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# Wigner systems from first principles (Miguel Escobar Azor / LCPQ / These). – 13 decembre 2022, 14H

## 13 December 2022; 14h00 - 17h00

**Miguel Escobar Azor,** LCPQ, seminary room, 3rd floor -3r1B4

**Supervisors** : Arjan Berger and Stefano Evangelisti

**Abstract:**

This thesis aims to study Wigner localization at very low densities. Almost a century ago, Eugene Wigner predicted that a system consisting of interacting electrons in a neutralizing uniform background would form a crystalline structure at sufficiently low density with the electrons localized at lattice sites. His argument can be understood by considering the dependence of the kinetic and repulsive energies on the Wigner-Seitz radius, which is the radius of a sphere that contains, on average, one electron or, equivalently, half the average distance between nearest-neighbor electrons. While the kinetic energy scales as the inverse of the square of the Wigner-Seitz radius, the repulsive energy scales as the inverse of the Wigner-Seitz radius. Consequently, in the low-density limit (large Wigner-Seitz radius), the Hamiltonian is dominated by the repulsive energy leading the electrons to localize in space. When many electrons are present, the electrons will localize at crystallographic sites forming a so-called Wigner crystal. More generally speaking, one speaks of Wigner localization whenever electrons localize due to the electron-electron repulsion being dominant with respect to the kinetic energy. For a few electron systems, one also speaks of a Wigner molecule. To study Wigner localization, we have performed three correlated studies. The first of them consists of confining two electrons to a ring. To do so, we work with M-equal evenly distributed Gaussian orbitals as basis set and compute the energies and wave function through exact diagonalization of the Hamiltonian, yielding exact results to a given basis set. We limited ourselves to two electrons since it is sufficient to observe the Wigner localization while simultaneously allowing us to obtain numerically exact results by exact diagonalization of the Hamiltonian. We have verified our approach and its implementation by comparing analytical results for one electron confined to a strictly 1D ring. For this case, we have also analyzed the Wigner localization by several possible indicators, notably the 2-body reduced density matrix, the localization tensor, and the particle-hole entropy. We have observed the Wigner localization in the two-body reduced density matrix and the localization tensor. Instead, the Wigner localization cannot easily be detected in the particle-hole entropy. Intending to move to higher dimensions, we have formulated the Clifford boundary conditions, which consist of the definition of a d-dimensional supercell and then modify its topology into a toroidal topology by joining opposite sides of the cell without deformation. This procedure yields a supercell with the topology of a d-Clifford torus which is a flat, closed d-dimensional real Euclidean space embedded in a complex d-dimensional Euclidean space. In this framework, a circle and a line are topologically equivalent. We will explore the system’s translational symmetry, allowing us to evaluate efficiently the one- and two-electron integrals. To validate our approach, we developed a semi-classical model that becomes exact at low densities and compared the energies of both approaches. Since strong electron correlation drives the Wigner localization, the ab-initio study of this phenomenon requires accurate quantum-chemistry approaches such as Full configuration interaction or multiconfigurational methods like complete active space self-consistent field to obtain highly accurate energies and wave functions in both the low-correlation and the high-correlation regimes. In this Second study and the first one, we will use the exact diagonalization of the Hamiltonian to obtain numerically exact results. Lastly, we will present the generalization of our approach to N-electron systems.