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

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

Orbital Magnetization in Insulators: Bulk vs. Surface

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 owes to conducting surface states. Simulations on a model Hamiltonian validate our theory

Irrelevance of the boundary on the magnetization of metals

The macroscopic current density responsible for the mean magnetization $\mathbf{M}$ of a uniformly magnetized bounded sample is localized near its surface. In order to evaluate $\mathbf{M}$ 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 $\mathbf{M}$ in insulators: therein, $\mathbf{M}$ admits an alternative expression, not based on currents. $\mathbf{M}$ 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 $\mathbf{M}$ value can be retrieved in terms of the bulk electron distribution only.Comment: Phys. Rev. Lett. to be published (http://journals.aps.org/prl/accepted/f107dYd2Yc11e65562463bc449e91e07bcccf9546

Lyddane-Sachs-Teller relationship in linear magnetoelectrics

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 \infty fulfill a generalized Lyddane-Sachs-Teller relationship. The rhs 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 an harmonic crystal.Comment: 4 pages, no figur

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