Electron crystallization using localized representation

The character of the ground state of the electron crystal—an electron gas with periodic density and/or spin density is investigated. Calculations for non-magnetic, ferromagnetic and anti-ferromagnetic electron crystals based on the Koster-Kohn variational principle for direct calculation of Wannier functions are presented. The Wannier function is approximated by a symmetrically orthonormalized Gaussian. The orbital exponent of the Gaussian is used as a variational parameter. The effect of the positive background is suitably taken into account. The results of our calculation support Wigner’s prediction of electron crystallization.     

Chemical pseudopotentials

The subject of so-called chemical or localized pseudopotentials will be reviewed. These pseudopotentials are based on the concept of Wannier functions and one can derive a self-consistent wave equation which such localized functions satisfy. As pointed out by Adams and by Gilbert, the optimum such functions are not orthonormalized, as was remarked by Wannier himself many years ago. These functions have been the subject of many successful chemical calculations by D.J. Bullet, which will be reviewed.     

Exponential decay properties of Wannier functions and related quantities

The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of -3/4 for Wannier functions, -1/2 for the density matrix and for energy matrix elements, and -1/2 or -3/2 for different constructions of nonorthonormal Wannier-like functions.     

Existence of the exponentially localised Wannier functions

A partial answer (Theorem 1 below) to a problem concerning analytic and periodic families of projections in Hilbert spaces is given. As a consequence the existence of exponentially localised Wannier functions corresponding to nondegenerated bands of arbitrary three-dimensional crystals is proved.     

Tight-binding models for ultracold atoms in optical lattices: general formulation and applications

Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients—one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum—are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.     

The Existence of Generalised Wannier Functions for One-Dimensional Systems

It is proved that the subspaces corresponding to bounded, isolated parts of the spectrum of (periodic and nonperiodic) one-dimensional Schr?dinger operators admit bases of exponentially localised functions (generalised Wannier functions). Received: 17 February 1997 / Accepted: 15 April 1997     

Collective excitations of a periodic Bose condensate in the Wannier representation

We study the dispersion relation of the excitations of a dilute Bose-Einstein condensate confined in a periodic optical potential and its Bloch oscillations in an accelerated frame. The problem is reduced to one-dimensionality through a renormalization of the s-wave scattering length and the solution of the Bogolubov-de Gennes equations is formulated in terms of the appropriate Wannier functions. Some exact properties of a periodic one-dimensional condensate are easily demonstrated: (i) the lowest band at positive energy refers to phase modulations of the condensate and has a linear dispersion relation near the Brillouin zone centre; (ii) the higher bands arise from the superposition of localized excitations with definite phase relationships; and (iii) the wavenumber-dependent current under a constant force in the semiclassical transport regime vanishes at the zone boundaries. Early results by Slater [Phys. Rev. 87, 807 (1952)] on a soluble problem in electron energy bands are used to specify the conditions under which the Wannier functions may be approximated by on-site tight-binding orbitals of harmonic-oscillator form. In this approximation the connections between the low-lying excitations in a lattice and those in a harmonic well are easily visualized. Analytic results are obtained in the tight-binding scheme and are illustrated with simple numerical calculations for the dispersion relation and semiclassical transport in the lowest energy band, at values of the system parameters which are relevant to experiment. Received 3 December 1999 and Received in final form 22 March 2000     

Partly occupied Wannier functions

We introduce a scheme for constructing partly occupied, maximally localized Wannier functions (WFs) for both molecular and periodic systems. Compared to the traditional occupied WFs the partly occupied WFs possess improved symmetry and localization properties achieved through a bonding-antibonding closing procedure. We demonstrate the equivalence between bonding-antibonding closure and the minimization of the average spread of the WFs in the case of a benzene molecule and a linear chain of Pt atoms. The general applicability of the method is demonstrated through the calculation of WFs for a metallic system with an impurity: a Pt wire with a hydrogen molecular bridge.     

Exponential localization of Wannier functions in insulators

The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under consideration. The same equivalence implies that Chern insulators cannot display exponentially localized Wannier functions. An explicit condition for the reality of the Wannier functions is identified.     

Short-range interaction in the exciton effective-mass Hamiltonian

The integral equation for excitons is considered in the weak binding limit. The expression of the short-range electron-hole interaction is provided in terms of the Wannier functions.     

A variational principle for band calculations in the kq-representation

A known variational principle satisfied by Wannier functions is extended to cover free electron-like solids. The extension is achieved by formulating the principle in the kq-representation where Wannier functions of a given band equal to Bloch functions of the same band.     

Constraints in the generation of photonic Wannier functions

We report on the generation of maximally localized photonic Wannier functions under constraints. This allows us to impress certain symmetry properties of the underlying Photonic Crystal onto the Wannier functions. This added flexibility enhances the utility of the Wannier function approach to Photonic Crystal circuits by providing deeper physical insight and making computations more efficient.     

Construction of maximally localized Wannier functions

We present a general method for constructing maximally localized Wannier functions. It consists of three steps: (i) picking a localized trial wave function, (ii) performing a full band projection, and (iii) orthonormalizing with the Löwdin method. Our method is capable of producing maximally localized Wannier functions without further minimization, and it can be applied straightforwardly to random potentials without using supercells. The effectiveness of our method is demonstrated for both simple bands and composite bands.     

Motion of domain walls under the action of spin-polarized current in a magnetic junction

The effect of spin-polarized current on a domain structure in a magnetic junction consisting of two ferromagnetic metallic layers separated by an ultrathin nonmagnetic layer is studied within a phenomenological theory. The magnetization of one ferromagnetic layer (layer 1) is assumed to be fixed, while that of the other ferromagnetic layer (layer 2) can be freely oriented both parallel and antiparallel to the magnetization of layer 1. Layer 2 can be split into domains. Charge transfer from layer 1 to layer 2 is not attended with spin scattering by the interface but results in spin injection. Due to s-d exchange interaction, injected spins tend to orient the magnetization in the domains parallel to layer 1. This causes the domain walls to move and “favorable” domains to grow. The average magnetization current injected into layer 2 and its contribution to the s-d exchange energy are found by solving the continuity equation for carriers with spins pointing up and down. From the minimum condition for the total magnetic energy of the junction, the parameters of the periodic domain structure in layer 2 are determined as functions of current through the junction and magnetic field. It is shown that the spin-polarized current can magnetize layer 2 up to saturation even in the absence of an external magnetic field. The associated current densities are on the order of 10⁵ A/cm². In the presence of the field, its effect can be compensated by such a high current. Current-induced magnetization reversal in the layer is also possible.     

The generating functions formalism for the analysis of spin response to the periodic trains of RF pulses. Echo sequences with arbitrary refocusing angles and resonance offsets

The generating functions (GF) formalism was applied for calculation of spin density matrix evolution under the influence of periodic trains of RF pulses. It was shown that in a general case, closed expression for the generating function can be found that allows in many cases to derive analytical expressions for the generating function of spin density matrix (magnetization, coherences). This approach was shown to be particularly efficient for the analysis of multi-echo sequences, where one has to average over various frequency isochromats. The explicit analytical expressions for the generating function for echo amplitudes in a Carr–Purcell–Meiboom–Gill (CPMG) echo sequence, a multiecho sequence with incremental phase of refocusing pulse, a gradient echo sequence including transient period were obtained for an arbitrary flip angle and an arbitrary resonance offset. Comparison of the theory and the spin-echo experiments was done, demonstrating a good agreement.     

Numerical calculation of localized (Wannier) functions from ab-initio Hartree-Fock wave functions

The Fourier transformation necessary to obtain localized (Wannier) functions from delocalized ab-initio crystal Hartree-Fock (Bloch) functions is carried out for the one-dimensional models of solids. Combinations of Wannier functions from different bands lead to even more localized functions in close analogy to localization in molecules.     

Ab initio many-body treatment of the electronic structure of metals

We propose and apply a combination of an ab initio (band-structure) calculation with a many-body treatment including screening effects. We start from a linearized muffin-tin orbital (LMTO) calculation to determine the Bloch functions for the Hartree one-particle Hamiltonian, from which we calculate the static susceptibility and dielectric function within the standard random phase approximation (RPA). From the Bloch functions we obtain maximally localized Wannier functions, using a method proposed by Marzari and Vanderbilt. Within this Wannier basis all relevant one-particle and unscreened and screened Coulomb matrix elements are calculated. This yields a multi-band Hamiltonian in second quantization with ab initio parameters, for which screening has been taken into account within the simplest standard approximation. Then, established methods of many-body theory are used. We apply this concept to a simple metal, namely lithium (Li). Here the maximally localized Wannier functions turn out to be of the sp³-orbital kind. Furthermore, only the on-site contributions of the screened Coulomb matrix elements are relevant, and a generalized, four-band Hubbard model is justified. The screened on-site Coulomb matrix elements are considerably smaller than the band width because of which it is sufficient to calculate the selfenergy in weak-coupling approximation. We compare results obtained within the screened Hartree-Fock approximation (HFA) and within the second-order perturbation theory (SOPT) in the Coulomb matrix elements for Li and find that many-body effects are small but not negligible even for this simple metal.     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Derivation of an optical potential for deuterons

A general interpolation scheme is described which allows to determine the different eigenvaluesE _n(κ) for a given value of κ by solving an eigenvalue problem of small rank. The elements of the corresponding matrix, not yet restricted by symmetry requirements, may be determined from calculated energy valuesE _n (κ) along the directions of high symmetry. In addition for different bands connected with one another a new set of Wannier functions is defined.