A system of trapped ions under the action of off--resonant standing--waves
can be used to simulate a variety of quantum spin models. In this work, we
describe theoretically quantum phases that can be observed in the simplest
realization of this idea: quantum Ising and XY models. Our numerical
calculations with the Density Matrix Renormalization Group method show that
experiments with ion traps should allow one to access general properties of
quantum critical systems. On the other hand, ion trap quantum spin models show
a few novel features due to the peculiarities of induced effective spin--spin
interactions which lead to interesting effects like long--range quantum
correlations and the coexistence of different spin phases.Comment: 11 pages, 13 figure
