Bethe-Salpeter equation and exciton spectrum of the transition metal dichalcogenides

Using the Bethe-Salpeter equation we derive the equation for the four component wave function of the interacting electron and hole in the transition metal dichalcogenides. We investigate the solutions of the equation for different potentials and different dimensionless interaction constants. For small interaction constant we obtain in the leading and next-to-leading order in the interaction constant the hamiltonian of the interacting electron and hole. We demonstrate that the obtained hamiltonian does not coincides with naive expansion of the hamiltonian of the two interacting Dirac particles. For the small interacting constant we found the energy of the excitons in the leading and next-to-leading orders in the interaction constant. We also investigate the dependence of the binding energies of the electron and hole on the interaction constant value for different types of the potentials. We found the critical values of the interaction constant for which the lowest binding energy reaches the valence band. We found the critical values of the interaction constant for the Coulomb potential and for the exponentially decreasing potential which does not contain a singularities.