Extension of Frohlich's method to 4-fermion interactions

الأصل · EN

Higher order terms of the transformed electron-phonon Hamiltonian, obtained by performing the Frohlich's transformation, are investigated. The influence of terms discarded by Frohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Frohlich-type transformation is performed, which yields, among others, an effective 4-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the 4-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with interaction coupling given by - DₖF⁶ / ωₖF⁵, where Dₖ is electron-phonon coupling, ωₖ is phonon energy and kF is Fermi momentum. The form of higher order terms of the original Frohlich-transformed He-ph are also found, up to terms proportional to the 6-th power of the coupling, that is up to those, which yield the effective 4-electron interactions.