A method for obtaining approximate solutions to the standard shell-model problem, as applied to collective states, is described which exploits two features of such problems. One is the completely general mathematical feature that the Hamiltonian is a simple polynomial in the generators of a Lie group. The second is the widespread physical feature that when the matrix element of a product of generators, taken between suitable low-lying states, is evaluated by sum rule techniques, the intermediate state sums are exhausted by a few terms (collective states). Such sum rule techniques can be applied to the Lie algebra, the Casimir operators of the algebra (whose eigenvalues specify the representations appropriate to the fermion system under study), to the condition that the Hamiltonian be diagonal in the space of its eigenstates, and to the equations of motion for the generators taken between such states. In this paper the technique is applied to an exactly soluble model introduced by Meshkov, Lipkin, and Glick. It is seen that the sum rules can be cut at various levels of approximation and with sufficient care in the way this is done that the resulting systems of non-linear equations among physical matrix elements constitutes a complete (or even overcomplete but reasonably self-consistent) dynamical scheme for the computation of a specified set of physical quantities. Even the crudest such approximation gives qualitatively reasonable results right through the phase transition exhibited by the model. The best approximation carried out yields all the physical quantities associated with the lowest four eigenstates of the system, i.e. up to three phonon-states in the vibrational regime. Essentially exact results for all coupling constants are obtained for the quantities normally calculated by the random phase approximation and good values for the remaining quantities. The strength of the method is seen to be that for systems with collective modes, the complexity of its application depends only on the structure of the group studied and on the number of physical states taken into account and is independent of the total number of particles treated. © 1969.
