A Succinct Diagram Approach to the Linear Optical Conductivity for an Electron-phonon System

It is well known that a proper combination of electron and phonon distributions can be included in the optical transition formalism by using the KC reduction identities and the state-dependent projectors [Kang and Choi, J. Phys. A: Math. Theor. 43, 165203 (2010)]. In the present work, the expression is shown to be represented in a simple diagram system, in which all the diagrams are grouped into topologically distinct ones. The criterion for identifying topologically equivalent diagrams originates from the proper combination of distribution functions. It is further shown that the linewidth function appearing in the energy denominator of the conductivity can be revived simply by using this diagram method, implying that the rules can help reduce the amount of calculation needed to formulate the conductivity.