  We investigate the energy dynamics in a generalized compass chain under an
external magnetic field. We show that the energy current operators act on three
contiguous sites in the absence of the magnetic field, and they are
incorporated with inhomogenous Dzyaloshinskii-Moriya interactions in the
presence of the magnetic field. These complex interactions remain the
Hamiltonian be an exactly solvable spin model. We study the effects of the
three-site interactions and the Dzyaloshinskii-Moriya interactions on the
energy spectra and phase diagram. The results have revealed that the energy
current of the pristine quantum compass model is conserved due to the
associated intermediate symmetries, and for other general cases such
characteristic does not exist.
