  We examine the impact of long-range Dzyaloshinskii-Moriya (DM) interaction in
the extended XY model on the phase diagram as well as the static and dynamical
properties of quantum and classical correlation functions. It is known that in
the nearest-neighbor XY model with DM interaction, the transition from the
gapless chiral phase to a gapped one occurs when the strengths of the DM
interaction and anisotropy coincide. We exhibit that the critical line gets
modified with the range of interactions which decay according to power-law.
Specifically, instead of being gapless in the presence of a strong DM
interaction, a gapped region emerges which grows with the increase of the
moderate fall-off rate (quasi-long range regime) in the presence of a
transverse magnetic field. The gapless chiral phase can also be separated from
a gapped one by the decay patterns of quantum mutual information and classical
correlation with distant sites of the ground state which are independent of the
fall-off rate in the gapless zone. We observe that the corresponding critical
lines that depend on the fall-off rate can also be determined from the
effective central charge involved in the scaling of entanglement entropy. We
illustrate that in a non-equilibrium setting, the relaxation dynamics of
classical correlation, the decay rate of total correlation, and the growth rate
of entanglement entropy can be employed to uncover whether the evolving
Hamiltonian and the Hamiltonian corresponding to the initial state are gapped
or gapless.
