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arXiv:2602.22397

Hidden $Z_{2}\times Z_{2}$ subspace symmetry protection for quantum scars

Ayush Sharma, Vikram Tripathi

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Feb 25, 2026• Announced Feb 27, 2026• 0 citations
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Abstract

We study the paradigmatic spin-1 XY chain under open boundary conditions, which hosts exact quantum many-body scars generated by an emergent Spectrum Generating Algebra (SGA). We show that the scar subspace possesses a symmetry-protected trivial (SPt) character that we attribute to a hidden $Z_{2}\times Z_{2}$ symmetry of another model, namely the commutant Hamiltonian, for which the scars are the ground states. We construct a Lieb-Schultz-Mattis (LSM) type twist operator, which, for scar states, takes the value $-1,$ and, for ergodic states, approaches zero in the thermodynamic limit. A complementary understanding of the stability of the scars under different perturbations is obtained by analyzing the Loschmidt echo and Quantum Fisher Information (QFI) of the scars. Finite-size scaling analysis of the QFI reveals that the scars are much more sensitive to perturbations as compared to the nearby thermal states. Based on the analysis of QFI and different LSM twist operators, we obtain a classification of different SGA-preserving and SGA-breaking perturbations.

Comments
33 pages, 19 figures. Comments are welcome

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Hidden Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ subspace symmetry protection for quantum scars
We study the paradigmatic spin-1 XY chain under open boundary conditions, which hosts exact quantum many-body scars generated by an emergent Spectrum Generating Algebra (SGA). We show that the scar subspace possesses a symmetry-protected trivial (SPt) character that we attribute to a hidden Z2×Z2Z_{2}\times Z_{2}Z2​×Z2​ symmetry of another model, namely the commutant Hamiltonian, for which the scars are the ground states. We construct a Lieb-Schultz-Mattis (LSM) type twist operator, which, for scar states, takes the value −1,-1,−1, and, for ergodic states, approaches zero in the thermodynamic limit. A complementary understanding of the stability of the scars under different perturbations is obtained by analyzing the Loschmidt echo and Quantum Fisher Information (QFI) of the scars. Finite-size scaling analysis of the QFI reveals that the scars are much more sensitive to perturbations as compared to the nearby thermal states. Based on the analysis of QFI and different LSM twist operators, we obtain a classification of different SGA-preserving and SGA-breaking perturbations.
We propose a framework that extends the notion of symmetry-protected topological properties beyond the ground-state paradigm to dynamically isolated subspaces formed by exceptional non-thermal energy eigenstates of non-integrable systems, known as quantum many-body scars (QMBS). We introduce the concept of a symmetry-protected topological (SPT) scar subspace -- a Hilbert subspace stabilized by a restricted spectrum-generating algebra (rSGA) while being protected by on-site, inversion, and time-reversal symmetries. QMBS often admit a non-interacting quasiparticle description, which enables matrix-product representations with small bond dimension. Although individual QMBS do not necessarily retain the protecting symmetries of the Hamiltonian, we show that the subspace formed by the symmetry-connected QMBS does retain them, giving rise to consistently emerging topological properties across the entire scar subspace. Using the spin-111 Affleck--Kennedy--Lieb--Tasaki (AKLT) model, we demonstrate that its bimagnon scar subspace reflects the topological properties of the SPT ground state, as evidenced by the appropriate bond-space symmetry representations, the expected topological response, and the numerically verified long-range string order. Our findings indicate that scar subspaces can inherit -- and in inhomogeneous cases systematically modify -- the topological character of the SPT ground state, offering a new and experimentally accessible platform for probing symmetry-protected topology beyond the ground-state regime.
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 special eigenstates that violate the eigenstate thermalization hypothesis while residing at finite energy density along with thermalizing eigenstates. The spin-1 XY model is known to host a family of such exceptional states originating from long-lived quasiparticle excitations that exhibit anomalously low entanglement entropy and long-time periodic revivals, resulting in weak ergodicity breaking. We study the stability of these scarred states against typical U(1) symmetry preserving perturbation in the XY chain. While perturbation theory can describe the deformed scar states at small system sizes, finite-size scaling of the perturbation matrix elements indicate that the scars ultimately thermalize in larger chains. Nonetheless, we demonstrate that the long-range order associated with the scars decays under the perturbation, and we estimate the relaxation timescale of oscillatory dynamics in certain local observables to be of order λ−2λ^{-2}λ−2, where λλλ is the perturbation strength.
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
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.
Formation of quantum scars in many-body systems provides a novel mechanism for enhancing coherence of weakly entangled states. At the same time, coherence of edge modes in certain symmetry-protected topological (SPT) phases can persist away from the ground state. In this work we show the existence of many-body scars and their implications on bulk coherence in such an SPT phase. To this end, we study the eigenstate properties and the dynamics of an interacting spin-1/21/21/2 chain with three-site ``cluster'' terms hosting a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ SPT phase. Focusing on the weakly interacting regime, we find that eigenstates with volume-law entanglement coexist with area-law entangled eigenstates throughout the spectrum. We show that a subset of the latter can be constructed by virtue of repeated cluster excitations on the even or odd sublattice of the chain, resulting in an equidistant ``tower'' of states, analogous to the phenomenology of quantum many-body scars. We further demonstrate that these scarred eigenstates support nonthermal expectation values of local cluster operators in the bulk and exhibit signatures of topological order even at finite energy densities. Studying the dynamics for out-of-equilibrium states drawn from the noninteracting ``cluster basis,'' we unveil that nonthermalizing bulk dynamics can be observed on long timescales if clusters on odd and even sites are energetically detuned. In this case, cluster excitations remain essentially confined to one of the two sublattices such that inhomogeneous cluster configurations cannot equilibrate and thermalization of the full system is impeded. Our work sheds light on the role of quantum many-body scars in preserving SPT order at finite temperature and the possibility of coherent bulk dynamics in models with SPT order beyond the existence of long-lived edge modes.