Projection-Reduction Approach to Optical Conductivities for an Electron-Phonon System and Their Diagram Representation

Utilizing state-dependent projection operators and the Kang-Choi reduction identities, we derive the linear, first, and second-order nonlinear optical conductivities for an electron system interacting with phonons. The lineshape functions included in the conductivity tensors satisfy “the population criterion” saying that the Fermi distribution functions for electrons and Planck distribution functions for phonons should be combined in multiplicative forms. The results also contain energy denominator factors enforcing the energy conservation as well as interaction factors describing electron-phonon interaction properly. Therefore, the phonon absorption and emission processes as well as photon absorption and emission processes in all electron transition processes can be presented in an organized manner and the results can be represented in diagrams that can model the quantum dynamics of electrons in a solid. 1. Introduction Studies of the optical transitions for electron systems interacting with a background are useful for examining the electronic properties of solids because the absorption lineshapes are sensitive to the type of scattering mechanisms affecting the carrier behavior. One of the approaches to this problem is the projection method [1–17], which is generally based on the response theory. The projection techniques can be divided into two kinds: single-electron projection techniques [1–9] and many-electron projection techniques [10–17]. A proper many-body theory should satisfy the so-called “many-body criteria,” one of the most difficult being “the population criterion,” meaning that the Fermi distribution functions for electrons and the Planck distribution functions for phonons should be combined in multiplicative forms because the electrons and phonons belong to different categories in quantum-statistical physics. The population factors satisfying the population criterion should be combined in a physically acceptable manner with the energy denominator factor to maintain energy conservation and with the interaction factor to interpret the occurrence of local fluctuations due to electron-phonon interactions in such a way that the transition can take place via implicit states. Theories based on the single-electron projection technique starting from the Kubo formalism have failed to meet this criterion. The many-electron projection techniques are classified into two categories; the first being state-independent or current-dependent [10–13] and the second being state-dependent [14–17]. The conductivity tensors [10, 11] produced using the isolation