We consider spin chain models with local Hamiltonians that display weak ergodicity breaking. In these models, the majority of the eigenstates are thermal, but there is a distinguished subspace of the Hilbert space in which ergodicity is broken. We achieve such a weak breaking by embedding selected integrable models into larger Hilbert spaces of otherwise chaotic models. The integrable subspaces do not have a tensor product structure with respect to any spatial bipartition, therefore our constructions differ from certain trivial embeddings. We consider multiple mechanisms for such an embedding, and we also review previous examples in the literature. Curiously, all our examples can be seen as perturbations of models with Hilbert space fragmentation, such that the perturbed models are not fragmented anymore.
We consider spin chain models with local Hamiltonians that display weak ergodicity breaking. In these models, the majority of the eigenstates are thermal, but there is a distinguished subspace of the Hilbert space in which ergodicity is broken. We achieve such a weak breaking by embedding selected integrable models into larger Hilbert spaces of otherwise chaotic models. The integrable subspaces do not have a tensor product structure with respect to any spatial bipartition, therefore our constructions differ from certain trivial embeddings. We consider multiple mechanisms for such an embedding, and we also review previous examples in the literature. Curiously, all our examples can be seen as perturbations of models with Hilbert space fragmentation, such that the perturbed models are not fragmented anymore.
We show that the combination of charge and dipole conservation-characteristic of fracton systems-leads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S-z) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S-z basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.
We show that the combination of charge and dipole conservation-characteristic of fracton systemsleads to an extensive fragmentation of the Hilbert space, which, in turn, can lead to a breakdown of thermalization.As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total S z ) and its associated dipole moment.First, we consider a minimal model including only three-site terms and find that the infinite temperature autocorrelation saturates to a finite value-showcasing nonthermal behavior.The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local S z basis, arising from the interplay of dipole conservation and local interactions.Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: The system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states.More generally, for any finite range of interactions, the system still exhibits nonthermal eigenstates appearing throughout the entire spectrum.We compare our results to charge and dipole moment-conserving random unitary circuit models for which we reach identical conclusions.
Systems exhibiting the Hilbert-space fragmentation are nonergodic, and their Hamiltonians decompose into exponentially many blocks in the computational basis. In many cases, these blocks can be labeled by eigenvalues of statistically localized integrals of motion (SLIOMs), which play a similar role in fragmented systems as local integrals of motion in integrable systems. While a nonzero perturbation eliminates all nontrivial conserved quantities from integrable models, we demonstrate for the $t\text{\ensuremath{-}}{J}_{z}$ chain that an appropriately chosen perturbation may gradually eliminate SLIOMs (one by one) by progressively merging the fragmented subspaces. This gradual recovery of ergodicity manifests as an extended critical regime characterized by multiple peaks of the fidelity susceptibility. Each peak signals a change in the number of SLIOMs and blocks, as well as an ultra-slow relaxation of local observables.
We propose two strategies to construct a family of nonintegrable spin chains with exactly solvable subspaces based on the idea of quasiparticle excitations from the matrix product vacuum state [Haegeman et al., Phys. Rev. B 88, 075133 (2013)]. The first one allows the boundary generalization, while the second one makes it possible to construct the solvable subspace with interacting quasiparticles. Each generalization is realized by removing the assumption made in the conventional method [Moudgalya et al., Phys. Rev. B 102, 085120 (2020)], which is the frustration-free condition or the local orthogonality, respectively. We found that the structure of the embedded equally spaced energy spectrum is not violated by the diagonal boundaries, as long as quasiparticles are identical and noninteracting in the invariant subspace. On the other hand, we show that there exists a one-parameter family of nonintegrable Hamiltonians, which shows the perfectly embedded energy spectrum of the integrable spin chain. Surprisingly, the embedded energy spectrum does not change by varying the free parameter of the Hamiltonian. The constructed models weakly break ergodicity, in which strong ETH is expected to be violated.
The discovery of quantum many-body scars (QMBS) both in Rydberg atom simulators and in the Affleck-Kennedy-Lieb-Tasaki spin-1 chain model, have shown that a weak violation of ergodicity can still lead to rich experimental and theoretical physics. In this review, we provide a pedagogical introduction to and an overview of the exact results on weak ergodicity breaking via QMBS in isolated quantum systems with the help of simple examples such as the fermionic Hubbard model. We also discuss various mechanisms and unifying formalisms that have been proposed to encompass the plethora of systems exhibiting QMBS. We cover examples of equally-spaced towers that lead to exact revivals for particular initial states, as well as isolated examples of QMBS. Finally, we review Hilbert space fragmentation, a related phenomenon where systems exhibit a richer variety of ergodic and non-ergodic behaviors, and discuss its connections to QMBS.
Non-ergodic quantum many-body dynamics offers a route to persistent quantum coherence far from equilibrium, beyond the conventional expectations of thermalizing statistical mechanics. Most known examples have been identified through intuition, analogy, or numerical search, rather than by systematic Hamiltonian design. This leaves few general methods for engineering robust non-ergodic dynamics in disorder-free interacting quantum many-body systems, especially in higher dimensions where enhanced connectivity generally disfavors ergodicity breaking. Here, we use an eigenstate-to-Hamiltonian construction approach to systematically engineer two-dimensional quantum spin Hamiltonians with tunable non-ergodic properties. Starting from a common parent model and input eigenstate, we construct two target Hamiltonians on the square lattice that differ in their interaction geometry: axial versus diagonal. We show that the resulting interaction geometry and coupling pattern control the Hilbert-space structure and dynamics. The axial model forms a connected Hilbert-space network and exhibits an anisotropy-driven crossover from ergodic to non-ergodic behavior through the suppression of resonances. In contrast, the diagonal model exhibits Hilbert-space fragmentation and, within its largest irreducible sector, signatures of quantum many-body scar dynamics including towers of highly localized, low-entanglement eigenstates, and long-lived coherent revivals. These results establish a direct link between Hamiltonian design, Hilbert-space network structure, and emergent non-ergodic dynamics.