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Pauli Spectrum and Stabilizer R\'enyi Entropy in Gapless Symmetry-Protected Topological Phases

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read The Pauli spectrum crosses at SPT transitions, tracking the exchange of dominant string-order correlations even when only a non-invertible duality is available.

desk verdict Solid numerical + duality paper: SRE marks SPT transitions coarsely, Pauli spectrum tracks string-order exchange, including via non-invertible maps for igSPT. read the letter →

arxiv 2607.03762 v1 pith:GKMKFQJ6 submitted 2026-07-04 cond-mat.str-el

Ying-Lin Li , Po-Yao Chang This is my paper · ORCID
classification cond-mat.str-el
keywords stabilizerRényientropyPaulispectrumsymmetry-protectedtopologicalphasesgaplessSPTnon-stabilizernessstringorderparametersnon-invertibledualityquantummagic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks whether quantum magic—how far a many-body ground state sits from the stabilizer manifold—can diagnose symmetry-protected topological phases, including gapless ones. Numerically, the stabilizer Rényi entropy only peaks near a transition and cannot tell the two phases apart. The full Pauli spectrum, however, shows a clean crossing of the dominant Pauli-string weights exactly at the transition; those weights are precisely the non-local string order parameters that distinguish the phases. For ordinary gapped and non-intrinsically gapless SPTs the crossing is produced by a local unitary (Clifford) duality; for intrinsically gapless SPTs the same spectral map is generated by a non-invertible Kramers–Wannier-type transformation. The result is a practical, operator-space diagnostic that works even when conventional local-unitary SPT entanglers do not exist.

What carries the argument

The Pauli spectrum (the probability distribution of squared Pauli-string expectation values) together with the duality maps—local-unitary cluster entanglers for ordinary SPTs and the non-invertible KW/KT composition for intrinsically gapless SPTs—that relabel those strings between dual phases.

What would settle it

Compute the full Pauli spectrum of the intrinsically gapless SPT model for larger sizes or open boundaries and check whether the string-order weights still cross exactly at the self-dual point once parity sectors mix or the ground state leaves the even sector.

Watch

Extended reading notes

Core claim

Across gapped, non-intrinsically gapless, and intrinsically gapless SPT models, the Pauli spectrum of the ground state exhibits a characteristic crossing of dominant string weights at the topological transition. That crossing records the exchange of the non-local string order parameters of the two phases. In the first two classes the exchange follows from an invertible local-unitary duality; in the intrinsically gapless case it is produced by a non-invertible duality. Stabilizer Rényi entropy alone only signals the transition by an extremum and cannot distinguish the phases.

Load-bearing premise

The claim for intrinsically gapless SPTs rests on the ground states remaining in the even-parity sector so the non-invertible duality still induces a clean Pauli-spectrum map.

Editorial extensions

If this is right

  • Stabilizer Rényi entropy can serve as a coarse, numerically accessible locator of SPT phase boundaries even in gapless systems.
  • Full Pauli spectra can diagnose which string-order sector dominates without prior knowledge of the order parameter.
  • Non-invertible dualities leave a readable imprint on the Pauli spectrum of even-parity ground states.
  • Quantum-magic diagnostics extend to intrinsically gapless SPT phases where local-unitary SPT entanglers do not exist.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar Pauli-spectrum crossings may appear at other non-invertible duality points in higher-dimensional topological phases.
  • Perfect Pauli sampling of matrix-product states could become a practical scan for unknown SPT candidates when string order parameters are hard to guess a priori.
  • The parity-sector restriction needed for the non-invertible map suggests open-boundary or projected calculations may be required to make the spectral duality fully unitary.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript studies stabilizer Rényi entropy (SRE) and the Pauli spectrum of ground states in one-dimensional SPT phases, covering the gapped cluster model, the non-intrinsically gapless cluster-Ising model, and an intrinsically gapless SPT (igSPT) model obtained from a Kennedy–Tasaki construction. Using MPS with perfect Pauli sampling (L up to 30–32) and exact small-L spectra, the authors show that M2 exhibits an extremum near symmetry-preserving transitions but cannot distinguish topologically distinct phases. The Pauli spectrum, by contrast, displays a crossing of weights of non-local string operators that track the exchange of string-order sectors. For the cluster-type models the crossing is explained by the Clifford dualities UCZ and PZUCZ; for the igSPT model it is attributed to a non-invertible KW-type map N=KT∘KWσ∘KT, with an obstruction to an invertible realization proved in Appendix B and the spectral map justified inside the even-parity sector in Appendix C.

Significance. The work supplies a concrete, operator-space diagnostic that goes beyond entanglement and beyond the scalar SRE. The numerical evidence is carefully controlled (convergence of sampling and bond dimension in Appendix A), the dualities for the cluster models are standard and cleanly applied, and the igSPT analysis correctly identifies the non-invertible character of the map while still obtaining a well-defined Pauli-spectrum relabeling. Explicit credit is due for the obstruction argument (Appendix B) and the parity-sector check that underpins Appendix C. If the interpretation holds, the Pauli spectrum becomes a practical probe of string-order exchange in both gapped and gapless SPT settings, complementary to conventional string correlators.

major comments (2)
  1. The central claim for the igSPT model (Sec. IV, paragraph preceding Eq. (13) and Appendix C) rests on the ground states remaining even under the relevant parity for all h∈[0,1] and the sizes Lunit=4,8,12,16. The authors state that this has been checked, yet the manuscript does not report the numerical values of the parity expectation or the projector weight. A short table or plot of ⟨P⟩ versus h for those sizes would make the spectral-map argument fully transparent and reproducible.
  2. In Sec. III B and Fig. 2(c) the non-local string weights along the critical line g1=2 are said to exhibit a crossing that signals the exchange of topological sectors of the Ising CFT. Because the correlators decay as power laws (Eq. (8)), the finite-L weights for different string lengths |n-m| are not O(1). The manuscript should clarify how the crossing is identified quantitatively (e.g., by comparing equal-length strings or by finite-size collapse) so that the diagnostic remains unambiguous when long-range order is absent.
minor comments (5)
  1. Abstract and Sec. V: “topological distinct” → “topologically distinct”; “persepctive” → “perspective”.
  2. Fig. 1(b), 2(c), 3(b): the gray background of residual Pauli weights is dense; a log-scale inset or a clearer separation of the highlighted string operators would improve readability.
  3. Eq. (2): the shift -N log 2 is conventional for M2 but should be stated explicitly for general α so that the definition is self-contained.
  4. Appendix A: the error bars in Fig. 4(b) are useful; a one-sentence statement of the maximum relative error used for the phase diagrams would help the reader assess the SRE extrema.
  5. References: a few recent works on SRE in critical and topological systems (e.g., Hoshino et al., arXiv:2503.13599; Nehra et al., arXiv:2512.16673) could be cited for context, though they are not essential to the claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: dualities are external literature facts; Pauli-spectrum crossings follow by construction from those maps acting on even-parity ground states, while SRE is an independent numerical diagnostic.

full rationale

The paper's central claims rest on three independently established dualities (U_CZ for the cluster SPT, P_Z U_CZ for the cluster Ising model, and the composite non-invertible N = KT ◦ KW_σ ◦ KT for the igSPT model). These maps are taken from the literature (cluster entangler, Kennedy-Tasaki, Kramers-Wannier) and re-derived only to the extent needed to exhibit the induced Pauli-string relabeling. Because the dualities send Pauli operators to Pauli operators (Clifford conjugation for the unitary cases; Clifford conjugation inside the even-parity projector for the non-invertible case, Appendix C), the probability distribution p_ψ(σ) of one ground state is simply a permutation of that of its dual. The observed numerical crossing of the particular string operators that serve as non-local order parameters is therefore the direct, expected consequence of those dualities, not a fitted or self-defined prediction. SRE is computed separately by perfect Pauli sampling of MPS ground states and is never used as an input to the duality argument. The only numerical assumption (even parity of the igSPT ground states for the sizes studied) is explicitly checked by the authors and does not close a definitional loop. Minor self-citations of related magic-resource papers are peripheral and non-load-bearing. No uniqueness theorem, ansatz, or fitted parameter is smuggled in to force the result. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard definitions of SRE/Pauli spectrum, known SPT dualities, and the numerical observation that ground states stay even-parity so the non-invertible map acts. No free parameters are fitted to produce the crossings; bond dimensions and sample counts are convergence controls. The only paper-specific modeling choice is the particular symmetry-preserving interpolation of the igSPT Hamiltonian.

free parameters (2)
  • MPS bond dimension χ
    Chosen (χ=50 for cluster models, χ=100 for igSPT) and checked for convergence; not fitted to produce the claimed crossings but controls numerical accuracy.
  • Number of Pauli samples N_samp
    Set to 10^5 after convergence plots; controls statistical error of SRE, not a physical free parameter.
assumptions (4)
  • domain assumption Stabilizer Rényi entropy M_α is a faithful, Clifford-invariant measure of non-stabilizerness for pure states (Leone et al. 2022).
    Used throughout Sec. II and all numerical sections as the definition of magic.
  • domain assumption The cluster entangler U_CZ (and P_Z U_CZ) is a Clifford unitary that maps the cluster Hamiltonian to the trivial paramagnet and swaps the corresponding string operators.
    Standard SPT duality invoked in Sec. III to explain the spectral crossing.
  • domain assumption On a periodic chain the Kramers–Wannier and Kennedy–Tasaki maps are non-invertible (contain parity projectors) yet still induce a Pauli-to-Pauli map inside the even-parity sector.
    Taken from recent non-invertible-symmetry literature and used in Sec. IV and Apps. B–C to explain the igSPT crossing.
  • ad hoc to paper Ground states of the studied igSPT Hamiltonian remain even under the relevant parity for 0≤h≤1 and the system sizes examined.
    Explicitly checked for L_unit=4,8,12,16 (text before Eq. 13 and App. C); required for the non-invertible map to act on the ground state.

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Pith. "Pith review of Pauli Spectrum and Stabilizer R\'enyi Entropy in Gapless Symmetry-Protected Topological Phases." pith.science (2026). https://pith.science/paper/GKMKFQJ6

@misc{pith2026260703762,
  author       = {Pith},
  title        = {Pith review of: Pauli Spectrum and Stabilizer R\'enyi Entropy in Gapless Symmetry-Protected Topological Phases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKMKFQJ6}},
  note         = {Machine review of arXiv:2607.03762}
}
read the original abstract

Quantum entanglement is widely used as a diagnostic of topological phases of matter. Beyond entanglement, non-stabilizerness captures a distinct aspect of quantum many-body states by quantifying their distance from the manifold of stabilizer states. In this work, we study the stabilizer R\'enyi entropy in symmetry protected topological (SPT) phases, including both gapped SPT, non-intrinsically gapless SPT, and intrinsically gapless SPT phases. Under symmetry preserving perturbations, we find numerically that the stabilizer R\'enyi entropy exhibits an extremum near the phase transition. However, the stabilizer R\'enyi entropy alone cannot distinguish different SPT phases. In contrast, the Pauli spectrum reveals a characteristic crossing structure at the transition point. This crossing reflects the exchange of dominant Pauli-string correlations associated with the non-local string order parameters of the two topological distinct phases. For gapped SPT and non-intrinsically gapless SPT phases, the crossing structure can be understood from a local-unitary duality that maps the Pauli spectrum between the two phases. For intrinsically gapless SPT phases, such a local-unitary mapping is absent. Instead, we find that the Pauli spectrum mapping is generated by a non-invertible duality transformation. These results show that although the stabilizer R\'enyi entropy provides only a coarse diagnostic of phase transitions, the Pauli spectrum contains finer information about the exchange of string order sectors. Our findings demonstrate that quantum magic offers a complementary perspective for characterizing both gapped and gapless SPT phases.

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