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arxiv: 2605.10320 · v1 · submitted 2026-05-11 · ❄️ cond-mat.other · cond-mat.dis-nn· cond-mat.stat-mech· cond-mat.str-el· quant-ph

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· Lean Theorem

Beyond Topological Invariants: Order Parameters from Dominant Fock-state Patterns

Pedro D. Sacramento, Tsz Hin Hui, Wing Chi Yu, Xiaodan Xia

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:20 UTC · model grok-4.3

classification ❄️ cond-mat.other cond-mat.dis-nncond-mat.stat-mechcond-mat.str-elquant-ph
keywords order parametersFock statestopological phasesSu-Schrieffer-Heeger modelBerezinskii-Kosterlitz-Thouless transitionmany-body systemsphase diagramsdisordered systems
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0 comments X

The pith

Patterns extracted from dominant Fock states yield order parameters that split each topological sector into two distinct phases in the extended Su-Schrieffer-Heeger model.

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

The paper introduces a general method to build order parameters by identifying recurring patterns among the most probable configurations in the ground-state wavefunction. In the extended Su-Schrieffer-Heeger chain, these real-space order parameters show that the standard winding number misses an internal division, so that each topological sector actually contains two separate phases. The same construction quantifies how deeply a system sits inside a phase and continues to work when disorder is added or when the model includes interactions, as demonstrated by its ability to locate the Berezinskii-Kosterlitz-Thouless transition in the spin-1/2 XXZ chain. A reader would care because the approach supplies a practical, local diagnostic that applies across both interacting and disordered many-body systems without needing extra fitting parameters.

Core claim

Order parameters constructed from generic patterns in the dominant Fock states of the many-body ground state provide a refined classification of phases. In the extended Su-Schrieffer-Heeger model the standard winding number is insufficient to distinguish all phases, and the new order parameters reveal that each topological sector splits into two distinct phases. The same parameters quantify the depth of a phase, remain robust under disorder, and serve as a finite-size diagnostic for the Berezinskii-Kosterlitz-Thouless transition in the interacting XXZ model.

What carries the argument

Extraction of generic patterns from the dominant Fock states of the ground-state wavefunction, which defines real-space order parameters that capture phase distinctions beyond non-local topological invariants.

If this is right

  • Each topological sector in the extended Su-Schrieffer-Heeger model splits into two phases that the new order parameters distinguish.
  • The order parameters quantify the depth of a phase in addition to locating its boundaries.
  • The construction remains effective for locating transitions in the presence of disorder.
  • The same order parameters provide a practical finite-size indicator for the Berezinskii-Kosterlitz-Thouless transition in the spin-1/2 XXZ chain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may supply a route to phase classification in models where standard topological invariants are known to be incomplete.
  • It could be tested on other interacting lattice models to check whether dominant-Fock-state patterns continue to resolve sub-structures missed by winding numbers or Chern numbers.
  • Because the construction is local and real-space based, it may extend naturally to open or driven systems where non-local invariants become ill-defined.

Load-bearing premise

That recurring patterns among the highest-probability Fock states of the ground-state wavefunction always supply a robust, system-independent order parameter that captures the essential physics.

What would settle it

Numerical computation of the proposed order parameters on finite extended Su-Schrieffer-Heeger chains across the parameter space; if the parameters do not detect additional transitions inside the sectors already separated by the winding number, or if they lose their ability to locate the BKT transition in the XXZ model under increasing system size, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.10320 by Pedro D. Sacramento, Tsz Hin Hui, Wing Chi Yu, Xiaodan Xia.

Figure 1
Figure 1. Figure 1: (a) shows the GS fidelity F( ⃗λ, ⃗λ ′ ) = [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The GS expectation values of the OPs in Eq. (9-12) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) shows our OPs in Eq. (9-11) for n = N 2 −1 as a function of D. The ensemble averages over 400 dis￾order realizations, for each D. The crossings between our OPs match qualitatively well with the transitions re￾ported in Ref. [37], which are determined from only 50 disorder realizations using the non-commutative winding number [38]. We remark that such consistency can only be reached in large systems bey… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The four left columns show the ground-state expectation values of the OPs in Eq. (9-12) with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The OPs in Eqs. (24-27) constructed from the scheme in Ref.[4, 6] as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The OPs in Eq. (13) as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We introduce a general scheme for constructing order parameters (OPs) by extracting generic patterns from the dominant Fock states of many-body ground states. While topological phases are traditionally characterized by non-local invariants, we demonstrate that our real-space OPs provide a more refined classification. In the extended Su-Schrieffer-Heeger model, we show that the standard winding number is insufficient to fully distinguish all phases; our OPs reveal a hidden sub-structure where each topological sector splits into two distinct phases. Beyond identifying the phase boundaries, these OPs quantify the depth of a phase, and remain robust in characterizing transitions in disordered systems. Furthermore, our approach provides a practical finite-size diagnostic for the Berezinskii-Kosterlitz-Thouless transition in the interacting spin-1/2 XXZ model. The presented framework offers a broadly applicable tool for uncovering the phase diagrams of diverse interacting and non-interacting quantum many-body systems.

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

2 major / 2 minor

Summary. The manuscript introduces a general scheme to construct order parameters by extracting generic patterns from the dominant Fock states of many-body ground-state wavefunctions. In the extended Su-Schrieffer-Heeger model, these order parameters are shown to split each topological sector (labeled by the standard winding number) into two distinct sub-phases. The approach is further applied to disordered systems, where it remains robust, and is used as a finite-size diagnostic for the Berezinskii-Kosterlitz-Thouless transition in the interacting spin-1/2 XXZ chain.

Significance. If the sub-phases identified by the Fock-state patterns correspond to physically distinct regimes, the method would supply a practical, real-space tool for refining phase diagrams in interacting and disordered many-body systems where conventional topological invariants are insufficient. The finite-size BKT diagnostic is a concrete strength. However, the significance is limited by the absence of cross-checks against independent observables, which leaves open whether the reported sub-structure reflects new physics or merely re-labels wavefunction features within the same phase.

major comments (2)
  1. [§4] §4 (extended SSH model results): The central claim that each winding-number sector splits into two distinct phases rests exclusively on changes in the identity and pattern of dominant Fock components. No data are presented demonstrating that any measurable quantity—many-body gap, dimerization order parameter, entanglement spectrum, or transport—exhibits a discontinuity, singularity, or qualitative change across the proposed intra-sector boundaries. Without such evidence the sub-phases cannot be distinguished from intra-phase variations.
  2. [§5] §5 (XXZ model): While the order parameters are asserted to provide a practical finite-size diagnostic for the BKT transition, the manuscript does not compare their scaling or crossing points against established diagnostics such as the spin stiffness or the entanglement entropy scaling. This comparison is required to establish that the Fock-pattern diagnostic adds information beyond existing methods.
minor comments (2)
  1. [§2] The notation for the order parameters (e.g., how the pattern-extraction threshold is defined) should be made fully explicit in §2 so that the construction can be reproduced without reference to the specific numerical implementation.
  2. [Figures 3–5] Figure captions for the phase diagrams should explicitly state the system sizes used and whether the intra-sector lines remain stable under finite-size scaling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address each major comment below and indicate the revisions we will make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: §4 (extended SSH model results): The central claim that each winding-number sector splits into two distinct phases rests exclusively on changes in the identity and pattern of dominant Fock components. No data are presented demonstrating that any measurable quantity—many-body gap, dimerization order parameter, entanglement spectrum, or transport—exhibits a discontinuity, singularity, or qualitative change across the proposed intra-sector boundaries. Without such evidence the sub-phases cannot be distinguished from intra-phase variations.

    Authors: We agree that additional cross-checks with conventional observables would strengthen the physical interpretation of the sub-phases. The order parameters we introduce are themselves constructed from the dominant Fock-state occupations, which are directly accessible in the ground-state wave function. In the revised manuscript we will add a discussion relating the intra-sector boundaries to changes in real-space dimerization correlations and the many-body gap, demonstrating that the Fock-pattern transitions coincide with qualitative reorganizations of the wave function. This revision will clarify that the sub-structure is not merely a re-labeling but reflects distinct ordering patterns within each topological sector, while preserving the method's utility in regimes where standard invariants are insufficient. revision: partial

  2. Referee: §5 (XXZ model): While the order parameters are asserted to provide a practical finite-size diagnostic for the BKT transition, the manuscript does not compare their scaling or crossing points against established diagnostics such as the spin stiffness or the entanglement entropy scaling. This comparison is required to establish that the Fock-pattern diagnostic adds information beyond existing methods.

    Authors: We appreciate the suggestion to benchmark against established diagnostics. In the revised manuscript we will include a direct comparison of the finite-size scaling behavior and crossing points of our Fock-state order parameters with those obtained from the spin stiffness and the entanglement entropy scaling in the XXZ chain. This addition will show that the Fock-pattern diagnostic provides a computationally lightweight alternative that relies only on the dominant components of the ground-state wave function, offering complementary information especially useful for larger systems or when full entanglement calculations become prohibitive. revision: yes

Circularity Check

0 steps flagged

No significant circularity; OPs constructed directly from ground-state Fock patterns without reduction to fitted inputs or self-citations

full rationale

The paper's central construction extracts generic patterns from the dominant Fock states of the many-body ground-state wavefunction to define real-space order parameters. This starts from the wavefunction itself and does not presuppose the phase labels or topological invariants it is later compared against. No equations reduce the new OPs to quantities defined by the authors' prior work, and the claims about sub-structure within winding-number sectors and the BKT diagnostic follow from direct application to the extended SSH and XXZ models. The derivation remains self-contained; any self-citations in the full text are not load-bearing for the core scheme.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The scheme rests on the assumption that dominant Fock states encode sufficient information for order-parameter construction; no explicit free parameters or invented entities are mentioned in the abstract, but the generality claim implies domain assumptions about ground-state dominance that are not derived here.

axioms (1)
  • domain assumption Dominant Fock states of the many-body ground state contain generic patterns sufficient to define order parameters that classify phases more finely than topological invariants.
    Invoked in the opening sentence of the abstract as the basis for the general scheme.

pith-pipeline@v0.9.0 · 5477 in / 1340 out tokens · 28093 ms · 2026-05-12T03:20:34.411017+00:00 · methodology

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    also has a counterpart phase, namelyW p 1 (W p 0 ) [see the rightmost plot in Fig. 5(b)]. Evidence 3: Opposite behavior of physical quantities betweenW e m andW p m phases.—In addition, theW p m and W e m phases exhibit distinct behavior in real-space, reinforcing the necessity for classifying them as different phases. Consider the OPs found in previous w...

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    Similarly,⟨O W2 ⟩and⟨O W−1 ⟩are always zero in the whole region

    phase. Similarly,⟨O W2 ⟩and⟨O W−1 ⟩are always zero in the whole region. Furthermore, when one enters theW e 1 (W e

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    e” carries the meaning of electron- like, while “p

    phase fromW e 0 (W e 1),⟨O W0 ⟩(⟨O W1 ⟩) drops from a finite value to zero abruptly. Such a sharp change not only appears in this original SSH model, but also in the extended SSH model. It also holds true for the generalized version of the SSH model, where an arbitrary number of further neighbor hoppings and winding numbers are present (Fig. 7). Nomenclat...

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