Permittivity from first principles

Dielectric properties of materials are generally introduced phenomenologically through empirical values of permittivity. While this approach is necessary for practical work, it must be recognized that it bypasses the question of whether it is possible to predict permittivity values from first principles and thus get a deeper grasp of the physics involved in bridging the microscopic system and its macroscopic properties. Theoretical frameworks to gain insight and to compute exactly permittivity values are desirable to understand the nuances that go into building this bridge. We introduce here such a theoretical model system to gain physical intuition on the microscopic origin of the permittivity. The system consists of electrons in a one-dimensional atomic chain in the presence of an external electric field, where each atom is a binding site. We first consider a single atom in an external field to study atomic polarization, justify the model, tune the parameters, and compare with perturbative approaches. We then consider the assembly of many such atoms in a periodic arrangement and study its band-structure, including explicitly the external electric field. Last, within this model we develop explicit formulas for the permittivity in terms of relevant physical parameters. Finally, we obtain a numerically value for the permittivity of the system for typical values of binding energy and electric field.