Exciton insulator states in materials with "Mexican hat" band structure dispersion and in Graphene

In the paper a theoretical study the both the quantized energies of excitonic states and their wave functions in graphene and in materials with "Mexican hat" band structure dispersion as well as in zinc-blende GaN is presented. An integral two-dimensional Schrödinger equation of the electron-hole pairing for a particles with electron-hole symmetry of reflection is exactly solved. The solutions of Schrödinger equation in momentum space in studied materials by projection the two-dimensional space of momentum on the three-dimensional sphere are found exactly. We analytically solve an integral two-dimensional Schrödinger equation of the electron-hole pairing for particles with electron-hole symmetry of reflection. In studied materials the electron-hole pairing leads to the exciton insulator states. Quantized spectral series and light absorption rates of the excitonic states which distribute in valence cone are found exactly. If the electron and hole are separated, their energy is higher than if they are paired. The particle-hole symmetry of Dirac equation of layered materials allows perfect pairing between electron Fermi sphere and hole Fermi sphere in the valence cone and conduction cone and hence driving the Cooper instability. The solutions of Coulomb problem of electron-hole pair does not depend from a width of band gap of graphene. It means the absolute compliance with the cyclic geometry of diagrams at justification of the equation of motion for a microscopic dipole of graphene where . The absorption spectrums for the zinc-blende GaN/(Al,Ga)N quantum well as well as for the zinc-blende bulk GaN are presented. Comparison with available experimental data shows good agreement.