We study the phase diagram and critical properties of quantum Ising chains with longrange ferromagnetic interactions decaying in a power-law fashion with exponent α, in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzerotemperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents α > 1 up to small deviations in some critical exponents. We also address the elusive regime α < 1, where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at α = 0. Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
