Probing the spin structure of the fractional quantum Hall magnetoroton with polarized Raman scattering

Starting from the Luttinger model for the band structure of GaAs, we derive an effective theory that describes the coupling of the fractional quantum Hall (FQH) system with photons in resonant Raman scattering experiments. Our theory is applicable in the regime when the energy of the photons $\omega_0$ is close to the energy gap $E_G$, but $|\omega_0-E_G|$ is much larger than the energy scales of the quantum Hall problem. In the literature, it is often assumed that Raman scattering measures the dynamic structure factor $S(\omega,\mathbf{k})$ of the FQH. However, in this paper, we find that the light scattering spectrum measured in the experiments are proportional to the spectral densities of a pair of operators which we identified with the spin-2 components of the kinetic part of the stress tensor. In contrast with the dynamic structure factor, these spectral densities do not vanish in the long-wavelength limit $k\to0$. We show that Raman scattering with circularly polarized light can measure the spin of the magnetoroton excitation in the FQH system. We give an explicit expression for the kinetic stress tensor that works on any Landau level and which can be used for numerical calculations of the spectral densities that enter the Raman scattering amplitudes. We propose that Raman scattering provides a way to probe the bulk of the $\nu=5/2$ quantum Hall state to determine its nature.