  We study transverse spin dynamics on a microscopic level by measuring
energy-resolved spin correlations in weakly interacting Fermi gases (WIFGs).
The trapped cloud behaves as a many-body spin-lattice in energy space with
effective long-range interactions, simulating a collective Heisenberg model. We
observe the flow of correlations in energy space in this quasi-continuous
system, revealing the connection between the evolution of the magnetization and
the localization or spread of correlations. This work highlights energy-space
correlation as a new observable in quantum phase transition studies of WIFGs,
decoding system features that are hidden in macroscopic measurements.
