Clifford boundary conditions for the study of periodic Coulomb systems*

We propose an original and general approach for the numerical study of periodic Coulomb systems. In our formalism we change the topology of a super-cell into the topology of a torus. However, in order to maintain the regular structure of the system, and, in particular, the angles, we use a Clifford torus. A Clifford torus is a real, flat and closed d-dimensional Euclidean space embedded in a complex d-dimensional Euclidean space. Moreover, we renormalize the distance between two points on the Clifford torus as the distance between these points in the embedding space of the torus and use this renormalized distance in the Coulomb potential. We have, so far, successfully applied this strategy to the calculation of Madelung constants and Wigner crystals. We also propose a new position operator that is compatible with periodic boundary conditions and satisfies all appropriate physical constraints. The position operator and the renormalized distance are consistent with each other.