We explore the Mott insulating state of single-band bosonic pairing Hamiltonians using analytical approaches and large scale density matrix renormalization group calculations. We focus on the second Mott lobe which exhibits a magnetic quantum phase transition in the Ising universality class. We use this feature to discuss the behavior of a range of physical observables within the framework of the 1D quantum Ising model and the strongly anisotropic Heisenberg model. This includes the properties of local expectation values and correlation functions both at and away from criticality. Depending on the microscopic interactions it is possible to achieve either antiferromagnetic or ferromagnetic exchange interactions and we highlight the possibility of observing the E8 mass spectrum for the critical Ising model in a longitudinal magnetic field.
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