Engineering 2D Ising Interactions in a Large (N$>$100) Ensemble of Trapped Ions*

Experimental progress in atomic, molecular, and optical physics has enabled exquisite control over ensembles of cold trapped ions. We have recently engineered long-range Ising interactions in a two-dimensional, 1-mK Coulomb crystal of hundreds of $^{9}$Be$^{+}$ ions confined within a Penning trap. Interactions between the $^{9}$Be$^{+}$ valence spins are mediated via spin-dependent optical dipole forces (ODFs) coupling to transverse motional modes of the planar crystal. A continuous range of inverse power-law spin-spin interactions from infinite (1/r$^{0})$ to dipolar (1/r$^{3})$ are accessible by varying the ODF drive frequency relative to the transverse modes. The ions naturally form a triangular lattice structure within the planar array, allowing for simulation of spin frustration using our generated antiferromagnetic couplings. We report progress toward simulating the ferromagnetic/antiferromagnetic transverse quantum Ising Hamiltonians in this large ensemble. We also report spectroscopy, thermometry, and sensitive displacement detection ($\sim $100 pm) via entanglement of valence spin and drumhead oscillations.