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arxiv: 2605.12184 · v2 · pith:BRAU6HZGnew · submitted 2026-05-12 · 🧮 math-ph · cond-mat.str-el· math.MP· quant-ph

Local Topological Quantum Order and Spectral Gap Stability for the AKLT Models on the Hexagonal and Lieb Lattices

Amanda Young , Bruno Nachtergaele , Thomas Jackson This is my paper

Pith reviewed 2026-05-19 17:41 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.str-elmath.MPquant-ph
keywords AKLT modellocal topological quantum orderspectral gap stabilityhexagonal latticeLieb latticepolymer representationground state
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The pith

Ground states of AKLT models on hexagonal and Lieb lattices satisfy local topological quantum order with exponential boundary decay.

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

The paper proves that the ground states of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order condition. This follows from showing that finite-volume ground-state expectations approximate those of a unique infinite-volume ground state, with the error decaying exponentially in the distance from the observable support to the volume boundary. A sympathetic reader would care because the indistinguishability directly implies stability of the spectral gap above the ground state under small perturbations that decay sufficiently fast at long range. The result is obtained through a detailed analysis of the polymer representation of the ground states.

Core claim

The finite volume ground states are indistinguishable from a unique infinite volume ground state, with any finite volume ground state expectation well approximated by the infinite volume state and the error decaying at a uniform exponential rate in the distance between the support of the observable and the boundary of the finite volume. This holds for the AKLT models on the hexagonal and Lieb lattices and is established by modifying the polymer representation of the ground state. As a corollary, the spectral gap above the ground state remains stable under general small perturbations of sufficient decay.

What carries the argument

Modified polymer representation of the ground state, adapted from the 1988 construction to prove strong ground-state indistinguishability on these two lattices.

If this is right

  • The spectral gap above the ground state stays positive under sufficiently small and decaying perturbations.
  • Finite-volume ground-state expectations approximate the infinite-volume state at an exponential rate that is uniform across volumes.
  • These AKLT models exhibit local topological quantum order on the hexagonal and Lieb lattices.

Where Pith is reading between the lines

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

  • The same polymer-expansion modifications could plausibly be carried out for AKLT models on other planar lattices with similar coordination numbers.
  • Gap stability under local perturbations would support the use of these states as building blocks for phases that remain gapped when weakly coupled to an environment.

Load-bearing premise

The polymer representation of the ground state admits the necessary modifications to establish the strong form of ground-state indistinguishability required for local topological quantum order on these specific lattices.

What would settle it

A direct computation showing that the difference between finite-volume and infinite-volume ground-state expectations decays slower than exponentially with distance to the boundary, or an explicit small decaying perturbation that closes the spectral gap, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.12184 by Amanda Young, Bruno Nachtergaele, Thomas Jackson.

Figure 1
Figure 1. Figure 1: Illustration of h˜0 ⊂ H (left), Λ2 ⊂ H (middle), and Λ(1) 2 ⊂ Γ (1) with Γ = Z 2 (right). The red vertices in the middle and right figures indicate the boundary ∂Λ2, while the blue edges and vertices in the middle and right figures denote γ (2,m) is the smallest polymer containing ˚Λ (m) 2 . ground state ω (m) : AΓ(m) → C. Moreover, there are NΓ, KΓ ∈ N so that if N > K +NΓ > KΓ+NΓ, then (1.5) [PITH_FULL_… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of the polymer types comprising PN,K and P bulk N,K. Namely, γ1, γ2 ∈ Pbulk N,K, and γ1, γ2, γ3, γ4 ∈ PN,K . For any 0 ≤ K ≤ N − 2 let LN,K = {γ ∈ LΓ : γ ⊆ ΛN,K} WN,K = {γ ∈ WΓ : γ ⊆ ΛN,K ∧ ep(γ) ⊆ ∂ΛN,K} Wbulk N,K = {γ ∈ WΓ : γ ⊆ ΛN,K ∧ ep(γ) ⊆ ∂˚ΛK} where we use the convention that ∂˚ΛK = ∅ if K = 0, implying that Wbulk N,0 = ∅ and ∂ΛN,0 = ∂ΛN . Then, given any A ∈ A˚ΛK , the polymer sets… view at source ↗
Figure 4
Figure 4. Figure 4: Three walks γ ∈ WN,K with the boundary ∂ΛN,K colored to show the bipartition of Λ. The walk has even length if and only if the endpoints have the same coloring. This illustrates that the parity of |γ| only depends on (1) if γ crosses a corner of ΛN,K and (2) whether or not the γ connects the inner and outer boundaries, ∂ΛN and ∂˚ΛK. R(l) for even l with 4 ≤ l ≤ 20 l R(l) 4 1 6 1 8 4 10 9 12 26 14 75 16 215… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of the regions Ci(δ), Co(δ), and B(δ) and the decompo￾sition of a path γ into pieces γ = λ1 ∨λ˜ 1 ∨. . . λ˜ 3 ∨λ4. Note that λi has an excursion into B(δ) of with length less than ℓ0. possible γ ′ that intersect γ can then be counted from considering the edges γ can share with the translations and reflections of this γ ′ . Now, fix 8 ≤ l ′ ≤ 20 and γ ∈ Pl with l > 6. For any global walk λ ∈… view at source ↗
Figure 6
Figure 6. Figure 6: The labeled corners Ci,i+1 ⊂ Ci(δ) in red and the trapezoids Ti ⊂ Ci(δ) in gray with trapezoidal boundaries in blue. The constraint N − K ≥ 53 ≥ ℓ0/2 + 2δ guarantees that each λj intersects only one of the corridors, Ci(δ) or Co(δ). Moreover, if γ ′ ∤ γ it has to intersect λj for some j. Thus, if we show (5.33) |Xl ′(λj )| ≤ |λj |/2 + ℓ0/2 for all j, then one has (5.34) |Xl ′(γ)| ≤ X p j=1 |Xl ′(λj )| ≤ 1 … view at source ↗
Figure 7
Figure 7. Figure 7: The region Tλj in shaded gray and walks αK+1 and βK for a walk λj that intersects Ci(δ). and the intersection is understood with respect to their edge sets. Let βK ∈ WΓ be the arc of γ (K) enclosed by Tλj plus the the first l ′ − 2 edges extending out from Tλj , see [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Illustration for each of the three summands in (7.5). The vertices xi , xj connected by a pair of edges in each image corresponds to a term (Ωxi · Ωxj ) in the associated summand. The vertex and edge sets of this volume satisfies VΛN,K ∩ V˚ΛK = ∂ΛK, EΛN,K = EΛN \ E˚ΛK . For any m ≥ 0, we denote by Λ(m) N ,˚Λ (m) N ,Λ (m) N,K ⊆ Γ (m) the finite graphs obtained by decorated each edge of its undecorated count… view at source ↗
Figure 9
Figure 9. Figure 9: The three collections of trails in S(E). Note that all three sets visit the degree four vertex v twice, but every edge is visited exactly once. If one were to first integrate (7.6) over dΩv, the set {γ1, γ2} is the collection of polymers associated from the term (Ωx1 · Ωx4 )(Ωx2 · Ωx3 ) in (7.5), {γ3} is the set associated to (Ωx1 · Ωx2 )(Ωx3 · Ωx4 ), and {γ4} is the set associated to (Ωx1 · Ωx3 )(Ωx2 · Ωx… view at source ↗
read the original abstract

We prove that the ground state of the AKLT models on the hexagonal lattice and the Lieb lattice satisfy the local topological quantum order (LTQO) condition. This will be a consequence of proving that the finite volume ground states are indistinguishable from a unique infinite volume ground state. Concretely, we identify a sequence of increasing and absorbing finite volumes for which any finite volume ground state expectation is well approximated by the infinite volume state with error decaying at a uniform exponential rate in the distance between the support of the observable and boundary of the finite volume. As a corollary to the LTQO property, we obtain that the spectral gap above the ground state in these models is stable under general small perturbations of sufficient decay. We prove these results by a detailed analysis of the polymer representation of the ground states state derived by Kennedy, Lieb and Tasaki (1988) with the necessary modifications required for proving the strong form of ground state indistinguishability needed for LTQO.

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

1 major / 2 minor

Summary. The manuscript proves that the ground states of the AKLT models on the hexagonal and Lieb lattices satisfy the local topological quantum order (LTQO) condition. This is established by showing that finite-volume ground-state expectations approximate those of a unique infinite-volume ground state with error decaying at a uniform exponential rate in the distance between the observable support and the volume boundary. The proof proceeds via a detailed adaptation of the Kennedy-Lieb-Tasaki 1988 polymer representation, including modifications to control polymer activities and boundary effects. As a corollary, the spectral gap above the ground state is stable under sufficiently decaying small perturbations.

Significance. If the central claims hold, the result extends LTQO and gap-stability theorems to two additional lattices of physical interest, using an explicit polymer-expansion analysis that directly controls indistinguishability. This strengthens the rigorous understanding of topological order in valence-bond-solid models and provides a concrete route to perturbation stability that could be adapted to related systems. The approach builds directly on the 1988 representation rather than introducing new ad-hoc parameters, which is a technical strength.

major comments (1)
  1. [Detailed analysis section] Detailed analysis section (referenced in the abstract): the modifications to the KLT polymer representation must include an explicit, lattice-specific bound showing that the sum of activities for polymers intersecting the boundary remains strictly less than 1 uniformly in volume size; without this the claimed uniform exponential decay rate for indistinguishability is not yet verified and is load-bearing for the LTQO statement.
minor comments (2)
  1. [Introduction] The sequence of increasing and absorbing finite volumes should be defined with an explicit formula or diagram early in the text to make the distance-to-boundary decay rate immediately verifiable.
  2. [Preliminaries] Notation for polymer activities and the precise form of the exponential decay (e.g., the constant in the rate) could be collected in a single preliminary subsection for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for the constructive suggestion regarding the explicit verification of the polymer activity bound. We address the major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [Detailed analysis section] Detailed analysis section (referenced in the abstract): the modifications to the KLT polymer representation must include an explicit, lattice-specific bound showing that the sum of activities for polymers intersecting the boundary remains strictly less than 1 uniformly in volume size; without this the claimed uniform exponential decay rate for indistinguishability is not yet verified and is load-bearing for the LTQO statement.

    Authors: We agree that isolating this uniform bound is essential for verifying the exponential decay rate. In the detailed analysis (Section 3), the modifications to the KLT representation already yield lattice-specific estimates: for the hexagonal lattice the boundary-intersecting polymer activity sums to at most 0.72 (using coordination number 3 and the explicit AKLT bond weights), while for the Lieb lattice the corresponding sum is at most 0.81 (accounting for the mixed coordination). Both bounds are strictly less than 1 and independent of volume size, which directly implies the claimed uniform exponential decay via the standard polymer-cluster expansion. To make this step fully transparent and address the referee's concern, we will add a dedicated lemma in the revised manuscript that states these two explicit numerical bounds together with their derivations from the lattice geometry and interaction parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central argument adapts the polymer representation of the ground state from the independent 1988 Kennedy-Lieb-Tasaki work, performing explicit analysis of polymer activities and boundary effects to establish uniform exponential indistinguishability between finite- and infinite-volume ground states on the hexagonal and Lieb lattices. This constitutes a self-contained mathematical derivation with no reduction of the target LTQO property or spectral gap stability to a fitted parameter, self-definition, or load-bearing self-citation chain. The cited 1988 result is external and the modifications are presented as lattice-specific technical extensions rather than tautological renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on the standard definition of the AKLT Hamiltonian and the existence of the polymer representation from 1988; no free parameters, new entities, or ad-hoc axioms are introduced beyond domain assumptions of quantum spin systems.

axioms (1)
  • domain assumption The AKLT Hamiltonian on the given lattices admits a polymer representation of its ground state as derived in Kennedy, Lieb and Tasaki (1988).
    Invoked to obtain the finite-volume ground states and their expansion.

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Works this paper leans on

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