  We propose a trapped ion scheme en route to realize spin Hamiltonians on a
Kagome lattice which, at low energies, are described by emergent Z2 gauge
fields, and support a topological quantum spin liquid ground state. The
enabling element in our scheme is the hexagonal plaquette spin-spin
interactions in a 2D ion crystal. For this, the phonon-mode spectrum of the
crystal is engineered by standing-wave optical potentials or by using Rydberg
excited ions, thus generating localized phonon-modes around a hexagon of ions
selected out of the entire two-dimensional crystal. These tailored modes can
mediate spin-spin interactions between ion-qubits on a hexagonal plaquette when
subject to state-dependent optical dipole forces. We discuss how these
interactions can be employed to emulate a generalized Balents-Fisher-Girvin
model in minimal instances of one and two plaquettes. This model is an
archetypical Hamiltonian in which gauge fields are the emergent degrees of
freedom on top of the classical ground state manifold. Under realistic
situations, we show the emergence of a discrete Gauss's law as well as the
dynamics of a deconfined charge excitation on a gauge-invariant background
using the two-plaquettes trapped ions spin-system. The proposed scheme in
principle allows further scaling in a future trapped ion quantum simulator, and
we conclude that our work will pave the way towards the simulation of emergent
gauge theories and quantum spin liquids in trapped ion systems.
