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arXiv:2512.06691 · Chinese Physics Letters

Spurious Strange Correlators in Symmetry-Protected Topological Phases

Wei-Liang Gao, Jie-Yu Zhang, Zheng-Xin Liu, Peng Ye

cond-mat.str-elcond-mat.mes-hallquant-phSubmitted Dec 7, 2025• Announced Dec 9, 2025• 0 citations
v1 · Dec 7, 2025v2 · Feb 6, 2026
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Abstract

Strange correlator is a powerful tool widely used in detecting symmetry-protected topological (SPT) phases. However, the result of strange correlator crucially relies on the adoption of the reference state. In this work, we report that an ill-chosen reference state can induce spurious long-range strange correlators in trivial SPT phases, leading to false positives in SPT diagnosis. Using matrix product state (MPS) representation, we trace the origin of these spurious signals in trivial SPT phases to the magnitude-degeneracy of the transfer matrix. We systematically classify three distinct mechanisms responsible for such degeneracy, each substantiated by concrete examples: (1) the presence of high-dimensional irreducible representations in the entanglement space; (2) a phase mismatch in symmetry representations between the target and reference states; and (3) long-range order arising from symmetry breaking. Our findings clarify the importance of the choice of proper reference states, providing a guideline to avoid pitfalls and correctly identify SPT order using strange correlators.

Journal reference
Chinese Physics Letters 43, 040703 (2026); https://cpl.iphy.ac.cn/article/cstr/32039.14.0256-307X.43.4.040703
DOI
10.1088/0256-307X/43/4/040703

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Strange correlator is a powerful tool widely used in detecting symmetry-protected topological (SPT) phases. However, the result of strange correlator crucially relies on the adoption of the reference state. In this work, we report that an ill-chosen reference state can induce spurious long-range strange correlators in trivial SPT phases, leading to false positives in SPT diagnosis. Using matrix product state (MPS) representation, we trace the origin of these spurious signals in trivial SPT phases to the magnitude-degeneracy of the transfer matrix. We systematically classify three distinct mechanisms responsible for such degeneracy, each substantiated by concrete examples: (1) the presence of high-dimensional irreducible representations in the entanglement space; (2) a phase mismatch in symmetry representations between the target and reference states; and (3) long-range order arising from symmetry breaking. Our findings clarify the importance of the choice of proper reference states, providing a guideline to avoid pitfalls and correctly identify SPT order using strange correlators.
``Strange'' correlators provide a tool to detect topological phases arising in many-body models by computing the matrix elements of suitably defined two-point correlations between the states under investigation and trivial reference states. Their effectiveness depends on the choice of the adopted operators. In this paper, we give a systematic procedure for this choice, discussing the advantages of choosing operators using the bulk-boundary correspondence of the systems under scrutiny. Via the scaling exponents, we directly relate the algebraic decay of the strange correlators with the scaling dimensions of gapless edge modes operators. We begin our analysis with lattice models hosting symmetry-protected topological phases and we analyze the sums of the strange correlators, pointing out that integrating their moduli substantially reduces cancellations and finite-size effects. We also analyze instances of systems hosting intrinsic topological order, as well as strange correlators between states with different nontrivial topologies. Our results for both translational and nontranslational invariant cases, and in the presence of on-site disorder and long-range couplings, extend the validity of the strange correlator approach for the diagnosis of topological phases of matter and indicate a general procedure for their optimal choice.
We investigate mixed states exhibiting nontrivial topological features, focusing on symmetry-protected topological (SPT) phases under various types of decoherence. Our findings demonstrate that these systems can retain topological information from the SPT ground state despite decoherence. In the ''doubled Hilbert space,'' we define symmetry-protected topological ensembles (SPT ensembles) and examine boundary anomalies in this space. We generalize the concept of the strange correlator, initially used to diagnose SPT ground states, to identify anomalies in mixed-state density matrices. Through exact calculations of stabilizer Hamiltonians and field theory evaluations, we show that nontrivial features of SPT states persist in two types of strange correlators: type-I and type-II. The type-I strange correlator reveals SPT information that can be efficiently detected and used experimentally, such as in preparing long-range entangled states. The type-II strange correlator encodes the full topological response of the decohered mixed state, reflecting the SPT state's pre-decoherence presence. Our work offers a unified framework for understanding decohered SPT phases from an information-theoretic perspective.
Strange correlators are useful tools for diagnosing symmetry-protected topological states from their bulk wave functions. We study strange correlators for one-dimensional fermionic symmetry-protected topological states using fixed-point wave functions, and show that a combination of strange correlators constructed using fermion annihilation operators and bosonic order parameters can fully diagnose the classification from such wave functions. By converting the strange correlator to a correlation in a one-dimensional statistical problem described by transfer matrices, we show that when the wave function is topologically nontrivial, the corresponding strange correlator exhibits a long-range-order behavior, which can be analyzed from the symmetry properties of the transfer matrices constructed from the fixed-point wave function. Our general argument is demonstrated by several concrete examples.
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.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
Entanglement asymmetry, introduced by F. Ares, S. Murciano and P. Calabrese, provides a density-matrix diagnostic of symmetry breaking and successfully captures the Landau data associated with a broken symmetry pattern. However, it is by now well established that gapped quantum many-body systems can exhibit phases which are not characterized solely by Landau symmetry breaking. A fundamental example is a symmetry-protected topological (SPT) phase, and the ordinary definition of entanglement asymmetry is insensitive to this topological information. In this work we introduce a refined quantity, which we call the twisted entropic order parameter, designed to detect SPT phases from reduced density matrices, particularly focusing on one-dimensional bosonic systems. The key ingredient in our construction is an ancilla degrees of freedom that coherently records the untwisted state and the twisted state associated to a one-ended topological defect of unbroken symmetry, so that the enlarged density matrix retains the charge carried by the defect endpoint. We demonstrate our proposal in concrete lattice models and further generalize it beyond ordinary group symmetries, establishing its ability to diagnose SPT phases. This provides a first step toward a unified entanglement-asymmetry framework for diagnosing quantum phases of matter.