Raman scattering through a metal-insulator transition

The exact solution for nonresonant $A_{1\mathrm{g}}$ and $B_{1\mathrm{g}}$ Raman scattering is presented for the simplest model that has a correlated metal-insulator transition, the Falicov-Kimball model, by employing dynamical mean-field theory. In the general case, the $A_{1\mathrm{g}}$ response includes nonresonant, resonant, and mixed contributions, and the $B_{1\mathrm{g}}$ response includes nonresonant and resonant contributions (we prove the Shastry-Shraiman relation for the nonresonant $B_{1\mathrm{g}}$ response), while the $B_{2\mathrm{g}}$ response is purely resonant. Three main features are seen in the nonresonant $B_{1\mathrm{g}}$ channel: (i) the rapid appearance of low-energy spectral weight at the expense of higher-energy weight; (b) the frequency range for this low-energy spectral weight is much larger than the onset temperature, where the response first appears; and (iii) the occurrence of an isosbestic point, which is a characteristic frequency where the Raman response is independent of temperature for low temperatures. Vertex corrections renormalize away all of these anomalous features in the nonresonant $A_{1\mathrm{g}}$ channel. The calculated results compare favorably to the Raman response of a number of correlated systems on the insulating side of the quantum-critical point (ranging from Kondo insulators to mixed-valence materials to underdoped high-temperature superconductors). We also show why the nonresonant $B_{1\mathrm{g}}$ Raman response is “universal” on the insulating side of the metal-insulator transition.