  We describe a family of lattice models that support a new class of quantum
magnetism characterized by correlated spin and bosonic ordering [Phys. Rev.
Lett. 112, 180405 (2014)]. We explore the full phase diagram of the model using
Matrix-Product-State methods. Guided by these numerical results, we describe a
modified variational ansatz to improve our analytic description of the
groundstate at low boson frequencies. Additionally, we introduce an
experimental protocol capable of inferring the low-energy excitations of the
system by means of Fano scattering spectroscopy. Finally, we discuss the
implementation and characterization of this model with current circuit-QED
technology.
