Recent studies of interacting systems of quantum spins, ultracold atoms and correlated fermions have shed a new light on how isolated many-body systems can avoid rapid equilibration to their thermal state. It has been shown that many such systems can "weakly" break ergodicity: they possess a small number of non-thermalising eigenstates and/or display slow relaxation from certain initial conditions, while majority of other initial states equilibrate fast, like in conventional thermalising systems. In this chapter, we provide a pedagogical introduction to weak ergodicity breaking phenomena, including Hilbert space fragmentation and quantum many-body scars. Central to these developments have been the tools based on quantum entanglement, in particular matrix product states and tangent space techniques, which have allowed to analytically construct non-thermal eigenstates in various non-integrable quantum models, and to explore semiclassical quantisation of such systems in the absence of a large-N or mean field limit. We also discuss recent experimental realisations of weak ergodicity breaking phenomena in systems of Rydberg atoms and tilted optical lattices.
We study disorder-induced ergodicity breaking transition in high-energy eigenstates of interacting spin-1/2 chains. Using exact diagonalization we introduce a cost function approach to quantitatively compare different scenarios for the eigenstate transition. We study ergodicity indicators such as the eigenstate entanglement entropy and the spectral level spacing ratio, and we consistently find that an (infinite-order) Kosterlitz-Thouless transition yields a lower cost function when compared to a finite-order transition. Interestingly, we observe that the transition point in finite systems exhi…
Quantum many-body scars have been put forward as counterexamples to the Eigenstate Thermalization Hypothesis. These atypical states are observed in a range of correlated models as long-lived oscillations of local observables in quench experiments starting from selected initial states. The long-time memory is a manifestation of quantum non-ergodicity generally linked to a sub-extensive generation of entanglement entropy, the latter of which is widely used as a diagnostic for identifying quantum many-body scars numerically as low entanglement outliers. Here we show that, by adding kinetic constr…
We consider a Bose-Josephson junction (BJJ) formed by a binary mixture of ultracold atoms to investigate the manifestation of coherent collective dynamics on ergodicity and quantum scars, unfolding the connection between them. By tuning the inter- and intra-species interaction, we demonstrate a rich variety of Josephson dynamics and transitions between them, which plays a crucial role in controlling the overall ergodic behavior. The signature of underlying classicality is revealed from the entanglement spectrum, which also elucidates the formation of quantum scars of unstable steady states and…
Unlike classical system, understanding ergodicity from phase space mixing remains unclear for interacting quantum systems due to the absence of phase space trajectories. By considering an interacting spin model known as kicked coupled top, we elucidate the manifestation of phase space dynamics on local ergodic behavior of its quantum counterpart and quantum scarring phenomena. A transition to chaos occurs by increasing the kicking strength, and in the mixed phase space, the islands of regular motions within the chaotic sea clearly exhibit deviation from ergodicity, which we quantify from entan…
We study ergodicity breaking in the clean Bose-Hubbard chain for small hopping strength. We see the existence of a non-ergodic regime by means of indicators as the half-chain entanglement entropy of the eigenstates, the average level spacing ratio, {the properties of the eigenstate-expectation distribution of the correlation and the scaling of the Inverse Participation Ratio averages.} We find that this ergodicity breaking {is different from many-body localization} because the average half-chain entanglement entropy of the eigenstates obeys volume law. This ergodicity breaking appears unrelate…
Many-body localization (MBL) behavior is analyzed {in an extended Bose-Hubbard model with quasiperiodic infinite-range interactions. No additional disorder is present. Examining level statistics and entanglement entropy of eigenstates we show that a significant fraction of eigenstates of the system is localized in the presence of strong interactions. In spite of this, our results suggest that the system becomes ergodic in the standard thermodynamic limit in which the energy of the system is extensive. At the same time, the MBL regime seems to be stable if one allows for a super-extensive scali…
We investigate many body localization in the presence of a single particle mobility edge. By considering an interacting deterministic model with an incommensurate potential in one dimension we find that the single particle mobility edge in the noninteracting system leads to a many body mobility edge in the corresponding interacting system for certain parameter regimes. Using exact diagonalization, we probe the mobility edge via energy resolved entanglement entropy (EE) and study the energy resolved applicability (or failure) of the eigenstate thermalization hypothesis (ETH). Our numerical resu…
Recent studies of quantum circuit models have theoretically shown that frequent measurements induce a transition in a quantum many-body system, which is characterized by the change of the scaling law of the entanglement entropy from a volume law to an area law. In order to propose a way for experimentally observing this measurement-induced transition, we present numerical analyses using matrix-product states on quench dynamics of a dissipative Bose-Hubbard model with controllable two-body losses, which has been realized in recent experiments with ultracold atoms. We find that when the strength…
We provide evidence that a clean kicked Bose-Hubbard model exhibits a many-body dynamically localized phase. This phase shows ergodicity breaking up to the largest sizes we were able to consider. We argue that this property persists in the limit of large size. The Floquet states violate eigenstate thermalization and then the asymptotic value of local observables depends on the initial state and is not thermal. This implies that the system does not generically heat up to infinite temperature, for almost all the initial states. Differently from many-body localization here the entanglement entrop…
We analyze the localization properties of the disordered Hubbard model in the presence of a synthetic magnetic field. An analysis of level spacing ratio shows a clear transition from ergodic to many-body localized phase. The transition shifts to larger disorder strengths with increasing magnetic flux. Study of dynamics of local correlations and entanglement entropy indicates that charge excitations remain localized whereas spin degree of freedom gets delocalized in the presence of the synthetic flux. This residual ergodicity is enhanced by the presence of the magnetic field with dynamical obse…
Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $ k $-ETH ($ k=1,2,\dots $), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, wh…
We find that a flat-band fermion system with interactions and without disorders exhibits non-thermalized ergodicity-breaking dynamics, an analog of many-body localization (MBL). In the previous works, we observed flat-band many-body localization (FMBL) in the Creutz ladder model. The origin of FMBL is a compact localized state governed by local integrals of motion (LIOMs), which are to be obtained explicitly. In this work, we clarify the dynamical aspects of FMBL. We first study dynamics of two-particles, and find that the states are not substantially modified by weak interactions, but the per…
