Long-range Heisenberg models in quasi-periodically 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 quasi-periodic 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 non-linear 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.