  We study decoherence in an infinite range Heisenberg model (IRHM) in the two
situations where the system is coupled to a bath of either local optical
phonons or global optical phonons. Using a non-perturbative treatment, we
derive an effective Hamiltonian that is valid in the regime of strong
spin-phonon coupling under non-adiabatic conditions. It is shown that the
effective Hamiltonian commutes with the IRHM and thus has the same eigenstates
as the IRHM. By analyzing the dynamics of the system using a quantum master
equation approach, we show that the quantum states of the IRHM system do not
decohere under Markovian dynamics when the spins interact with local phonons.
For interactions with global phonons, the off-diagonal matrix elements of the
system's reduced density matrix, obtained for non-Markovian dynamics, do not
indicate decoherence only when states with the same $S^z_T$ (i.e., eigenvalue
for the z-component of the total spin) are considered.
