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The Journal of Chemical Physics

Signatures of Wigner localization in one-dimensional systems

Alejandro Díaz-Márquez, Stefano Battaglia, Gian Luigi Bendazzoli, Stefano Evangelisti, Thierry Leininger, J. A. Berger

Published Mar 22, 2018• 27 citations
Publisher

Abstract

We propose a simple and efficient approach to study Wigner localization in one-dimensional systems using ab initio theory. In particular, we propose a suitable basis for the study of localization which consists of equally spaced overlapping gaussians. We illustrate our approach with full-configuration interaction which yields exact results for a given basis set. With our approach, we were able to study up to 8 electrons with full-configuration interaction. Finally, we propose the total-position spread tensor and the total electron entropy as convenient quantities to obtain signatures of Wigner localization.

DOI
10.1063/1.5017118

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We propose a simple and efficient approach to study Wigner localization in one-dimensional systems using ab initio theory. In particular, we propose a suitable basis for the study of localization which consists of equally spaced overlapping gaussians. We illustrate our approach with full-configuration interaction which yields exact results for a given basis set. With our approach, we were able to study up to 8 electrons with full-configuration interaction. Finally, we propose the total-position spread tensor and the total electron entropy as convenient quantities to obtain signatures of Wigner localization.
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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.
In this work we investigate Wigner localization at very low densities by means of the exact diagonalization of the Hamiltonian. This yields numerically exact results. In particular, we study a quasi-one-dimensional system of two electrons that are confined to a ring by three-dimensional gaussians placed along the ring perimeter. To characterize the Wigner localization we study several appropriate observables, namely the two-body reduced density matrix, the localization tensor and the particle-hole entropy. We show that the localization tensor is the most promising quantity to study Wigner localization since it accurately captures the transition from the delocalized to the localized state and it can be applied to systems of all sizes.
Exact ground-state properties are presented by combining the diagonalization in the Fock space (and taking all hopping integrals and all two-site interactions) with the ab initio optimization of the Wannier functions. Electrons are essentially localized for the interatomic distance $R\ensuremath{\sim}2.0 \AA{}$ for s-like states, when the quasiparticle mass is divergent. The momentum distribution dispersion is proposed to define the localization order parameter. Dimerization and zero-point energies are also discussed. The method provides convergent results for N> 8N>~8N> 8 atoms.
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.
We introduce a diagrammatic quantum field formalism for the evaluation of normalized expectation values of operators, and suitable for systems with localized electrons. It is used to develop a convergent series expansion for the energy in powers of overlap integrals of single-particle orbitals. This method gives intuitive and practical rules for writing down the expansion to arbitrary order of overlap and can be applied to any spin configuration and to any dimension. Its applicability for systems with well-localized electrons is illustrated with examples, including the two-dimensional Wigner crystal and spin singlets in the low-density electron gas.