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

Bond-Alternation-Induced Topological Quantum Gaussian Transition and  Topological Quantum Crossover in Ising Chains with the Dzyaloshinskii-Moriya  Interaction