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The phase diagram of the quantum compass ladder model is investigated through numerical density matrix renormalization group based on infinite matrix product state algorithm and analytic effective perturbation theory. For this model we obtain two symmetry-protected topological phases, protected by a [Formula: see text] symmetry, and a topologically-trivial Z 2-symmetry-breaking phase. The symmetry-protected topological phases-labeled by symmetry fractionalization-belong to different topological classes, where the complex-conjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z 2-symmetry-breaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a non-local string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the local-order parameter is related to the magnetization exponent [Formula: see text].
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