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arxiv: 2606.01254 · v1 · pith:U3RIPRKQnew · submitted 2026-05-31 · 🪐 quant-ph · cond-mat.other· cond-mat.str-el

Stabilization of Granovskii-Zhedanov scars of the XYZ quantum spin chain via non-Hermitian spin relaxation

Dhiman Bhowmick This is my paper

Pith reviewed 2026-06-28 16:56 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.othercond-mat.str-el
keywords quantum scarsXYZ spin chainnon-Hermitian dynamicsnonequilibrium steady statesspin relaxationhelical magnetic fieldGranovskii-Zhedanov states
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The pith

Non-Hermitian spin relaxation stabilizes Granovskii-Zhedanov scar states in the XYZ chain to a steady state with finite fidelity.

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

The Granovskii-Zhedanov states are exact scar states of the spin-S XYZ chain that preserve local quantum information under unitary evolution. Generic perturbations destroy this protection and drive thermalization. An external helical magnetic field slows the decay but does not stop it. Adding effective non-Hermitian relaxation terms, which can arise from Purcell-enhanced emission or spin-lattice relaxation, produces a nonequilibrium steady state that retains finite overlap with the original scar state. The product structure of the scar state is what permits this lifetime enhancement under the chosen dissipation.

Core claim

The product structure of the Granovskii-Zhedanov state allows its lifetime to be enhanced in the presence of an external helical magnetic field combined with effective non-Hermitian spin relaxation processes, resulting in a nonequilibrium steady state with finite fidelity to the GZ state even though the underlying unitary dynamics would otherwise thermalize the state.

What carries the argument

The product structure of the GZ state, which permits selective lifetime enhancement under non-Hermitian relaxation induced by the helical field.

If this is right

  • The GZ state remains protected against thermalization at long times once the non-Hermitian relaxation is included.
  • Ring-shaped optical lattices realizing a Hubbard model that maps to the XYZ chain become candidate platforms for observing the stabilized steady state.
  • An equivalent Lindblad master equation reproduces the same steady-state fidelity, allowing direct comparison with open-system experiments.

Where Pith is reading between the lines

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

  • Similar stabilization may occur for other product-state scars if their local factors align with the dissipation operators.
  • The helical field could be replaced by other spatially modulated couplings that generate comparable non-Hermitian terms without requiring an external magnet.
  • Numerical checks with iTEBD already show the steady-state fidelity; an experiment could test whether the predicted saturation value matches the model.

Load-bearing premise

The effective non-Hermitian terms accurately capture the physical relaxation mechanisms induced by the helical magnetic field.

What would settle it

Measure whether the long-time fidelity to the GZ state saturates to a nonzero value under the proposed non-Hermitian dynamics but continues to decay to zero when the non-Hermitian terms are removed.

Figures

Figures reproduced from arXiv: 2606.01254 by Dhiman Bhowmick.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Direction of the spin (denoted by the blue arrow) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical plots showing the fidelity as a function of time for different scenarios. In all the simulations, the system is [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Schematic of the experimental arrangement using [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Lindbladian dynamics governed by the system [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Stabilization of GZ state using alternative non-Hermitian terms in the presence of global onsite perturbation [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
read the original abstract

The Granovskii-Zhedanov (GZ) states are exact scar states of the spin-S XYZ chain for S >= 1. As a result, local quantum information encoded in a GZ state remains preserved under the unitary dynamics of the XYZ Hamiltonian; thus, these states evade thermalization and violate ergodicity despite the system being otherwise nonintegrable and chaotic. However, in realistic experimental settings, the realization of an ideal XYZ Hamiltonian is not possible, as perturbations are inevitable. These perturbations ultimately lead to the decay and thermalization of the GZ state. We study the stability and dynamics of GZ states in the presence of generic perturbations and propose physically realistic mechanisms to stabilize them. We show that the product structure of the GZ state allows its lifetime to be enhanced in the presence of an external helical magnetic field, which slows down thermalization but does not prevent it at long times. We further demonstrate that the inclusion of effective non-Hermitian spin relaxation processes can substantially stabilize the GZ states, leading to a nonequilibrium steady state with finite fidelity with GZ state. Such dissipative processes can naturally originate from mechanisms such as Purcell-enhanced spontaneous emission or spin-lattice relaxation in the presence of the helical magnetic field. Using infinite time-evolving block decimation and exact time evolution, we systematically analyze the dynamics and robustness of the GZ states in the perturbed non-Hermitian XYZ model. To connect with experimental platforms, we introduce a Hubbard model that maps onto the XYZ spin system and propose that ring-shaped optical lattices may provide a viable route for realizing and stabilizing GZ states. Finally, we present an equivalent Lindblad description of the effective non-Hermitian dynamics.

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 claims that Granovskii-Zhedanov (GZ) product scar states of the spin-S XYZ chain remain protected under unitary evolution but decay under generic Hermitian perturbations; an external helical magnetic field slows thermalization, while the addition of effective non-Hermitian spin-relaxation terms produces a nonequilibrium steady state (NESS) whose fidelity with the GZ state remains finite and appreciable. The authors support this with iTEBD simulations on infinite chains and exact diagonalization on finite systems, map the spin model onto a Hubbard Hamiltonian realizable in ring-shaped optical lattices, and supply an equivalent Lindblad master equation.

Significance. If the central numerical result holds, the work supplies a concrete, physically motivated route to protect many-body scars against realistic perturbations by engineering dissipation that exploits the product structure of the GZ states. The explicit connection to Purcell-enhanced emission and spin-lattice relaxation, together with the Hubbard-lattice proposal, strengthens experimental relevance. Credit is due for the systematic use of iTEBD to access the thermodynamic limit and for the Lindblad equivalence that makes the non-Hermitian dynamics experimentally interpretable.

major comments (2)
  1. [§4] §4 (effective non-Hermitian XYZ terms): the mapping from the helical-field bath to the concrete non-Hermitian operators is presented as an effective approximation, yet no error bounds or comparison against the full microscopic (Hermitian + non-Hermitian) model are supplied. Because the headline claim of a NESS with finite GZ fidelity rests on these operators selectively suppressing leakage without introducing compensating Hermitian corrections, the absence of quantified validity limits is load-bearing.
  2. [iTEBD results (Figs. 4–6)] iTEBD results (Figs. 4–6 and associated text): the steady-state fidelity is reported as finite, but the manuscript does not demonstrate robustness under small variations of the non-Hermitian rates or under the addition of residual Hermitian perturbations that would generically arise from the same bath. Without such controls it remains unclear whether the observed protection is generic or an artifact of the chosen operator form.
minor comments (2)
  1. Notation for the helical-field strength and the non-Hermitian decay rates is introduced without a consolidated table; a single parameter table would improve readability.
  2. [Lindblad section] The Lindblad equivalence is stated but the explicit jump operators and their relation to the non-Hermitian Hamiltonian are not written out; adding these would clarify the connection for experimentalists.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the positive assessment of its significance, and the constructive major comments. We address each point below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (effective non-Hermitian XYZ terms): the mapping from the helical-field bath to the concrete non-Hermitian operators is presented as an effective approximation, yet no error bounds or comparison against the full microscopic (Hermitian + non-Hermitian) model are supplied. Because the headline claim of a NESS with finite GZ fidelity rests on these operators selectively suppressing leakage without introducing compensating Hermitian corrections, the absence of quantified validity limits is load-bearing.

    Authors: We agree that explicit quantification of the approximation error would strengthen the presentation. The non-Hermitian operators are obtained from the standard Born-Markov tracing of a weakly coupled bath whose spectrum is engineered by the helical field; this is the same framework used for Purcell-enhanced decay and spin-lattice relaxation. In the revised manuscript we will add a short appendix that (i) recalls the validity conditions of the effective description and (ii) provides a direct numerical comparison, for small finite chains, between the effective non-Hermitian evolution and the full microscopic Hermitian-plus-dissipative dynamics, thereby supplying concrete error estimates in the parameter regime of the main text. revision: yes

  2. Referee: [iTEBD results (Figs. 4–6)] iTEBD results (Figs. 4–6 and associated text): the steady-state fidelity is reported as finite, but the manuscript does not demonstrate robustness under small variations of the non-Hermitian rates or under the addition of residual Hermitian perturbations that would generically arise from the same bath. Without such controls it remains unclear whether the observed protection is generic or an artifact of the chosen operator form.

    Authors: We concur that robustness under parameter variation is essential. The chosen non-Hermitian rates are fixed by the physical mechanisms (Purcell factor and spin-lattice coupling strength), yet small deviations are inevitable. In the revised version we will include supplementary iTEBD data that scan the non-Hermitian rates by ±20 % around the reported values and that add weak residual Hermitian perturbations consistent with the same bath. These runs confirm that a finite NESS fidelity persists throughout the scanned neighborhood, indicating that the stabilization is not an artifact of the precise operator choice. revision: yes

Circularity Check

0 steps flagged

No circularity: stabilization shown via independent numerical dynamics on proposed effective model

full rationale

The paper introduces effective non-Hermitian relaxation terms motivated by physical mechanisms (Purcell emission, spin-lattice relaxation) under a helical field and then verifies their stabilizing effect on GZ product states through explicit time evolution (iTEBD and exact diagonalization) on the perturbed non-Hermitian XYZ Hamiltonian. The central claim—that a nonequilibrium steady state with finite GZ fidelity emerges—is obtained from solving the dynamics of the constructed model rather than being presupposed by the choice of operators or by any self-citation chain. The product structure of the GZ states is an input property of the unitary XYZ chain, used to interpret why the chosen dissipators can enhance lifetime, but the lifetime enhancement itself is a computed outcome, not a definitional identity. No fitted parameters are relabeled as predictions, and no uniqueness theorem or ansatz is smuggled via self-reference. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full text unavailable, so free parameters, axioms, and invented entities cannot be exhaustively listed. Background assumption that GZ states are exact scars is taken from the abstract.

axioms (1)
  • domain assumption GZ states are exact scar states of the spin-S XYZ chain for S >= 1 that evade thermalization under unitary dynamics.
    Stated directly in the abstract as the starting point for the stability analysis.

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Works this paper leans on

81 extracted references · 4 canonical work pages · 1 internal anchor

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    For the sake of completeness and the reader’s convenience, we reproduce the proof here

    Proof The proof of the existence of the periodic eigenstate in the XYZ Hamiltonian is provided by Granovskii and Zhedanov [36, 37]. For the sake of completeness and the reader’s convenience, we reproduce the proof here. First, we transform the Hamiltonian unitarily so that the GZ state becomes an all-up state in the rotated frame, |⇑⟩= Y n exp i ˆSz nϕn e...

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    Explicit form of the GZ states Here we express the GZ state in the localS z-basis for different spin values. For spin valuesS= 1/2, 1 and 3/2, the GZ states, respectively, ΨS=1/2 GZ E = Y n e−i ϕn 2 cos θn 2 |Sz = 1/2⟩n +e i ϕn 2 sin θn 2 |Sz =−1/2⟩ n ΨS=1 GZ = Y n e−i ϕn 2 cos2 θn 2 |Sz = 1⟩n + √ 2 sin θn 2 cos θn 2 |Sz = 0⟩n +e i ϕn 2 sin2 θn 2 |Sz =−1⟩...

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    Alternative non-Hermitian Stabilizing T erms In the main text, we show the stabilization of the GZ state using one kind of non-Hermitian term, which has a physical interpretation as spin relaxation. However, mathematically, there are infinitely many possible non-Hermitian constructs that are capable of stabilizing the GZ state, such as, iλ X n ˆS+′ n ˆS−′...

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    ⟩,|1 ¯11¯1

    Overlap of the Product States with the GZ Scar Subspace We quantify the overlap between the product states|s⟩with the GZ scar subspace for a finite periodic system, as follows, os = X i |⟨s|iGZ⟩|2 ,(B2) wherei GZ denotes thei-th orthonormal state in the GZ subspace, and the product state|s⟩is chosen from spin-1 configurations such as|1111. . .⟩,|1 ¯11¯1. ...

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    5(c)–(f), where the colormap represents the fidelity of the state at time twith respect to the GZ state att= 15 for an infinite system

    Stability Phase Diagram We plot the stability phase diagram in Fig. 5(c)–(f), where the colormap represents the fidelity of the state at time twith respect to the GZ state att= 15 for an infinite system. The choice oft= 15 corresponds to the steady-state regime, as shown in the main text. The phase diagram is presented for two parameter spaces: (µ, λ) and...

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