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The coupling of spins to long-wavelength bosonic modes is a prominent meansto engineer long-range spin-spin interactions, and has been realized in avariety of platforms, such as atoms in optical cavities and trapped ions. Todate, much of the experimental focus has been on the realization of long-rangeIsing models, but generalizations to other spin models are highly desirable. Inthis work, we explore a previously unappreciated connection between therealization of an XY model by off-resonant driving of single sideband of bosonexcitation (i.e.~a single-beam M{\o}lmer-S{\o}rensen scheme) and aboson-mediated Ising simulator in the presence of a transverse field. Inparticular, we show that these two schemes have the same effective Hamiltonianin suitably defined rotating frames, and analyze the emergent effective XY spinmodel through truncated Magnus series and numerical simulations. In addition toXY spin-spin interactions that can be non-perturbatively renormalized from thenaive Ising spin-spin coupling constants, we find an effective transverse fieldthat is dependent on the thermal energy of the bosons, as well as otherspin-boson couplings that cause spin-boson entanglement not to vanish at anytime. In the case of a boson-mediated Ising simulator with transverse field, wediscuss the crossover from transverse-field Ising-like to XY-like spin behavioras a function of field strength.

Boson-mediated quantum spin simulators in transverse fields: XY model and spin-boson entanglement