Sum-rules for Raman scattering off strongly correlated electron systems

We investigate sum-rules applying to the Raman intensity in a strongly correlated system close to the Mott transition. Quite generally, it can be shown that, provided the frequency integration is performed up to a cutoff smaller than the upper Hubbard band, a sum-rule applies to the non-resonant Raman signal of a doped Mott insulator, resulting in an integrated intensity which is proportional to the doping level. We provide a detailed derivation of this sum-rule for the t-J model, for which the frequency cutoff can be taken to infinity and an unrestricted sum-rule applies. A quantitative analysis of the sum-rule is also presented for the d-wave superconducting phase of the t-J model, using slave boson methods. The case of the Hubbard model is studied in the framework of dynamical mean-field theory, with special attention to the cutoff dependence of the restricted sum-rule, and also to the intermediate coupling regime. The sum-rule investigated here is shown to be consistent with recent experimental data on cuprate superconductors, reporting measurements of Raman scattering intensities on an absolute scale.