  The quantum dynamics of the intrinsic metric profoundly influence the neutral
excitations in the fractional quantum Hall system, as established by Haldane in
2011 \cite{Haldane2011}, and further evidenced by a recent two-photon
experiment \cite{Liang2024}. Despite these advancements, a comprehensive
understanding of the dynamic properties of these excitations, especially at
long wavelengths, continues to elude interest. In this study, we employ
tensor-network methods to investigate the neutral excitations of the Laughlin
and Moore-Read states on an infinite cylinder. This investigation deepens our
understanding of the excitation spectrum in regions where traditional methods
do not work effectively. The spectral functions for both states reveal the
presence of $S=-2$ geometric excitations. For the first time, we unveil the
complex spectra of both neutral fermion and bosonic Girvin-MacDonald-Platzman
modes within the excitation continuum by calculating the three-particle density
response function for the Moore-Read state. Our findings support the hypothesis
of emergent supersymmetry and highlight the potential for detecting neutral
fermions in future experiments.
