Two- and three-particle complexes with logarithmic interaction: Compact wave functions for two-dimensional excitons and trions

Assuming a logarithmic interaction between constituent particles, compact and locally accurate wave functions that describe bound states of the two-particle neutral, and three-particle charged complexes in two dimensions are designed. Prime examples of these complexes are excitons and trions that appear in monolayers of transition-metal dichalcogenides (TMDCs). In the case of excitons, these wave functions led to five to six correct decimal digits in the energy and the diamagnetic shifts. In addition, it is demonstrated that they can be used as zero-order approximations to study magnetoexcitons via perturbation theory in powers of the magnetic field strength. For the trion, making a comparison with experimental data for concrete TMDCs, we established that the logarithmic potential leads to binding energies $\ensuremath{\lesssim}30$% greater than experimental ones. Finally, the structure of the wave function at small and large distances was established for excitons whose carriers interact via the Rytova-Keldysh potential.