Spurious Strange Correlators in Symmetry-Protected Topological Phases

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. Focusing on 1D gapped bosonic/spin systems described by matrix product states (MPS), 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 (abbreviated as \emph{irrep}) in the eigenspace corresponding to the entanglment spectrum (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.

💡 Research Summary

The paper investigates a subtle but crucial pitfall in the use of strange correlators (SCs) for diagnosing symmetry‑protected topological (SPT) phases. An SC is defined as the ratio ⟨Ω|O_i O_j|Φ⟩/⟨Ω|Φ⟩, where |Φ⟩ is the target many‑body state and |Ω⟩ is a chosen reference state. The conventional wisdom holds that a non‑trivial SPT yields long‑range (power‑law or constant) SCs, while a trivial SPT gives exponentially decaying SCs. The authors show that this criterion fails if the reference state is not carefully selected: even a completely trivial product state can produce a long‑range SC, leading to false‑positive identification of SPT order.

Focusing on one‑dimensional gapped bosonic or spin systems that admit a matrix‑product‑state (MPS) description, the authors formulate SCs in terms of transfer matrices. They prove two central theorems: (1) if the connected SC (the ordinary SC minus the product of one‑point SCs) remains non‑zero at infinite separation, then almost any pair of local operators will also give a long‑range ordinary SC; (2) the long‑range behavior of the connected SC is entirely governed by the magnitude‑degeneracy of the largest‑modulus eigenvalue of the reference‑state transfer matrix M_ω. If the leading eigenvalue is non‑degenerate (D = 1) all SCs decay exponentially; if it is D‑fold degenerate (D > 1) there always exist operators that generate a long‑range connected SC, and in fact almost every operator pair does so.

In SPT phases the virtual bonds of the MPS carry a projective representation V(g) of the protecting symmetry G. Schur’s lemma guarantees that for non‑trivial projective classes the eigenvalues of M_ω become magnitude‑degenerate, which explains why genuine SPTs produce long‑range SCs. However, the same degeneracy can arise in trivial phases, and the authors identify three distinct mechanisms:

1. High‑dimensional irreducible representations (irreps). When the virtual space contains a non‑abelian irrep of dimension D > 1, the maximal eigenvalue of M_ω can be D‑fold degenerate even though the phase is topologically trivial. The spin‑2 AKLT chain (SO(3) symmetric) provides a concrete example: after grouping two sites to form an SO(3)‑invariant reference state, the transfer matrix becomes proportional to the identity in a three‑dimensional subspace, yielding a spurious SC ⟨S⁺ S⁻⟩_S = −3.

2. Phase mismatch between target and reference representations. If |Φ⟩ and |Ω⟩ transform under different one‑dimensional linear representations of G, the transfer matrix acquires a relative phase factor A(g) = α(g) e^{−iθ(g)}. This phase can cause magnitude‑degeneracy even when both states are symmetric, producing false long‑range SCs.

3. Symmetry‑breaking long‑range order. When the target state itself breaks the protecting symmetry, the transfer matrix naturally develops multiple leading eigenvalues, and the SC inherits the ordinary long‑range order of the broken phase.

To avoid these spurious signals, the authors propose practical guidelines. The reference state must (i) transform under the same one‑dimensional linear representation as the target (eliminating mechanism 2) and (ii) preserve the full global symmetry (eliminating mechanism 3). For mechanism 1, they suggest an algorithmic procedure: decompose the virtual space into trivial (1D) and high‑dimensional sectors, compute the signal vector and noise covariance, and construct an optimal reference vector ω_opt using a Moore‑Penrose inverse that maximizes the signal‑to‑noise ratio. If a single‑site ω_opt does not achieve |λ_triv| > |λ_high|, one can block multiple sites to enlarge the physical Hilbert space until the gap opens.

Overall, the paper provides a rigorous mathematical foundation for when strange correlators faithfully detect SPT order and when they can be misleading. By exposing the three mechanisms that generate spurious long‑range SCs and offering concrete, implementable criteria for choosing reference states, the work equips both theorists and experimentalists with the tools needed to avoid false positives and to reliably identify genuine SPT phases.