Minimum energy configurations for interacting dipoles in simplex, orthoplex and hypercube crystals

Polytope structures composed by classical dipoles living on their vertices in R d present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics, as well as to discuss the energetics of the global minimum energy for the corresponding particles, interacting via dipole–dipole forces. As a theoretical benchmark for such investigations, we consider here three big families of d -dimensional polytopes: the d -simplex, the d -orthoplex and the d -hypercube. A dipole in O ( d ) is placed in each vertex. After numerically minimizing the total interaction energy, we shall observe that in both the d -simplex and the d -orthoplex, the energy surprisingly follows a simple linear functional behavior with the dimension d . The corresponding dipoles arrange themselves in orientations whose angles can be simple fractions of π , or incommensurate with π within numerical precision. We elucidate the ground-state of the simplex and orthoplex cases up to d = 7 , as well as the tesseract and the 5 -cube in great detail. • Multi-dimensional crystals with classical dipoles at vertices. • Minimum energy configuration is obtained by numerical tools. • Interesting behavior of the dipole system as a function of dimensionality.