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arxiv: 2512.06691 · v2 · submitted 2025-12-07 · ❄️ cond-mat.str-el · cond-mat.mes-hall· quant-ph

Spurious Strange Correlators in Symmetry-Protected Topological Phases

Wei-Liang Gao , Jie-Yu Zhang , Zheng-Xin Liu , Peng Ye This is my paper

Pith reviewed 2026-05-17 01:28 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallquant-ph
keywords strange correlatorssymmetry-protected topological phasesmatrix product statestransfer matrix degeneracyfalse positivesSPT diagnosis1D spin systems
0
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The pith

An ill-chosen reference state creates spurious long-range strange correlators in trivial phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that strange correlators, a tool for detecting symmetry-protected topological phases, produce results that depend critically on the reference state chosen for the calculation. In one-dimensional gapped bosonic and spin systems represented by matrix product states, certain reference states generate artificial long-range correlations even when the target phase is trivial. The authors trace these false signals to magnitude degeneracy in the transfer matrix. They classify three concrete mechanisms that produce this degeneracy and supply explicit examples for each. The work therefore supplies a practical warning and classification to prevent misdiagnosis of topological order.

Core claim

An ill-chosen reference state can induce spurious long-range strange correlators in trivial SPT phases. Focusing on 1D gapped bosonic/spin systems described by matrix product states, the origin of these spurious signals is traced to the magnitude-degeneracy of the transfer matrix. Three distinct mechanisms are classified: the presence of high-dimensional irreducible representations in the entanglement spectrum, a phase mismatch in symmetry representations between the target and reference states, and long-range order arising from symmetry breaking.

What carries the argument

Magnitude-degeneracy of the transfer matrix in the matrix product state description, which permits non-decaying strange correlators to appear in trivial phases.

Load-bearing premise

The systems are one-dimensional gapped bosonic or spin chains represented by matrix product states.

What would settle it

A numerical or analytic calculation in a known trivial 1D MPS phase that still shows long-range strange correlators after the reference state is chosen to eliminate all three listed sources of transfer-matrix degeneracy.

Figures

Figures reproduced from arXiv: 2512.06691 by Jie-Yu Zhang, Peng Ye, Wei-Liang Gao, Zheng-Xin Liu.

Figure 1
Figure 1. Figure 1: The formulation of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: AKLT models construction: Two spin-S/2 particles project onto spin-S at each site; adjacent sites form spin singlets. |AKLT⟩ := ⊗ N+1P |α⟩ ⊗ |bond⟩ N ⊗ |β⟩. (C1) Where P represents the projection onto spin-S subspace, ‘bond’ refers to the spin singlet state, and |α⟩ and |β⟩ represent the boundary degrees of freedom. AKLT state is very suitable to written in MPS form: |AKLT⟩ = X σ1,··· ,σN Tr h A [1] σ1 · ·… view at source ↗
read the original 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. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that strange correlators, a common diagnostic for symmetry-protected topological (SPT) phases, can produce spurious long-range signals in trivial phases when an inappropriate reference state is chosen. Restricting to 1D gapped bosonic/spin systems represented by matrix product states (MPS), the authors trace these artifacts to magnitude degeneracy of the transfer matrix and partition the degeneracy into three mechanisms—high-dimensional irreducible representations in the entanglement spectrum, phase mismatch between symmetry representations of target and reference states, and symmetry-breaking long-range order—each illustrated by explicit examples.

Significance. If the central claim holds, the work is significant for the field because strange correlators are widely used to detect SPT order in both numerical and analytical studies; identifying concrete mechanisms that generate false positives and supplying explicit MPS examples provides a practical guideline for reference-state selection that can reduce diagnostic errors in 1D systems.

minor comments (2)
  1. [Abstract] Abstract: 'entanglment spectrum' is a typographical error and should read 'entanglement spectrum'.
  2. The manuscript would benefit from a short table or flowchart summarizing the three mechanisms, their signatures in the transfer-matrix spectrum, and the corresponding reference-state choices that avoid them.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee's summary accurately reflects our central claim that ill-chosen reference states can induce spurious long-range strange correlators in trivial SPT phases via magnitude degeneracy of the MPS transfer matrix, along with the three classified mechanisms supported by explicit examples.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's analysis of spurious strange correlators in trivial SPT phases is self-contained within its stated scope of 1D gapped bosonic/spin systems represented by MPS. It directly traces the issue to magnitude degeneracy of the transfer matrix and partitions this degeneracy into three concrete, independently illustrated mechanisms (high-dimensional irreps in the entanglement spectrum, phase mismatch between symmetry representations, and symmetry-breaking long-range order), each backed by explicit examples. No load-bearing step reduces a claimed result to a fitted input, self-definition, or unverified self-citation chain; the central claim follows from the enumerated mechanisms without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the established MPS representation of 1D gapped systems and the standard role of transfer matrices; no new free parameters or postulated entities are introduced.

axioms (2)
  • domain assumption 1D gapped bosonic and spin systems admit faithful matrix product state representations
    Invoked when tracing spurious signals to transfer-matrix properties.
  • domain assumption Strange correlators require a reference state whose symmetry representation interacts with the target state
    Stated directly in the abstract as the basis for the diagnostic.

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Reference graph

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    19 Suppose the whole symmetry group of the system Hamiltonian isG, and a subgroupK⊂Gis spontaneously broken

    Symmetry Breaking Case In the main text, we claim that if the MPS target state is the symmetric ground state of a symmetry breaking Hamiltonian (this state can be selected by linear combining bases of all ground state subspaces), then we can always choose the reference state |Ω⟩= N i |ω⟩i, such that the largest-magitude eigenvalue of the transfer matrixM ...

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