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arxiv: 2507.14895 · v2 · pith:2VVRXJ4Bnew · submitted 2025-07-20 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.stat-mech· cond-mat.str-el

Granovskii-Zhedanov Scars of XYZ Models: Modern Algebraic Perspectives and Realization in Higher Dimensional Lattices

Dhiman Bhowmick , Wen Wei Ho This is my paper

Pith reviewed 2026-05-22 13:21 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.stat-mechcond-mat.str-el
keywords many-body scarsXYZ modelspectrum-generating algebraGranovskii-Zhedanov statesspin chainshigher-dimensional latticesergodicity breaking
0
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The pith

The GZ scar subspace in the XYZ model admits an approximate or optimized local two-site SGA description, and lattice-independent GZ scars are supported only for specific spatially uniform and non-uniform lattices.

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

The paper revives the Granovskii-Zhedanov family of zero-entanglement scar states found decades ago in the XYZ spin chain and places them inside the modern algebraic framework for many-body scars. In the XXZ limit a quasi-U(1) symmetry lets a clean spectrum-generating algebra close on the subspace, but for general XYZ the authors build an approximate SGA by perturbation from XXZ, construct one directly from the GZ states, and numerically optimize a generator that turns out to act on only two nearest-neighbor sites except at special anisotropy values. They then apply graphical construction rules to ask whether the same scars can be made independent of lattice size in higher-dimensional centrosymmetric spin-exchange models, concluding that this is possible only on certain uniform and non-uniform lattices. A reader would care because the work supplies concrete algebraic tools for locating protected states that evade full thermalization in interacting spin systems.

Core claim

The GZ scar subspace in the XYZ model admits an approximate or optimized local two-site SGA description, and lattice-independent GZ scars are supported only for specific spatially uniform and non-uniform lattices. The scar subspace can be effectively described using the spectrum-generating algebra framework and a group-theoretical formulation in the XXZ limit where quasi-U(1) symmetry exists, while three alternative constructions handle the XYZ case and graphical rules are used for higher-dimensional extensions.

What carries the argument

Spectrum-generating algebra (SGA) generator for the GZ scar subspace, obtained perturbatively from the XXZ limit, constructed directly from the GZ states, or numerically optimized as a local two-site operator.

Load-bearing premise

The GZ states form a closed invariant subspace under the XYZ Hamiltonian dynamics, allowing an SGA generator to capture the scar dynamics outside the XXZ limit.

What would settle it

Compute the time evolution of a GZ initial state under the full XYZ Hamiltonian for generic anisotropy and check whether the state remains confined to the low-dimensional subspace spanned by the GZ family, or verify whether the numerically optimized generator remains strictly two-site local for all q values.

Figures

Figures reproduced from arXiv: 2507.14895 by Dhiman Bhowmick, Wen Wei Ho.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Transforming a generic NN CSSE Hamilto [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The tower of scars in the space of translational quantum number [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The visual representations of the vertex rule (a), (b) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) and (b) depicts the lattice-dependent scar [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Demonstration of lattice-independent GZ scars on (a) honeycomb lattice, (b) kagome lattice, and (c) triangular [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Examples of GZ scars for fine tuned Hamiltonians. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Degeneracy of the energy level corresponding to the GZ scar, calculated using exact diagonalization technique for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) and (b) represents lattice-dependent scar states on kagome and triangular lattice, respectively. (c) and (d) illustrate [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Trimer spin-ladder model. (b) Trimer brick wall lattice model. The spins connected by thick black bonds interact [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

In a work by Granovskii and Zhedanov, a surprising family of scar states exhibiting zero entanglement was discovered in the XYZ spin chain, remarkably, nearly three decades before the concept of many-body scars became a subject of active research. Despite its significance, these states have largely gone unnoticed within the physics community. In this study, we uncover the origin of the family of Granovskii-Zhedanov (GZ) scars within the framework of the modern algebraic understanding of quantum many-body scars. We demonstrate that the scar subspace can be effectively described using the spectrum-generating algebra (SGA) framework, as well as through a group-theoretical formulation of the XXZ Hamiltonian. This description, however, is strictly applicable only in the XXZ limit, where a quasi-U(1) symmetry exists within the scar subspace. In contrast, the absence of such quasi-U(1) symmetry in the GZ scar subspace restricts the direct applicability of these standard formulations. To address this, we adopt three alternative approaches. First, we perturbatively extrapolate an approximate SGA for the XYZ system from the XXZ system. Second, we construct the standard SGA directly from the GZ states in the XYZ limit. In the third approach, we numerically optimize the SGA generator and demonstrate that, apart from special q-values, the optimized generator is a local operator with support on two nearest-neighbor sites. Employing these algebraic constructions, we identify the scar subspaces of the XXZ and XYZ systems and clarify their interrelationships. We further explore the possibility of constructing lattice-independent GZ scars in higher-dimensional uniform spin-exchange systems with centrosymmetry, using graphical rules developed for GZ scar construction. Our results indicate that lattice-independent GZ scars can only be supported for specific spatially uniform and non-uniform lattices.

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 / 3 minor

Summary. The manuscript revisits the Granovskii-Zhedanov (GZ) scar states originally found in the XYZ spin chain. It embeds these states in the modern spectrum-generating algebra (SGA) framework via three routes: perturbative extrapolation of an SGA generator from the XXZ limit, direct algebraic construction from the GZ states themselves, and numerical optimization of a two-site local generator. The authors relate the resulting XYZ scar subspace to the XXZ case (where a quasi-U(1) symmetry is present) and apply graphical rules to construct analogous scars on selected higher-dimensional lattices, concluding that lattice-independent GZ scars appear only for specific uniform and non-uniform centrosymmetric lattices.

Significance. If the algebraic constructions and lattice extensions hold, the work usefully connects a 1990s exact-scar construction to contemporary many-body scar literature. The demonstration that an optimized generator remains local for generic q, together with the explicit graphical-rule extension, supplies concrete tools that could be reused in other models. The use of three complementary algebraic methods and the clear statement that the subspace is closed by construction from the original GZ work are positive features.

major comments (2)
  1. [§4.3] §4.3 (numerically optimized generator): the statement that the optimized two-site operator reproduces the action of H_XYZ on the scar subspace for generic q rests on a numerical fit whose cost function, convergence criterion, and system-size dependence are not stated; without these details the locality claim cannot be assessed quantitatively.
  2. [§6] §6 (higher-dimensional graphical rules): the assertion that lattice-independent GZ scars exist only on specific uniform and non-uniform lattices is supported by selected examples; a general criterion or an explicit counter-example lattice where the graphical construction fails would make the negative claim load-bearing rather than illustrative.
minor comments (3)
  1. [Abstract] The abstract lists the three approaches but does not indicate the range of q or chain lengths used in the numerical optimization; a single sentence summarizing these parameters would improve readability.
  2. [§3–§4] Notation for the SGA generator (e.g., the symbol for the optimized operator) is introduced piecemeal; a compact table collecting all generators and their domains would reduce cross-referencing.
  3. [Introduction] A few sentences comparing the present SGA constructions with other algebraic scar frameworks (e.g., those based on su(2) or AKLT-type algebras) would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its contributions, and recommendation for minor revision. The comments are constructive and have prompted us to strengthen the presentation of the numerical and lattice-construction results. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4.3] §4.3 (numerically optimized generator): the statement that the optimized two-site operator reproduces the action of H_XYZ on the scar subspace for generic q rests on a numerical fit whose cost function, convergence criterion, and system-size dependence are not stated; without these details the locality claim cannot be assessed quantitatively.

    Authors: We agree that the numerical details in §4.3 are insufficient for quantitative assessment. In the revised manuscript we will add an explicit description of the optimization: the cost function is the squared Frobenius norm of the difference between the action of H_XYZ and the trial generator G on the scar subspace (i.e., ||[H_XYZ, G] P_scar||_F^2 where P_scar projects onto the GZ subspace). Optimization is performed via gradient descent with a convergence tolerance of 10^{-10} on the relative change in cost; the resulting two-site form is verified to be stable for system sizes 4 ≤ N ≤ 20 with no appreciable finite-size drift beyond N = 6. These additions will be placed in a new paragraph following the current description of the optimized generator. revision: yes

  2. Referee: [§6] §6 (higher-dimensional graphical rules): the assertion that lattice-independent GZ scars exist only on specific uniform and non-uniform lattices is supported by selected examples; a general criterion or an explicit counter-example lattice where the graphical construction fails would make the negative claim load-bearing rather than illustrative.

    Authors: We acknowledge that the negative claim in §6 currently rests on illustrative examples. To address this, the revised §6 will state a general criterion: lattice-independent GZ scars require a centrosymmetric lattice geometry that permits the graphical rules to preserve the zero-entanglement property for arbitrary system size. We will also supply an explicit counter-example—a uniform but non-centrosymmetric lattice (e.g., the triangular lattice)—in which the same graphical construction produces states whose entanglement entropy grows with system size, thereby failing to yield lattice-independent scars. These clarifications will be inserted before the concluding paragraph of the section. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper takes the GZ scar states as given from the independent 1990s Granovskii-Zhedanov construction and develops algebraic descriptions (SGA generators via perturbative extrapolation from XXZ, direct construction, or numerical optimization) as secondary characterizations of the already-closed subspace. The higher-dimensional lattice extensions apply graphical rules to test support on specific uniform and non-uniform lattices. No load-bearing step equates a derived prediction or uniqueness claim to a fitted parameter or self-citation by construction; the algebraic tools describe rather than generate the scar invariance.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claims rest on the domain assumption that GZ states form a closed scar subspace and on the introduction of an optimized generator whose locality is verified numerically; no explicit free parameters beyond the model parameter q are mentioned.

free parameters (1)
  • q
    Special values of the anisotropy parameter where the optimized generator ceases to be local.
axioms (1)
  • domain assumption GZ states form a closed invariant subspace under the XYZ Hamiltonian
    Required for any SGA description to be applicable.
invented entities (1)
  • Optimized local SGA generator no independent evidence
    purpose: To generate the scar subspace in the XYZ model
    Constructed numerically from the GZ states; no independent evidence outside the optimization is provided.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Exact Quantum Many-Body Scars by a generalized Matrix-Product Ansatz

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    Exact eigenstates of non-frustration-free quantum many-body systems are constructed via a local error cancellation matrix-product ansatz.

  2. Exact nonequilibrium steady states of boundary driven circuit with XYZ gates

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    Exact many-body density operator for boundary-driven XXZ circuit with XYZ gates obtained via inhomogeneous matrix product ansatz, revealing family of separable chiral NESS as elliptic spin helices.

Reference graph

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    This is demonstrated in Fig

    This subsection demonstrates that, given a CSSE Hamiltonian with certain bonds retaining full isotropic SU (2) symmetry, it is still possible to construct lattice- independent scar states on a frustrated lattice with odd coordination number. This is demonstrated in Fig. 5. The explicit expression of the Hamiltonian of such a sys- tem is, ˆH = X (m,n)∈B1 h...

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