Tensor network strategies for calculating biexcitons and trions in monolayer 2D materials beyond the ground state

Recently in [Phys. Rev. B 99, 241301(R) (2019)] tensor networks build upon logical circuits were briefly introduced to retrieve exciton and biexciton states. Compared to a conventional approach the tensor network methods scales logarithmic instead of linear in the grid points of the Brioullin zone and linear instead of exponential in the number of electrons and holes. This enables calculations with higher precision on the full Brioullin zone than previously possible. In this paper extensive details for an efficient implementation and the corresponding mathematical background are presented. In particular this includes applications and results for excitons, trions and biexcitons (for monolayer MoS$_2$ as example), going beyond the initial brief introduction. Furthermore strategies for calculating selective excited bound states and tests of common approximations are discussed making use of the high accuracy full Brioullin zone treatment of the tensor network method.