We study Bose-Hubbard models in a family of diamond necklace lattices with $n$ central sites. The single-particle spectrum of these models presents compact localized states (CLSs) that occupy the up and down sites of each diamond. By performing an appropriate basis rotation, the fragmentation of the many-boson Hilbert space becomes apparent in the adjacency graph of the Hamiltonian, showing disconnected subsectors with a wide range of dimensions. The models present a conserved quantity related to the occupation of the single-particle CLSs that uniquely identifies the different subsectors of the many-boson Hilbert space. Due to the fragmentation of the Hilbert space, the distribution of entanglement entropies of the system presents a nested-dome structure. We find weak thermalization through subsector-restricted entanglement evolution and a wide range of entanglement entropy scalings from area-law to logarithmic growth. Additionally, we observe how the distinguishability between the different domes increases with the number of central sites and we explain the mechanism behind this fact by analyzing the graph structure of the Hamiltonian.
We study Bose-Hubbard models in a family of diamond necklace lattices with n central sites. The single-particle spectrum of these models presents compact localized states (CLSs) that occupy the up and down sites of each diamond. By performing an appropriate basis rotation, the fragmentation of the many-boson Hilbert space becomes apparent in the adjacency graph of the Hamiltonian, showing disconnected subsectors with a wide range of dimensions. The models present a conserved quantity related to the occupation of the single-particle CLSs that uniquely identifies the different subsectors of the many-boson Hilbert space. Due to the fragmentation of the Hilbert space, the distribution of entanglement entropies of the system presents a nested-dome structure. We find weak thermalization through subsector-restricted entanglement evolution and a wide range of entanglement entropy scalings from area-law to logarithmic growth. Additionally, we observe how the distinguishability between the different domes increases with the number of central sites and we explain the mechanism behind this fact by analyzing the graph structure of the Hamiltonian.
We demonstrate that a complete class of flat-band lattices with underlying commutative local symmetries exhibit a locally fragmented Hilbert space. The equitable partition theorem ensures distinct parities for the compact localized states (CLSs) present in this class of flat-band lattices and the extended eigenstates of the system. In the presence of on-site bosonic interactions, such models exhibit a conserved quantity, the parity of the number of particles located in all the CLSs in a unit cell. As a consequence, the Hilbert space presents local fragmentation, which is only revealed upon rotating the basis of the Hamiltonian that decouples the CLSs at the single-particle level. We find that the fragmentation is strong and also robust to the addition of long-range interactions. As an example, we numerically analyze the fragmentation of the one-dimensional pyrochlore chain, which exhibits nonintegrable sectors, effective single-particle sectors, and frozen states. We also show that the entanglement entropies form a nested-dome structure typical of these fragmented systems and that thermalization is restricted to each subsector.
We study Hilbert-space fragmentation and thermalization in a one-dimensional dipole-conserving Bose-Hubbard chain. By analyzing the structure of the Hamiltonian matrix in the Fock basis, we show that the system exhibits weak Hilbert-space fragmentation. We further construct an exponentially large family of frozen product states and derive analytical upper and lower bounds on their number. Using exact diagonalization, we examine the consequences of weak fragmentation for eigenstate half-chain entanglement, density relaxation dynamics, and level statistics. All these quantities reveal a transition from a weak eigenstate thermalization regime to a nonergodic regime with increasing on-site interaction strength. These results show that weak Hilbert-space fragmentation \textit{does not} preclude quantum chaos or thermalization, and provides a minimal platform for studying the interplay of dipole conservation, weak fragmentation, and ergodicity breaking.
While isolated quantum systems generally thermalize after long-time evolution, there are several exceptions defying thermalization. A notable mechanism of this nonergodicity is the Hilbert space fragmentation (HSF), where the Hamiltonian matrix splits into an exponentially large number of sectors due to the presence of nontrivial conserved quantities. Using ultracold gases, here, we experimentally investigate the one-dimensional Bose-Hubbard system with neither disorder nor tilt potential, which has been predicted to exhibit HSF caused by a strong interatomic interaction. Specifically, we analyze far-from-equilibrium dynamics starting from a charge density wave of doublons (atoms in doubly occupied sites) in a singlon- and doublon-resolved manner to reveal a slowing down of the relaxation in a strongly interacting regime. We find that the numbers of singlons and doublons are conserved during the dynamics, indicating HSF as a mechanism of the observed slow relaxation. Our results provide an experimental confirmation of the conserved quantities responsible for HSF.
The relaxation behaviour of isolated quantum systems taken out of equilibrium is among the most intriguing questions in many-body physics. Quantum systems out of equilibrium typically relax to thermal equilibrium states by scrambling local information and building up entanglement entropy. However, kinetic constraints in the Hamiltonian can lead to a breakdown of this fundamental paradigm due to a fragmentation of the underlying Hilbert space into dynamically decoupled subsectors in which thermalisation can be strongly suppressed. Here, we experimentally observe Hilbert space fragmentation (HSF) in a two-dimensional tilted Bose-Hubbard model. Using quantum gas microscopy, we engineer a wide variety of initial states and find a rich set of manifestations of HSF involving bulk states, interfaces and defects, i.e., d = 2, 1 and 0 dimensional objects. Specifically, uniform initial states with equal particle number and energy differ strikingly in their relaxation dynamics. Inserting controlled defects on top of a global, non-thermalising chequerboard state, we observe highly anisotropic, sub-dimensional dynamics, an immediate signature of their fractonic nature. An interface between localized and thermalising states in turn displays dynamics depending on its orientation. Our results mark the first observation of HSF beyond one dimension, as well as the concomitant direct observation of fractons, and pave the way for in-depth studies of microscopic transport phenomena in constrained systems
Abstract The relaxation behaviour of isolated quantum systems taken out of equilibrium is among the most intriguing questions in many-body physics 1 . Quantum systems out of equilibrium typically relax to thermal equilibrium states by scrambling local information and building up entanglement entropy. However, kinetic constraints in the Hamiltonian can lead to a breakdown of this fundamental paradigm owing to a fragmentation of the underlying Hilbert space into dynamically decoupled subsectors in which thermalization can be strongly suppressed 2–5 . Here we experimentally observe Hilbert space fragmentation in a two-dimensional tilted Bose–Hubbard model. Using quantum gas microscopy, we engineer a wide variety of initial states and find a rich set of manifestations of Hilbert space fragmentation involving bulk states, interfaces and defects, that is, two-, one- and zero-dimensional objects. Specifically, uniform initial states with equal particle number and energy differ strikingly in their relaxation dynamics. Inserting controlled defects on top of a global, non-thermalizing chequerboard state, we observe highly anisotropic, subdimensional dynamics, an immediate signature of their fractonic nature 6–9 . An interface between localized and thermalizing states in turn shows dynamics depending on its orientation. Our results mark the observation of Hilbert space fragmentation beyond one dimension, as well as the concomitant direct observation of fractons, and pave the way for in-depth studies of microscopic transport phenomena in constrained systems.
While the eigenstate entanglement entropy has been extensively studied for fermionic systems, much less is known about bosonic systems. Here, we study the entanglement entropy of mid-spectrum eigenstates of Bose-Hubbard models, focusing on weakly disordered models with and without particle-number conservation, and contrasting them with the translationally-invariant model. We analyze the volume-law and O(1) contributions to the entanglement entropy via the averages over mid-spectrum eigenstates and the corresponding distributions. We derive the volume-law coefficient of the entanglement entropy by generalizing the mean-field approach from [Phys. Rev. Lett. 119, 220603 (2017)] to many-body systems with a tunable local bosonic cutoff, which agrees with previous analytical and numerical results from [Phys. Rev. B 110, 235154 (2024)]. We show that the volume-law contribution to the entanglement entropy does not change upon breaking translational invariance via on-site disorder. We then numerically study the role of the subleading O(1) contribution to the entanglement entropy. We find that, in the particle-number conserving case, it exhibits a nontrivial dependence on the particle-number density and the local bosonic cutoff, while without particle-number conservation, results suggest the emergence of a universal O(1) contribution beyond the random pure state predictions.