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arXiv:2604.11015

Scar subspaces stabilized by algebraic closure: Beyond equally-spaced spectra and exact solvability

Chihiro Matsui

cond-mat.stat-mechquant-phSubmitted Apr 13, 2026• Announced Apr 14, 2026• 0 citations• Revised Apr 21, 2026 (v2)
v1 · Apr 13, 2026v2 · Apr 21, 2026
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Abstract

We construct a class of quantum many-body systems hosting an $\mathfrak{su}(3)$-invariant scar subspace, extending the conventional paradigm of quantum many-body scars beyond equally spaced spectra and single-directional tower structures. Our construction is based on local constraints that realize an algebraic closure within the scar subspace. As a result, the spectrum in the subspace is no longer equally spaced, but instead forms a multidirectional lattice structure parametrized by multiple independent quantum numbers. This leads to qualitatively new dynamical signatures: instead of single-frequency revivals, the system exhibits multifrequency oscillations governed by integer linear combinations of distinct energy scales. Importantly, the stability of the scar subspace does not rely on exact solvability of individual eigenstates. We show that algebraic closure preserves the invariant subspace even under perturbations that render the eigenstates analytically intractable, thereby realizing quantum many-body scars on an unsolvable reference state. Our results identify algebraic closure as a unifying mechanism underlying scar subspaces beyond the conventional $\mathfrak{su}(2)$ paradigm, and open a route toward richer nonthermal dynamics in nonintegrable quantum systems.

Comments
8 pages, 2 figures; Minor corrections to improve consistency of definitions and notation

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We construct a class of quantum many-body systems hosting an su(3)\mathfrak{su}(3)su(3)-invariant scar subspace, extending the conventional paradigm of quantum many-body scars beyond equally spaced spectra and single-directional tower structures. Our construction is based on local constraints that realize an algebraic closure within the scar subspace. As a result, the spectrum in the subspace is no longer equally spaced, but instead forms a multidirectional lattice structure parametrized by multiple independent quantum numbers. This leads to qualitatively new dynamical signatures: instead of single-frequency revivals, the system exhibits multifrequency oscillations governed by integer linear combinations of distinct energy scales. Importantly, the stability of the scar subspace does not rely on exact solvability of individual eigenstates. We show that algebraic closure preserves the invariant subspace even under perturbations that render the eigenstates analytically intractable, thereby realizing quantum many-body scars on an unsolvable reference state. Our results identify algebraic closure as a unifying mechanism underlying scar subspaces beyond the conventional su(2)\mathfrak{su}(2)su(2) paradigm, and open a route toward richer nonthermal dynamics in nonintegrable quantum systems.
Quantum many-body scars are nonthermal excited eigenstates of nonintegrable Hamiltonians, which could support coherent revival dynamics from special initial states when scars form an equally spaced tower in the energy spectrum. For open quantum systems, engineering many-body scarred dynamics by a controlled coupling to the environment remains largely unexplored. Here, we provide a general framework to exactly embed quantum many-body scars into the decoherence-free subspaces of Lindblad master equations. The dissipative scarred dynamics manifest persistent periodic oscillations for generic initial states, and can be practically utilized to prepare scar states with potential quantum metrology applications. We construct the Liouvillian dissipators with the local projectors that annihilate the whole scar towers, and utilize the Hamiltonian part to rotate the undesired states out of the null space of dissipators. We demonstrate our protocol through several typical models hosting many-body scar towers and propose an experimental scheme to observe the dissipative scarred dynamics based on digital quantum simulations and resetting ancilla qubits.
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 are non-thermal eigenstates embedded in the spectra of otherwise non-integrable Hamiltonians. Paradigmatic examples often appear as quasiparticle towers of states, such as the maximally ferromagnetic spin-1/2 states, also known as Dicke states. A distinguishing feature of quantum many-body scars is that they admit multiple local "parent" Hamiltonians for which they are exact eigenstates. In this work, we show that the locality of such parent Hamiltonians strongly constrains the relative placement of these states within the energy spectrum. In particular, we prove that if the full set of Dicke states are exact eigenstates of an extensive local Hamiltonian, then their energies must necessarily be equally spaced. Our proof builds on recent results concerning parent Hamiltonians of the WWW state, together with general algebraic structures underlying such quasiparticle towers. We further demonstrate that this equal spacing property extends to local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and expander graphs. Hamiltonians with kkk-local interactions and a bounded number of interaction terms per site are also encompassed by our proof. On the same classes of graphs, we additionally establish equal spacing for towers constructed from multi-site quasiparticles on top of product states. For the towers considered here, an immediate corollary of the equal spacing property is that any state initialized entirely within the quantum many-body scar manifold exhibits completely frozen entanglement dynamics under any local Hamiltonian for which those scars are exact eigenstates. Overall, our results reveal a stringent interplay between locality and the structure of quantum many-body scars.
Exact stabilizer scars in two-dimensional U(1)U(1)U(1) lattice gauge theory
The complexity of highly excited eigenstates is a central theme in nonequilibrium many-body physics, underpining questions of thermalization, classical simulability, and quantum information structure. In this work, considering the paradigmatic Rokhsar-Kivelson model, we connect quantum many-body scarring in Abelian lattice gauge theories to an emergent stabilizer structure. We identify a distinct class of scarred eigenstates, termed sublattice scars, originating from gauge-invariant zero modes that form exact stabilizer states. Remarkably, although the underlying Hamiltonian is not a stabilizer Hamiltonian, its eigenspectrum intrinsically hosts exact stabilizer eigenstates. These sublattice scars exhibit vanishing stabilizer Rényi entropy together with finite, highly structured entanglement, enabling efficient classical simulation. Exploiting their stabilizer structure, we construct explicit Clifford circuits that prepare these states in a two-dimensional lattice gauge model. Our results demonstrate that the scarred subspace of the Rokhsar-Kivelson spectrum forms an intrinsic stabilizer manifold, revealing a direct connection between stabilizer quantum information, lattice gauge constraints, and quantum many-body scarring.
Motivated by recent experimental observations of coherent many-body revivals in a constrained Rydberg atom chain, we construct a weak quasilocal deformation of the Rydberg-blockaded Hamiltonian, which makes the revivals virtually perfect. Our analysis suggests the existence of an underlying nonintegrable Hamiltonian which supports an emergent SU(2)-spin dynamics within a small subspace of the many-body Hilbert space. We show that such perfect dynamics necessitates the existence of atypical, nonergodic energy eigenstates-quantum many-body scars. Furthermore, using these insights, we construct a toy model that hosts exact quantum many-body scars, providing an intuitive explanation of their origin. Our results offer specific routes to enhancing coherent many-body revivals and provide a step toward establishing the stability of quantum many-body scars in the thermodynamic limit.
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.