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arXiv:2505.09346 · Physical review. B./Physical review. B

Tunable Hilbert space fragmentation and extended critical regime

Mateusz Lisiecki, J. Bonča, Marcin Mierzejewski, J. Herbrych, Patrycja Łydżba

Published Nov 13, 2025• 1 citation
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Abstract

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.

DOI
10.1103/fysk-w1vs

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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\ensuremath-Jzt\text{\ensuremath{-}}{J}_{z}t\ensuremath-Jz​ 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.
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Ergodicity breaking in isolated systems has emerged as an important frontier in the study of quantum many-body physics. While generic Hamiltonians are expected to obey the eigenstate thermalization hypothesis (ETH), recent studies on Hilbert space fragmentation (HSF) have revealed possible robust nonthermal behavior even in disorder-free systems. Although numerous models exhibiting strong HSF are already known, existing analyses are typically model dependent, and a general organizing principle remains elusive. In this work, we introduce a simple mechanism for achieving strong HSF in one-dimensional integer spin chains, which we term "peak-valley (PV) fragmentation". The key idea is to devise a simple local rule which ensures the spin states in the computational basis can be labeled by a set of emergent good quantum numbers corresponding to the heights and depths of alternating peaks and valleys in a geometrical representation. We demonstrate that some known examples of strong HSF models, as well as their variants which break the HSF property, can be understood within the framework of PV fragmentation. Our approach also enables systematic construction of new fragmented models in higher-spin systems, and allows us to identify higher-order HSF models.
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.
Hilbert space fragmentation is an ergodicity-breaking phenomenon, in which the Hamiltonian shatters into exponentially many dynamically disconnected sectors. In many fragmented systems, these sectors can be labeled by statistically localized integrals of motion, which are nonlocal operators. We study the paradigmatic nearest-neighbor pair hopping model exhibiting the so-called strong fragmentation. We show that this model hosts local integrals of motion (LIOMs), which correspond to frozen density modes with long wavelengths. The latter modes become subdiffusive when longer-range pair hoppings are allowed. Finally, we make a connection with a tilted (Stark) chain. Contrary to the dipole-conserving effective models, the tilted chain is shown to support either a Hamiltonian or dipole moment as an LIOM. Numerical results are obtained from a numerical algorithm, in which finding LIOMs is reduced to a data compression problem.
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.
The phenomenon of Hilbert space fragmentation, whereby dynamical constraints fragment Hilbert space into many disconnected sectors, provides a simple mechanism by which thermalization can be arrested. However, little is known about how thermalization occurs in situations where the constraints are not exact. To study this, we consider a situation in which a fragmented 1d chain with pair-flip constraints is coupled to an ergodicity-restoring thermal bath at its boundary. We numerically observe an exponentially long thermalization time in Hamiltonian dynamics, manifested in both entanglement dynamics and the relaxation of local observables. To understand this, we study an analogous model of random unitary circuit dynamics, whose thermalization time we prove scales exponentially with system size. Slow thermalization in this model is shown to be a consequence of strong bottlenecks in configuration space, which restrict how the system can explore Hilbert space, and demonstrate a new way of producing anomalously slow thermalization dynamics.