Quasiparticles and Polaronic Effects in Crystals, Fermi Gasses and Fermi Liquids

We consider the formation of polarons in three different situations: (1) the formation of a polaron in the crystal due to the electron-phonon interaction, (2) the formation of a polaron in the presence of a quiescent Fermi sea, and the (3) the formation of polarons in an interacting Fermi gas, and Fermi liquids, including charge and spin polarization or magnetic polarization. We develop the perturbation diagram approach to the polaron problem and show that the series of maximally crossed and repeated one-phonon (Tamm–Dancoff approximation) diagrams leads to the series of the ladder diagrams which is equivalent to the well-known Bethe–Salpeter equation. We show that the polaron mass cannot exceed the triple mass of an electron and the bound state for the electron-optical phonon interaction exists for any strength of the electron-phonon coupling constant in 3D, and the bound state always exists in 2D case even for the electron-acoustic phonon interaction. We consider the polaron effect in the presence of a quiescent Femi sea and show that the polaron effect leads both to the appearance of the insulating gap and the formation of the quasiparticles with the dispersion law similar to the quasiparticle spectrum in the BCS model of superconductivity. We show that the polarization effect in the case of interacting Fermi gas leads to the polaron caused by the exchange-correlation hole and the electron spectrum again is an insulating with the dispersion law of quasiparticles similar to the BCS model. We show that the interaction potential in polaronic effect in the case of Femi liquid is the same as the Landau interaction function for Fermi liquid. Moreover, we also consider spin and magnetic polaronic effects in Fermi liquids. We show that in the case of repulsive Hubbard model in the spin channel appears the spin gap with the spectrum of quasiparticles again similar to the BCS model of quasiparticles.