  We have studied the phase diagram and entanglement of the one dimensional
Ising model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the
quantum renormalization group (QRG) approach to get the stable fixed points,
critical point and the scaling of coupling constants. This model has two
phases, antiferromagnetic and saturated chiral ones. We have shown that the
staggered magnetization is the order parameter of the system and DM interaction
produces the chiral order in both phases. We have also implemented the exact
diagonalization (Lanczos) method to calculate the static structure factors. The
divergence of structure factor at the ordering momentum as the size of systems
goes to infinity defines the critical point of the model. Moreover, we have
analyzed the relevance of the entanglement in the model which allows us to shed
insight on how the critical point is touched as the size of the system becomes
large. Nonanalytic behavior of entanglement and finite size scaling have been
analyzed which is tightly connected to the critical properties of the model. It
is also suggested that a spin-fluid phase has a chiral order in terms of new
spin operators which are defined by a nonlocal transformation.
