The Emergence of the Hexagonal Lattice in Two-Dimensional Wigner Fragments.

At very low density, the electrons in a uniform electron gas spontaneously break symmetry and form a crystalline lattice called a Wigner crystal. But which type of crystal will the electrons form? We report a numerical study of the density profiles of fragments of Wigner crystals from first principles. To simulate Wigner fragments, we use Clifford periodic boundary conditions and a renormalized distance in the Coulomb potential. Moreover, we show that high-spin restricted open-shell Hartree-Fock theory becomes exact in the low-density limit. We are thus able to accurately capture the localization in two-dimensional Wigner fragments with many electrons. No assumptions about the positions where the electrons will localize are made. The density profiles we obtain emerge naturally when we minimize the total energy of the system. We clearly observe the emergence of the hexagonal crystal structure, which has been predicted to be the ground-state structure of the two-dimensional Wigner crystal.

Multichannel Dyson Equation: Coupling Many-Body Green's Functions.

We present the multichannel Dyson equation that combines two or more many-body Green's functions to describe the electronic structure of materials. In this work we use it to model photoemission spectra by coupling the one-body Green's function with the three-body Green's function. We demonstrate that, unlike methods using only the one-body Green's function, our approach puts the description of quasiparticles and satellites on an equal footing. We propose a multichannel self-energy that is static and only contains the bare Coulomb interaction, making frequency convolutions and self-consistency unnecessary. Despite its simplicity, we demonstrate with a diagrammatic analysis that the physics it describes is extremely rich. Finally, we present a framework based on an effective Hamiltonian that can be solved for any many-body system using standard numerical tools. We illustrate our approach by applying it to the Hubbard dimer and show that it is exact both at 1/4 and 1/2 filling.

Solution to the Thomson Problem for Clifford Tori with an Application to Wigner Crystals.

In its original version, the Thomson problem consists of the search for the minimum-energy configuration of a set of point-like electrons that are confined to the surface of a two-dimensional sphere (S2) that repel each other according to Coulomb's law, in which the distance is the Euclidean distance in the embedding space of the sphere, i.e., R3. In this work, we consider the analogous problem where the electrons are confined to an n-dimensional flat Clifford torus Tn with n = 1, 2, 3. Since the torus Tn can be embedded in the complex manifold Cn, we define the distance in the Coulomb law as the Euclidean distance in Cn, in analogy to what is done for the Thomson problem on the sphere. The Thomson problem on a Clifford torus is of interest because supercells with the topology of a Clifford torus can be used to describe periodic systems such as Wigner crystals. In this work, we numerically solve the Thomson problem on a square Clifford torus. To illustrate the usefulness of our approach, we apply it to Wigner crystals. We demonstrate that the equilibrium configurations we obtain for large numbers of electrons are consistent with the predicted structures of Wigner crystals. Finally, in the one-dimensional case, we analytically obtain the energy spectrum and the phonon dispersion law.

Mapping of Hückel zigzag carbon nanotubes onto independent polyene chains: Application to periodic nanotubes.

The electric polarizability and the spread of the total position tensors are used to characterize the metallic vs insulator nature of large (finite) systems. Finite clusters are usually treated within the open boundary condition formalism. This introduces border effects, which prevent a fast convergence to the thermodynamic limit and can be eliminated within the formalism of periodic boundary conditions. Recently, we introduced an original approach to periodic boundary conditions, named Clifford boundary conditions. It considers a finite fragment extracted from a periodic system and the modification of its topology into that of a Clifford torus. The quantity representing the position is modified in order to fulfill the system periodicity. In this work, we apply the formalism of Clifford boundary conditions to the case of carbon nanotubes, whose treatment results in a particularly simple zigzag geometry. Indeed, we demonstrate that at the Hückel level, these nanotubes, either finite or periodic, are formally equivalent to a collection of non-interacting dimerized linear chains, thus simplifying their treatment. This equivalence is used to describe some nanotube properties as the sum of the contributions of the independent chains and to identify the origin of peculiar behaviors (such as conductivity). Indeed, if the number of hexagons along the circumference is a multiple of three, a metallic behavior is found, namely a divergence of both the (per electron) polarizability and total position spread of at least one linear chain. These results are in agreement with those in the literature from tight-binding calculations.

The Wigner localization of interacting electrons in a one-dimensional harmonic potential.

In this work, we study the Wigner localization of interacting electrons that are confined to a quasi-one-dimensional harmonic potential using accurate quantum chemistry approaches. We demonstrate that the Wigner regime can be reached using small values of the confinement parameter. To obtain physical insight in our results, we analyze them with a semi-analytical model for two electrons. Thanks to electronic-structure properties such as the one-body density and the particle-hole entropy, we are able to define a path that connects the Wigner regime to the Fermi-gas regime by varying the confinement parameter. In particular, we show that the particle-hole entropy, as a function of the confinement parameter, smoothly connects the two regimes. Moreover, it exhibits a maximum that could be interpreted as the transition point between the localized and delocalized regimes.

Wigner localization in two and three dimensions: An ab initio approach.

In this work, we investigate the Wigner localization of two interacting electrons at very low density in two and three dimensions using the exact diagonalization of the many-body Hamiltonian. We use our recently developed method based on Clifford periodic boundary conditions with a renormalized distance in the Coulomb potential. To accurately represent the electronic wave function, we use a regular distribution in space of Gaussian-type orbitals and we take advantage of the translational symmetry of the system to efficiently calculate the electronic wave function. We are thus able to accurately describe the wave function up to very low density. We validate our approach by comparing our results to a semi-classical model that becomes exact in the low-density limit. With our approach, we are able to observe the Wigner localization without ambiguity.

The localization spread and polarizability of rings and periodic chains.

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta's, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities.

Potential Energy Surfaces without Unphysical Discontinuities: The Coulomb Hole Plus Screened Exchange Approach.

In this work, we show the advantages of using the Coulomb hole plus screened exchange (COHSEX) approach in the calculation of potential energy surfaces (PES). In particular, we demonstrate that, unlike perturbative GW and partial self-consistent GW approaches, such as eigenvalue self-consistent GW and quasi-particle (QP) self-consistent GW, the COHSEX approach yields smooth PES without irregularities and discontinuities. Moreover, we show that the ground-state PES obtained from the Bethe-Salpeter equation (BSE), within the adiabatic connection fluctuation dissipation theorem, built with QP energies obtained from perturbative COHSEX on top of Hartree-Fock (BSE@COHSEX@HF) yield very accurate results for diatomic molecules close to their equilibrium distance. When self-consistent COHSEX QP energies and orbitals are used to build the BSE equation, the results become independent of the starting point. We show that self-consistency worsens the total energies but improves the equilibrium distances with respect to BSE@COHSEX@HF. This is mainly due to the changes in the screening inside the BSE.