  A systematic analysis is performed for quantum phase transitions in a
bond-alternative one-dimensional Ising model with a Dzyaloshinskii-Moriya (DM)
interaction by using the fidelity of ground state wave functions based on the
infinite matrix product states algorithm. For an antiferromagnetic phase, the
fidelity per lattice site exhibits a bifurcation, which shows spontaneous
symmetry breaking in the system. A critical DM interaction is inversely
proportional to an alternating exchange coupling strength for a quantum phase
transition. Further, a finite-entanglement scaling of von Neumann entropy with
respect to truncation dimensions gives a central charge c = 0.5 at the critical
point.
