interaction between O 2p orbitals and unoccupied cation d states. We also explain the variations, along the series, of the individual contributions to Z* arising from the occupied band manifolds., Comment: 12 pages Latex, plus 2 Postscript figures

The ferroelectric polarization of perovskite oxides is much larger than implied by displacement of static ionic charges. We use an explicitly correlated scheme to investigate the phenomenon; charge transport is evaluated as a geometric quantum phase. Both covalence and electron-electron interaction enhance polarization in the weakly correlated regime. At higher values of the electron-electron interaction, the system undergoes 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, oxygen transports a positive charge., Comment: REVTeX 10 pages + 3 PostScript figures appended to the file