  The quantum magnetism in a three-singlet model (TSM) with singlet crystalline
electric field (CEF) states interacting on a lattice is investigated, motivated
by its appearance in compounds with 4f^2 and 5f^2 electronic structure.
Contrary to conventional (semi-classical) magnetism there are no preformed
moments above the ordering temperature Tm. They appear spontaneously as induced
or excitonic moments due to singlet-singlet mixing at Tm. In most cases the
transition is of second order, however for large matrix elements between the
excited states it turns into a first order transition at a critical point.
Furthermore we derive the excitonic mode spectrum and its quantum critical soft
mode behaviour which leads to the criticality condition for induced order as
expressed in terms of the control parameters of the TSM and discuss the
distinctions to the previously known two-singlet case. We also derive the
temperature dependence of order parameters for second and first order
transitions and the exciton spectrum in the induced magnetic phase.
