Many-body phenomena in semiconductors and cluster expansion approach

In recent years the attention of many researchers is riveted to the study the optical properties of two-dimensional semiconductors such as transition-metal dichalcogenide (TMD or TMDC) monolayers (MLs) which are looking highly promising both from the fundamental point of view and the practical application prospects. The complex band structure and reduced screening coulomb interaction formed there facilitate strong correlation phenomena. These correlation effects in terms of optical response reveal themselves through the specific optical resonances such as excitons, trions, etc., as well as strong optical nonlinearity. In this work we aim to demonstrate some initial steps of comprehensive computational machinery being developed to take into account many-particle effects which are widely observed in a huge number of experiments on TMD MLs conducted at present. As a starting point to describe the corresponding dynamics we address the Heisenberg equation of motion. Processing the quantity of interest we come to an infinite hierarchy of equations. In order to properly truncate this system we use the cluster expansion technique that allows together with several tricks to competently construct some kind of perturbative series for the analyzed equations. Subsequently, these expansions can be treated numerically. In particular, within the current letter we present lower two-particle corrections which contribute to interband polarization.