The Hubbard model and optics: reflectivity of 1D and 2D systems

The Hubbard model (HM) was proposed around the middle of the last century as a model of the behavior of correlated electrons. It has so far been solved only in the 1D case. The HM is used in condensed matter physics in work on high T-c superconductors, organic conductors, graphene, ... It is known that reflectivity is related to the electrical conductivity, and both quantities can be measured. This paper is divided into several sections. It begins with a brief introduction to the HM, and the links between the reflectivity and conductivity of a material with its parameters. In the second and third parts, the explicit calculations of the reflectivity of a 1D and 2D square rectangular lattices will be presented. Expressions for the reflectivity thus obtained are tied to the material parameters via previous results of the author on the conductivity of a 1D lattice. We attempt to find a set of material parameters in both cases of lattices, for which the reflectivity becomes close to zero, or equal to an arbitrary constant. If such a set of material parameters could be found, it could have interesting applications in material science such as invisibility.