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arXiv:2603.01859

Local integrals of motion encoded in a few eigenstates

J. Pawłowski, P. Łydżba, M. Mierzejewski

cond-mat.str-elquant-phSubmitted Mar 2, 2026• Announced Mar 3, 2026• 0 citations
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Abstract

Many properties of a quantum system can be obtained from just a single eigenstate of its Hamiltonian. For example, a single eigenstate can be used to determine whether a system is integrable or chaotic and, in the latter case, to establish its thermal properties. Focusing on the XXZ model, we show that the local integrals of motion, which lie at the heart of integrability, can also be estimated from a small number of eigenstates. Moreover, as the system size increases, fewer eigenstates are required, so that in the thermodynamic limit, the integrals of motion can be obtained from a vanishingly small fraction of all eigenstates. Interestingly, this property does not extend to integrals of motion arising solely from Hilbert space fragmentation, as found in the folded XXZ model, where the majority of eigenstates has to be used. This represents one of the few fundamental differences known between integrability and Hilbert space fragmentation.

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Many properties of a quantum system can be obtained from just a single eigenstate of its Hamiltonian. For example, a single eigenstate can be used to determine whether a system is integrable or chaotic and, in the latter case, to establish its thermal properties. Focusing on the XXZ model, we show that the local integrals of motion, which lie at the heart of integrability, can also be estimated from a small number of eigenstates. Moreover, as the system size increases, fewer eigenstates are required, so that in the thermodynamic limit, the integrals of motion can be obtained from a vanishingly small fraction of all eigenstates. Interestingly, this property does not extend to integrals of motion arising solely from Hilbert space fragmentation, as found in the folded XXZ model, where the majority of eigenstates has to be used. This represents one of the few fundamental differences known between integrability and Hilbert space fragmentation.
Systems where all energy eigenstates are localized are known to display an emergent local integrability, in the sense that one can construct an extensive number of operators that commute with the Hamiltonian and are localized in real space. Here we show that emergent local integrability does not require a set of localized eigenstates to be complete. Given a set of localized eigenstates comprising a nonzero fraction $(1\ensuremath{-}{p}_{f})$ of the full many-body spectrum, one can construct an extensive number of integrals of motion which are local in the sense that they have nonzero weight in a compact region of real space, in the thermodynamic limit. However, these modified integrals of motion have a ``global dressing'' whose weight vanishes as $\ensuremath{\sim}{p}_{f}$ as ${p}_{f}\ensuremath{\rightarrow}0$. In this sense, the existence of a nonzero fraction of localized eigenstates is sufficient for emergent local integrability. We discuss the implications of our findings for systems where the spectrum contains delocalized states, for systems with projected Hilbert spaces, and for the robustness of quantum integrability.
The eigenstate thermalization hypothesis conjectures that for a thermodynamically large system in one of its energy eigenstates, the reduced density matrix describing any finite subsystem is determined solely by a set of relevant conserved quantities. In a chaotic quantum system, only the energy is expected to play that role and hence eigenstates appear locally thermal. Integrable systems, on the other hand, possess an extensive number of such conserved quantities and therefore the reduced density matrix requires specification of all the corresponding parameters (generalized Gibbs ensemble). However, here we show by unbiased statistical sampling of the individual eigenstates with a given finite energy density that the local description of an overwhelming majority of these states of even such an integrable system is actually Gibbs-like, i.e., requires only the energy density of the eigenstate. Rare eigenstates that cannot be represented by the Gibbs ensemble can also be sampled efficiently by our method and their local properties are then shown to be described by appropriately truncated generalized Gibbs ensembles. We further show that the presence of these rare eigenstates differentiates the model from the chaotic case and leads to the system being described by a generalized Gibbs ensemble at long time under a unitary dynamics following a sudden quench, even when the initial state is a typical (Gibbs-like) eigenstate of the prequench Hamiltonian.
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.
The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/21/21/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.
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.
Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-1/2 XXZ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.