Long-range Heisenberg models in quasiperiodically driven crystals of trapped ions

We introduce a theoretical scheme for the analog quantum simulation of long-range XYZ models using current trapped-ion technology. In order to achieve fully tunable Heisenberg-type interactions, our proposal requires a state-dependent dipole force along a single vibrational axis, together with a combination of standard resonant and detuned carrier drivings. We discuss how this quantum simulator could explore the effect of long-range interactions on the phase diagram by combining an adiabatic protocol with the quasiperiodic drivings, and test the validity of our scheme numerically. At the isotropic Heisenberg point, we show that the long-range Hamiltonian can be mapped onto a nonlinear sigma model with a topological term that is responsible for its low-energy properties, and we benchmark our predictions with matrix-product-state numerical simulations.