Two-Level Systems in Metals

As all of you know, the metal electrons constitute a degenerate Fermi system. Its excitation energy ranges from zero to several electron volts. Then, one may ask what is the energy scale of the metal electrons. In the case of static perturbations acting on the electrons, it is of the order of the Fermi energy ɛF. For example, suppose that an impurity potential V(r) is placed in the jellium of the electrons. The energy shift due to this perturbation may be expanded in V: $$\Delta {\rm{E = }}{{\rm{c}}_1}{v_0}\; + \;{c_2}{v_0}^2{\rm{\rho + }}\;{{\rm{c}}_3}{v_0}^3{{\rm{\rho }}^2}\; + \;,$$ where V0 is the matrix element of V(r), which is assumed to be independent of the wave numbers, ρ is the density of the electron states. Thus, the expansion parameter is V0ρ, which is about V0/ɛF. This means that the energy scale of the electrons is the Fermi energy. On the other hand, when the perturbation is dynamical and local, such as the s-d exchange model, excitation modes of low energy come into play and give rise to an infrared divergence. We call this fact the Fermi surface effect.