  In this paper, we generalize the results of S. Oh (Physics Letters A. 644-647
\textbf{373 }) to Dzyaloshinski-Moriya model under nonuniform external magnetic
field to investigate the relation between entanglement, geometric phase (or
Berry phase) and quantum phase transition. We use quaternionic representation
to relate the geometric phase to the quantum phase transition. For small values
of DM parameter, the Berry phase is more appropriate than the concurrence
measure, while for large values, the concurrence is a good indicator to show
the phase transition. On the other hand, by increasing the DM interaction the
phase transition occurs for large values of anisotropy parameter. In addition,
for small values of magnetic field the concurrence measure is appropriate
indicator for quantum phase transition, but for large values of magnetic field
the Berry phase shows a sharp changes in the phase transition points. The
results show that the Berry phase and concurrence form a complementary system
from phase transition point of view.
