Quantum spherical spin models

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 $\mathrm{SU}\left(\mathcal{N}\right)$ Heisenberg ferromagnet, and the model without momenta shares the critical phenomena of an $\mathrm{SU}\left(\mathcal{N}\right)$ Heisenberg antiferromagnet.