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 symmetryprotected topological phases, protected by a [Formula: see text] symmetry, and a topologicallytrivial Z 2symmetrybreaking phase. The symmetryprotected topological phaseslabeled by symmetry fractionalizationbelong to different topological classes, where the complexconjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z 2symmetrybreaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a localorder parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a nonlocal string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the localorder parameter is related to the magnetization exponent [Formula: see text].
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