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arXiv:2507.14895

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

Dhiman Bhowmick, Wen Wei Ho

Published Jul 20, 2025• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2507.14895

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Asymmetric decay of quantum many-body scars in XYZ quantum spin chains

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Quantum many-body scars are atypical energy eigenstates of chaotic quantum many-body systems that prevent certain special non-equilibrium initial conditions from thermalizing. We point out that quantum many-body scars exist for any nearest-neighbor spin-$S$ XYZ quantum spin chain, and arise in the form of an infinite family of highly excited yet nonentangled product-state eigenstates, which define periodic textures in spin space. This set of scars, discovered originally by Granovskii and Zhedanov in 1985, encompasses both the experimentally relevant 'spin helices' for XXZ chains and more complicated helix-like states constructed from Jacobi elliptic functions for generic XYZ chains. An appealing feature of Granovskii-Zhedanov scars is that they are well-defined in the semiclassical limit $S \to \infty$, which allows for a systematic and analytical treatment of their dynamical instability to perturbations of the Hamiltonian. Using time-dependent spin-wave theory, we predict that upon perturbing along certain directions in Hamiltonian space, Granovskii-Zhedanov scars exhibit a dramatic asymmetry in their decay: depending on the sign of the perturbation, the decrease of their contrast is either slow and linear, or fast and exponential in time. This asymmetry can be traced to the absence (presence) of imaginarity in the spectrum of the Bogoliubov Hamiltonian governing quantum fluctuations about the scar, which corresponds to the absence (presence) of a non-zero Lyapunov exponent for the limiting classical trajectory. Numerical simulations using matrix product states (MPS) and infinite time-evolving block decimation (iTEBD) confirm that our prediction remains valid even far from the semiclassical limit. Our findings challenge existing theories of how quantum-many body scars relax.

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

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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.

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We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024

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It has been shown [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that three families of highly symmetric states are many-body scars for any spin-$\frac{1}{2}$ fermionic Hamiltonian of the form ${H}_{0}+OT$, where $T$ is a generator of an appropriate Lie group. One of these families consists of the well-known $\ensuremath{\eta}$-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine tuning. The explicit ${H}_{0}+OT$ decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write all the generators $T$ that can be used as building blocks for designing new models with scars, most interestingly including the spin-orbit coupled hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the ``revivals.'' A full numerical study of an extended two-dimensional $tJU$ model explicitly illustrates the novel properties of the invariant scars and supports our findings.

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Quantum many-body scars (QMBS) represent a mechanism for weak ergodicity breaking, characterized by the coexistence of atypical non-thermal eigenstates within an otherwise thermalizing many-body spectrum. In this work, we revisit the spin-$1$ $XY$ model on a periodic chain and construct several new families of exact scar eigenstates embedded within its extensively degenerate manifolds that owe their origins to an interplay of $U(1)$ magnetization conservation and chiral symmetries. We go beyond previously studied towers of states and first identify a novel set of interference-protected eigenstates resembling Fock space cage states, where destructive interference confines the wave function to sparse subgraphs of the Fock space. These states exhibit subextensive entanglement entropy, and when subjected to a transverse magnetic field, form equally spaced ladders whose coherent superpositions display long-lived fidelity oscillations. We further reveal a simpler organizing principle behind these nonthermal states by using the commutant algebra framework, in particular by showing that they are simultaneous eigenstates of non-commuting local operators. Moreover, in doing so, we uncover two more novel families of exact scars: a tower of volume-entangled states, and a set of mirror-dimer states with some free local degrees of freedom. Our results illustrate the power and interplay of interference-based and algebraic mechanisms of non-ergodicity, offering systematic routes to identifying and classifying QMBS in generic many-body quantum systems.

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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.
Quantum many-body scars are atypical energy eigenstates of chaotic quantum many-body systems that prevent certain special non-equilibrium initial conditions from thermalizing. We point out that quantum many-body scars exist for any nearest-neighbor spin-SSS XYZ quantum spin chain, and arise in the form of an infinite family of highly excited yet nonentangled product-state eigenstates, which define periodic textures in spin space. This set of scars, discovered originally by Granovskii and Zhedanov in 1985, encompasses both the experimentally relevant 'spin helices' for XXZ chains and more complicated helix-like states constructed from Jacobi elliptic functions for generic XYZ chains. An appealing feature of Granovskii-Zhedanov scars is that they are well-defined in the semiclassical limit S→∞S \to \inftyS→∞, which allows for a systematic and analytical treatment of their dynamical instability to perturbations of the Hamiltonian. Using time-dependent spin-wave theory, we predict that upon perturbing along certain directions in Hamiltonian space, Granovskii-Zhedanov scars exhibit a dramatic asymmetry in their decay: depending on the sign of the perturbation, the decrease of their contrast is either slow and linear, or fast and exponential in time. This asymmetry can be traced to the absence (presence) of imaginarity in the spectrum of the Bogoliubov Hamiltonian governing quantum fluctuations about the scar, which corresponds to the absence (presence) of a non-zero Lyapunov exponent for the limiting classical trajectory. Numerical simulations using matrix product states (MPS) and infinite time-evolving block decimation (iTEBD) confirm that our prediction remains valid even far from the semiclassical limit. Our findings challenge existing theories of how quantum-many body scars relax.
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.
We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024
Quantum many-body scars provide an exception to generic thermalising dynamics, but their relation to integrability remains unclear. Here we apply the periodic-orbit framework to the integrable XYZ/XXZ spin-1/2 chain, developing an energy-resolved approach that enables us to locate and track periodic orbits across the spectrum. We identify families of scarred orbits and resolve their supporting eigenstates in terms of Bethe-ansatz quantum numbers, revealing a common string core dressed by zero-momentum magnon pairs. This allows us to reconstruct both the scarred eigenstates and the associated towers directly within the Bethe ansatz, and explain the equidistant tower spacing analytically from the decoupling of zero-momentum magnon pairs in the Bethe equations, providing a microscopic realisation of scar phenomenology in an integrable setting. We further show that this structure persists upon breaking integrability with a transverse field, with the periodic orbits continuing to govern the dynamics. Our results establish a direct connection between the periodic-orbit picture of quantum scarring and the algebraic structure of integrable models, showing that key features of scar dynamics can be understood analytically within the Bethe ansatz framework.
It has been shown [K. Pakrouski et al., Phys. Rev. Lett. 125, 230602 (2020)] that three families of highly symmetric states are many-body scars for any spin-12\frac{1}{2}21​ fermionic Hamiltonian of the form H0+OT{H}_{0}+OTH0​+OT, where TTT is a generator of an appropriate Lie group. One of these families consists of the well-known $\ensuremath{\eta}$-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine tuning. The explicit H0+OT{H}_{0}+OTH0​+OT decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write all the generators TTT that can be used as building blocks for designing new models with scars, most interestingly including the spin-orbit coupled hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the ``revivals.'' A full numerical study of an extended two-dimensional tJUtJUtJU model explicitly illustrates the novel properties of the invariant scars and supports our findings.
Additional quantum many-body scars of the spin-111 XYXYXY model with Fock-space cages and commutant algebras
Quantum many-body scars (QMBS) represent a mechanism for weak ergodicity breaking, characterized by the coexistence of atypical non-thermal eigenstates within an otherwise thermalizing many-body spectrum. In this work, we revisit the spin-111 XYXYXY model on a periodic chain and construct several new families of exact scar eigenstates embedded within its extensively degenerate manifolds that owe their origins to an interplay of U(1)U(1)U(1) magnetization conservation and chiral symmetries. We go beyond previously studied towers of states and first identify a novel set of interference-protected eigenstates resembling Fock space cage states, where destructive interference confines the wave function to sparse subgraphs of the Fock space. These states exhibit subextensive entanglement entropy, and when subjected to a transverse magnetic field, form equally spaced ladders whose coherent superpositions display long-lived fidelity oscillations. We further reveal a simpler organizing principle behind these nonthermal states by using the commutant algebra framework, in particular by showing that they are simultaneous eigenstates of non-commuting local operators. Moreover, in doing so, we uncover two more novel families of exact scars: a tower of volume-entangled states, and a set of mirror-dimer states with some free local degrees of freedom. Our results illustrate the power and interplay of interference-based and algebraic mechanisms of non-ergodicity, offering systematic routes to identifying and classifying QMBS in generic many-body quantum systems.