Quantum many-body scars (QMBS) constitute a subtle violation of ergodicity through a set of non-thermal eigenstates, referred to as scar states, which are embedded in an otherwise thermal spectrum. In a broad class of known examples, these scar states admit a simple interpretation: they are magnon excitations of fixed momentum on top of a ferromagnetic background. In this paper we prove that any Hamiltonian hosting such ``ferromagnetic scar states'' necessarily admits a structural decomposition into a Zeeman term and an ``annihilator'' that annihilates the entire scar manifold. Moreover, we show that this annihilator must itself decompose into a sum of terms built from local projectors that locally annihilate the scar states. This architecture is closely related to the Shiraishi--Mori construction, and our main theorem establishes that an appropriate generalization of that construction is in fact essentially exhaustive for this class of scar states.
Quantum many-body scars (QMBS) constitute a subtle violation of ergodicity through a set of non-thermal eigenstates, referred to as scar states, which are embedded in an otherwise thermal spectrum. In a broad class of known examples, these scar states admit a simple interpretation: they are magnon excitations of fixed momentum on top of a ferromagnetic background. In this paper we prove that any Hamiltonian hosting such ferromagnetic scar states'' necessarily admits a structural decomposition into a Zeeman term and an annihilator'' that annihilates the entire scar manifold. Moreover, we show that this annihilator must itself decompose into a sum of terms built from local projectors that locally annihilate the scar states. This architecture is closely related to the Shiraishi--Mori construction, and our main theorem establishes that an appropriate generalization of that construction is in fact essentially exhaustive for this class of scar states.
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 W 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 k-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.
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 highly excited eigenstates of non-integrable Hamiltonians which violate the eigenstate thermalization hypothesis and are embedded in a sea of thermal eigenstates. We provide a general mechanism to construct partially integrable models with arbitrarily large local Hilbert space dimensions, which host exact many-body scars. We introduce designed integrability-breaking terms to several exactly solvable spin chains, whose integrable Hamiltonians are composed of the generators of the Temperley-Lieb algebra. In the non-integrable subspace of these models, we identify a special kind of product states -- the generalized spin helix states as exact quantum many-body scars, which lie in the common null space of the non-Hermitian generators of the Temperley-Lieb algebra and are annihilated by the integrability-breaking terms. Our constructions establish an intriguing connection between integrability and quantum many-body scars, meanwhile provide a systematic understanding of scarred Hamiltonians from the perspective of non-Hermitian projectors.
Additional quantum many-body scars of the spin-1XY 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-1XY 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.
Many systems that host exact quantum many-body scars (towers of energy-equidistant low entanglement eigenstates) are governed by a Hamiltonian that splits into a Zeeman term and a sum of local terms that annihilate the scar subspace. We show that this unifying structure also applies to models, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) model or the PXP model of Rydberg-blockaded atoms, which were previously believed to evade this characterization. To fit these models into the local annihilator framework, we need to fractionalize their degrees of freedom and enlarge the associated Hilbert space. The embedding of the original system in a larger space elucidates the structure of their scar states, which are characterized as the common kernel of local annihilators. This simplifies their construction and reveals close analogies with lattice gauge theories.
Quantum many-body scars (QMBS) have attracted considerable interest due to their role in weak ergodicity breaking in many-body systems. We present a general construction that embeds stabilizer states as QMBS of local Hamiltonians. The method relies on a notion of factorizability of Pauli strings on a lattice, which is used to convert stabilizer elements into local, few-body operators that annihilate the stabilizer state. This enables the systematic construction of parent Hamiltonians with zero-energy stabilizer QMBS typically near the middle of the spectrum. The method reproduces several known results in a unified framework, including recent examples of volume-law entangled QMBS, such as the ``rainbow'' QMBS and the entangled antipodal Bell pair state. We also apply the framework to construct examples of stabilizer QMBS with a more complex entanglement structure, such as the cluster state, the toric code state, and a volume-law entangled state we dub the antipodal toric code (ATC) state. Exact diagonalization confirms our results and reveal the stabilizer states as exact eigenstates of their parent Hamiltonian.