  Non-local orders, entanglement entropy, and quantum fidelity are investigated
in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya
interaction by employing the infinite matrix product state representation with
the infinite time evolving block decimation method. Directly computing two
distinct types of finite string correlations for very large lattice distances,
in contrast to an extrapolated extreme value for finite size chains, reveals
two topologically ordered phases. As the bond alternation varies, a topological
quantum phase transition with continuously variable critical exponents along
the phase boundary occurs between the two Haldane phases for the
Dzyaloshinskii-Moriya interaction stronger than the Ising interaction, while a
topological quantum crossover between them happens through an intermediate
antiferromagentic phase demonstrated with the quantum fidelity for the
Dzyaloshinskii-Moriya interaction weaker than the Ising interaction. The
critical exponents of the order parameters and the central charges from the
entanglement entropy quantify the universality classes of the phase transition
points. Anisotropic Heisenberg types of spin chains with bond alternations are
finally discussed to share the same criticality.
