Extension of the Frohlich method to 4-fermion interactions

Higher-order terms of the transformed electron-phonon Hamiltonian He-ph, obtained by performing the Frohlich 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 four-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the four-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with the interaction coupling given by -D-kF(6)/omega(5)(kF), where D-k is electron-phonon coupling, omega(k) is the phonon energy, and k(F) is the Fermi momentum. The form of higher-order terms of the original Frohlich-transformed He-ph is also found, up to terms proportional to the sixth power of the coupling-that is, up to those which yield the effective four-electron interactions.