The generalized Gibbs ensemble (GGE) was introduced ten years ago to describe observables in isolated integrable quantum systems after equilibration. Since then, the GGE has been demonstrated to be a powerful tool to predict the outcome of the relaxation dynamics of few-body observables in a variety of integrable models, a process we call generalized thermalization. This review discusses several fundamental aspects of the GGE and generalized thermalization in integrable systems. In particular, we focus on questions such as: which observables equilibrate to the GGE predictions and who should play the role of the bath; what conserved quantities can be used to construct the GGE; what are the differences between generalized thermalization in noninteracting systems and in interacting systems mappable to noninteracting ones; why is it that the GGE works when traditional ensembles of statistical mechanics fail. Despite a lot of interest in these questions in recent years, no definite answers have been given. We review results for the XX model and for the transverse field Ising model. For the latter model, we also report original results and show that the GGE describes spin-spin correlations over the entire system. This makes apparent that there is no need to trace out a part of the system in real space for equilibration to occur and for the GGE to apply. In the past, a spectral decomposition of the weights of various statistical ensembles revealed that generalized eigenstate thermalization occurs in the XX model (hard-core bosons). Namely, eigenstates of the Hamiltonian with similar distributions of conserved quantities have similar expectation values of few-spin observables. Here we show that generalized eigenstate thermalization also occurs in the transverse field Ising model. is known as the diagonal ensemble density matrix [74], and ρ[n] DE = |c n | 2 . This has been shown to occur in numerical studies of nonintegrable systems [74,86,87,88]. Degeneracies

Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g = log 10 (tH/t Th ), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t Th and the Heisenberg time tH, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t Th ≈ tH, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.

We study the bipartite von Neumann entanglement entropy and matrix elements of local operators in the eigenstates of an interacting integrable Hamiltonian (the paradigmatic spin-1/2 XXZ chain), and contrast their behavior with that of quantum chaotic systems. We find that the leading term of the average (over all eigenstates in the zero magnetization sector) eigenstate entanglement entropy has a volume-law coefficient that is smaller than the universal (maximal entanglement) one in quantum chaotic systems. This establishes the entanglement entropy as a powerful measure to distinguish integrable models from generic ones. Remarkably, our numerical results suggest that the volume-law coefficient of the average entanglement entropy of eigenstates of the spin-1/2 XXZ chain is the same as, or very close to, the one for translationally invariant quadratic fermionic models. We also study matrix elements of local operators in the eigenstates of the spin-1/2 XXZ chain at the center of the spectrum. For the diagonal matrix elements, we show evidence that the support does not vanish with increasing system size, while the average eigenstate to eigenstate fluctuations vanish in a power law fashion. For the off-diagonal matrix elements, we show that they follow a distribution that is close to (but not quite) log-normal, and that their variance is a well-defined function of ω = Eα − E β ({Eα} are the eigenenergies) whose magnitude scales as 1/D, where D is the Hilbert space dimension.

In a seminal paper [Phys. Rev. Lett. 71, 1291], Page proved that the entanglement entropy of typical pure states is Styp ln DA − (1/2)D 2 A /D, for 1 DA ≤ √ D, where DA and D are the Hilbert space dimensions of the subsystem and the system, respectively. Typical pure states are hence (nearly) maximally entangled. We develop tools to compute the average entanglement entropy S of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models, ln DA − (ln DA)2 / ln D. Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then S < ln DA in the thermodynamic limit, i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal, i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.Introduction. The concept of entanglement is a cornerstone in modern quantum physics. Different measures of entanglement have been extensively used to probe the structure of pure quantum states [1], and they have started to be measured in experiments with ultracold atoms in optical lattices [2,3]. Here, we are interested in the bipartite entanglement entropy (referred to as the entanglement entropy) in fermionic lattice systems. In such systems, an upper bound for the entanglement entropy of a subsystem A (smaller than its complement) is S max = ln D A , where D and D A are the dimensions of the Hilbert space of the system and of the subsystem, with D A ≤ √ D (see Fig. 1 for an example for spinless fermions). Note that ln D A ∝ V A , where V A is the number of sites in A, i.e., this upper bound scales with the "volume" of A. (When A is larger than its complement, the Hilbert space of the complement is the one that determines S.) Almost twenty-four years ago, motivated by the puzzle of information in black hole radiation [4], Page proved [5] that typical (with respect to the Haar measure) pure states nearly saturate that bound (the correction is exponentially small) [6][7][8][9][10]. Their reduced density matrices are thermal at infinite temperature [11][12][13].In stark contrast with typical pure states, ground states and low-lying excited states of local Hamiltonians are known to exhibit an area-law entanglement [1]. Namely, their entanglement entropy scales with the area of the boundary of the subsystem. On the other hand, most eigenstates of local Hamiltonians at nonzero energy densities above the ground state are expected to have a volume-law entanglement entropy (with the exception of many-body localized systems [14,15]). Within the eigenstate thermalization hypothesis (ETH) [16][17][18], one expects volume-law entanglement in all eigenstates (excluding those at the edges of the spectrum) of quantum chaotic Hamiltonians [19][20][21][22][23], with those in the center of the spectrum exhibiting maximal entanglement [...

In quantum statistical mechanics, it is of fundamental interest to understand how close the bipartite entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.Introduction.-Entanglement in few-body quantum systems is a topic that triggered special interest since the birth of quantum mechanics [1,2]. In recent years, interest has shifted towards entanglement in systems with many degrees of freedom, which, e.g., is relevant to current problems in the fields of condensed matter, quantum information, and quantum gravity [3]. In condensed matter and quantum information, the concept of entanglement has played an essential role in designing efficient numerical algorithms [4][5][6], in understanding quantum phase transitions [7][8][9], and in characterizing the dynamics after quantum quenches [10][11][12]. Also, studies with ultracold atoms in optical lattices [13,14] have begun the experimental exploration of entanglement in and out of equilibrium. An important aspect of these experimental systems is that the size of the subsystem of interest is not necessarily a vanishing fraction of the size of the entire system, in contrast to traditional statistical mechanics.Despite considerable theoretical efforts [15][16][17][18][19][20][21][22][23][24][25][26][27][28], rigorous understanding of the behavior of the bipartite entanglement entropy in eigenstates of generic (quantum chaotic) Hamiltonians (with ground states being an exception [29][30][31][32][33]) is lacking. The expectation is that typical eigenstates at high temperature are (nearly) maximally entangled. This follows from the result by Page [34], who proved that, for a bipartition of a system into subsystem A and its complement B, the average entanglement entropy of random pure states is

Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized

Long-range order in quantum many-body systems is usually associated with equilibrium situations. Here, we experimentally investigate the quasicondensation of strongly-interacting bosons at finite momenta in a far-from-equilibrium case. We prepare an inhomogeneous initial state consisting of one-dimensional Mott insulators in the center of otherwise empty one-dimensional chains in an optical lattice with a lattice constant d. After suddenly quenching the trapping potential to zero, we observe the onset of coherence in spontaneously forming quasicondensates in the lattice. Remarkably, the emerging phase order differs from the ground-state order and is characterized by peaks at finite momenta ±(π/2)( /d) in the momentum distribution function.

We investigate the expansion of bosons and fermions in a homogeneous lattice after a sudden removal of the trapping potential using exact numerical methods. As a main result, we show that in one dimension, both bosonic and fermionic Mott insulators expand with the same velocity, irrespective of the interaction strength, provided the expansion starts from the ground state of the trapped gas. Furthermore, their density profiles become identical during the expansion; the asymptotic density dynamics is identical to that of initially localized, noninteracting particles, and the asymptotic velocity distribution is flat. The expansion velocity for initial correlated Mott insulating states is therefore independent of the interaction strength and particle statistics. Interestingly, this nonequilibrium dynamics is sensitive to the interaction driven quantum phase transition in the Bose-Hubbard model; while being constant in the Mott phase, the expansion velocity decreases in the superfluid phase and vanishes for large systems in the noninteracting limit. These results are compared to the setup of a recent experiment [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)], where the trap opening was combined with an interaction quench from infinitely strong interactions to finite values. In the latter case, the interaction quench breaks the universal dynamics in the asymptotic regime and the expansion depends on the interaction strength. We carry out an analogous analysis for a two-component Fermi gas, with similar observations. In addition, we study the effect of breaking the integrability of hard-core bosons in different ways; while the fast ballistic expansion from the ground state of Mott insulators in one dimension remains unchanged for finite interactions, we observe strong deviations from this behavior on a two-leg ladder even in the hard-core case. This change in dynamics bares similarities with the dynamics in the dimensional crossover from one to two dimensions observed in the aformentioned experimental study.