The vast majority of symmetry-protected topological (SPT) states are difficult to detect, which often leads to their misidentification as ordinary or topologically trivial phases. In this work, we propose a general framework for detecting these hidden topological states. We distinguish the ordinary matter state from the topological phase by exploiting the boundary effects in space (via surface behaviors on engineered edge) and time (via strange correlators) according to the principle of bulk-edge correspondence. As a concrete example, we study the dimerized spin-1/2 Heisenberg model on a square lattice using quantum Monte Carlo simulations, focusing on its paramagnetic dimer phase and edge states. The dimer phase has been widely regarded as topologically trivial due to its gapped edge state on conventional edges. However, the model can also be viewed as two-dimensional antiferromagnetically coupled typical ladders, which suggests an SPT state adiabatically connected to the one-dimensional Haldane phase. We resolve this puzzle and demonstrate that the dimer phase is indeed a quasi-one-dimensional SPT state by measuring generalized strange correlators introduced in this work and by showing that the nontrivial gapless edge state on a zigzag edge is ferromagnetically ordered, resulting from effective ferromagnetic interactions between degenerate spinons liberated on each side of the cut. Furthermore, we show that the ordered edge state gives rise to an extraordinary surface critical behavior at the ($2+1$)-dimensional O(3) bulk critical points of the model, which contradicts theoretical predictions based on classical-quantum mapping. Overall, we establish a standard detection method for uncovering topological phases that masquerade as ordinary states of matter.

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• Quantum criticality
• Quantum phase transitions
• Symmetry protected topological states
• Magnetic systems
• Quantum Monte Carlo