  Linear Paul traps have been used recently to simulate the transverse field
Ising model with long-range spin-spin couplings. We study the intrinsic effects
of phonon creation (from the initial phonon ground state) on the spin-state
probability and spin entanglement for such quantum spin simulators. While it
has often been assumed that phonon effects are benign because they play no role
in the pure Ising model, they can play a significant role when a transverse
field is added to the model. We use a many-body factorization of the quantum
time-evolution operator of the system, adiabatic perturbation theory and exact
numerical integration of the Schr\"odinger equation in a truncated spin-phonon
Hilbert space followed by a tracing out of the phonon degrees of freedom to
study this problem. We find that moderate phonon creation often makes the
probabilities of different spin states behave differently from the static spin
Hamiltonian. In circumstances in which phonon creation is minor, the spin
dynamics state probabilities converge to the static spin Hamiltonian prediction
at the cost of reducing the spin entanglement. We show how phonon creation can
severely impede the observation of kink transitions in frustrated spin systems
when the number of ions increases. Many of our results also have implications
for quantum simulation in a Penning trap.
