Electronic and magnetic properties of small one-dimensional Wigner crystals from an ab initio approach
We present an \emph{ab initio} method to study the electronic and magnetic properties of small one-dimensional Wigner crystals. In particular, we focus on the calculation of the electronic charge distribution and the exchange coupling constant. Our theoretical studies are motivated by the experimental observation of few-electron Wigner crystals in a carbon nanotube [Science 364, 870 (2019)]. We model the experimental setup by confining electrons in a one-dimensional potential well with infinite side barriers. We represent the Hamiltonian of the system in a basis of Slater determinants and perform full configuration interaction to ensure we capture all the electron correlation for a given basis set. As the one-particle basis set we use particle-in-a-box wave functions which by construction satisfy the boundary conditions. With our approach, we obtain accurate electronic density profiles of small one-dimensional Wigner crystals. These profiles clearly show the localisation of the electrons. Finally, we present a simple approach to obtain the exchange coupling constant by mapping our \textit{ab initio} method on a Heisenberg Hamiltonian. We illustrate our approach on a Wigner dimer. We obtain excellent agreement with a result in the literature.