Symmetries of a mean-field spin model

Thermodynamic limit evolution of a closed quantum Heisenberg-type spin model with mean-field interactions is characterized by classifying all the symmetries of the equations of motion. It is shown that parameters of the model induce a structure in the Hilbert space by partitioning it into invariant subspaces, decoupled by the underlying Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. All possible partitions are classified in terms of a $3\times3$ matrix of effective, thermodynamic limit coupling constants. It is found that there are either 1, 2, 4, or O(N) invariant subspaces. The BBGKY hierarchy decouples into the corresponding number of anti-Hermitian operators on each subspace. These findings imply that equilibration and the equilibrium in this model depend on the initial conditions.