Detecting underlying symmetry-protected topological phases via strange correlators and edge engineering

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 (AF) coupled usual 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.

💡 Research Summary

In this paper the authors introduce a comprehensive framework for uncovering symmetry‑protected topological (SPT) phases that are often hidden behind seemingly trivial bulk properties. The approach combines two complementary diagnostics: (i) generalized strange correlators, which probe the bulk SPT order without physically cutting the system, and (ii) engineered edge geometries that reveal characteristic boundary modes in space. As a concrete testbed they study the dimerized spin‑½ Heisenberg model on a square lattice, a system that has long been regarded as a conventional paramagnetic dimer phase with gapped edges. By re‑interpreting the lattice as a set of antiferromagnetically coupled two‑leg ladders, they argue that the dimer phase is in fact a quasi‑one‑dimensional (Q1D) Haldane‑type SPT state, adiabatically connected to the well‑known Haldane phase of a spin‑1 chain.

The authors first generalize the strange correlator CSC(r,r′)=⟨Ω|ϕ(r)ϕ(r′)|Ψ⟩/⟨Ω|Ψ⟩ to accommodate both even and odd topological sectors. They construct two distinct trivial product states |Ω⟩ using valence‑bond (VB) patterns that are topologically even (columnar) or odd (staggered). The local operator ϕ is chosen as a spin‑1 operator built from pairs of neighboring spin‑½’s. This yields two correlators, C_E^SC and C_O^SC, which are evaluated with stochastic series expansion quantum Monte‑Carlo (SSE‑QMC) in the valence‑bond basis. Finite‑size scaling shows that both correlators approach non‑zero constants at the maximal separation L/2, demonstrating long‑range “strange order” characteristic of non‑trivial SPT phases in both sectors.

Next, the authors engineer two types of cuts along the lattice. Cut 1 severs a VB that connects two effective spin‑1 objects, thereby liberating a pair of spin‑½ spinons at the newly created edge. These spinons experience an emergent ferromagnetic interaction, leading to a gapless ferromagnetically ordered edge mode along a zig‑zag boundary. Cut 2, by contrast, does not cut any VB and therefore produces a conventional gapped edge. QMC measurements of edge spin–spin correlations confirm that the cut‑1 edge exhibits distance‑independent correlations, i.e., true long‑range ferromagnetic order, while the cut‑2 edge remains non‑magnetic.

The presence of an ordered edge has profound consequences at the bulk quantum critical point separating the dimer (Q1DH) phase from the antiferromagnetic Néel phase. This transition belongs to the (2+1)‑dimensional O(3) universality class. Classical‑quantum mapping would forbid an “extraordinary” surface critical behavior for a (1+1)‑dimensional surface with continuous symmetry; only an extraordinary‑log scenario is allowed. However, the authors find that the ordered ferromagnetic surface drives a genuine extraordinary surface critical behavior, with surface magnetization and correlation exponents that deviate from both ordinary and extraordinary‑log predictions. This constitutes a clear quantum‑mechanical violation of the standard surface‑criticality classification.

Overall, the paper delivers three major contributions: (1) a generalized strange‑correlator methodology capable of diagnosing SPT order in two‑dimensional Q1D systems; (2) a concrete demonstration that the dimerized Heisenberg model hosts a hidden Q1D Haldane SPT phase, revealed through a ferromagnetic zig‑zag edge; and (3) the discovery that such an edge induces an unexpected extraordinary surface critical regime at the bulk O(3) critical point. The work not only resolves a long‑standing puzzle about the topological nature of the dimer phase but also provides a practical detection protocol that can be applied to a broad class of quantum materials, including Cu‑based compounds (e.g., BaCuSi₂O₆) and engineered quantum simulators.