Theory of Magnetoroton Bands and Three-Point Correlations in Fractional Quantum Hall States of Moiré Materials

The recent realization of fractional Chern insulators in moiré materials has introduced a new ingredient, a periodic lattice potential, to the study of quantum Hall phases. We examine how the magnetoroton collective modes of fractional quantum Hall (FQH) states are modified by external periodic potentials, highlighting qualitative differences from the conventional case that are manifested in THz response properties. We show that the excitation-energy shifts are governed by the three-point correlation functions of the FQH ground state and give a universal expansion for these functions in liquid states within the lowest Landau level. This framework is then used to analyze the stability of FQH states at different filling factors, to explore potential connections to the superconductivity observed in twisted MoTe₂, and to construct a family of parent Hamiltonians of zero-energy eigenstates of three-body interaction models.