  A recently introduced class of quantum spherical spin models is considered in
detail. Since the spherical constraint already contains a kinetic part, the
Hamiltonian need not have kinetic term. As a consequence, situations with or
without momenta in the Hamiltonian can be described, which may lead to
different symmetry classes. Two models that show this difference are analyzed.
Both models are exactly solvable and their phase diagram is analyzed. A
transversal external field leads to a phase transition line that ends in a
quantum critical point. The two considered symmetries of the Hamiltonian
considered give different critical phenomena in the quantum critical region.
The model with momenta is argued to be analog to the large-N limit of an SU(N)
Heisenberg ferromagnet, and the model without momenta shares the critical
phenomena of an SU(N) Heisenberg antiferromagnet.
