Motivated by recent experiments with ultracold quantum gases in optical lattices we study the decay of the staggered moment in the one-dimensional Fermi-Hubbard model starting from a perfect N\'eel state using exact diagonalization and the infinite-system-size time-evolving-block-decimation method. This extends previous work in which the same problem has been addressed for pure spin Hamiltonians. As a main result, we show that the relaxation dynamics of the double occupancy and of the staggered moment are different. The former is controlled by the nearest-neighbor tunneling rate while the latter is much slower and strongly dependent on the interaction strength, indicating that spin excitations are important. This difference in characteristic energy scales for the fast charge dynamics and the much slower spin dynamics is also reflected in the real-time evolution of nearest-neighbor density and spin correlations. A very interesting time dependence emerges in the von Neumann entropy, which at short times increases linearly with a slope proportional to the tunneling matrix element while the long-time growth of entanglement is controlled by spin excitations. Our predictions for the different relaxation dynamics of the staggered moment and the double occupancy should be observable in state-of-the-art optical lattice experiments. We further compare time averages of the double occupancy to the expectation values in both the canonical and diagonal ensembles, which quantitatively disagree with each other on finite systems. We relate the question of thermalization to the eigenstate thermalization hypothesis.
Motivated by recent experiments with ultracold quantum gases in optical lattices we study the decay of the staggered moment in the one-dimensional Fermi-Hubbard model starting from a perfect N'eel state using exact diagonalization and the infinite-system-size time-evolving-block-decimation method. This extends previous work in which the same problem has been addressed for pure spin Hamiltonians. As a main result, we show that the relaxation dynamics of the double occupancy and of the staggered moment are different. The former is controlled by the nearest-neighbor tunneling rate while the latter is much slower and strongly dependent on the interaction strength, indicating that spin excitations are important. This difference in characteristic energy scales for the fast charge dynamics and the much slower spin dynamics is also reflected in the real-time evolution of nearest-neighbor density and spin correlations. A very interesting time dependence emerges in the von Neumann entropy, which at short times increases linearly with a slope proportional to the tunneling matrix element while the long-time growth of entanglement is controlled by spin excitations. Our predictions for the different relaxation dynamics of the staggered moment and the double occupancy should be observable in state-of-the-art optical lattice experiments. We further compare time averages of the double occupancy to the expectation values in both the canonical and diagonal ensembles, which quantitatively disagree with each other on finite systems. We relate the question of thermalization to the eigenstate thermalization hypothesis.
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.
Thermalization in strongly correlated fermionic systems remains a central open problem in quantum many-body physics. In this work, we investigate the real-time dynamics and the approach to thermalization in the two-dimensional Hubbard model, a paradigmatic framework for correlated electrons, relevant to high-temperature superconductivity and ultracold quantum simulation. Focusing on the half-filled square lattice, we monitor the time evolution of the double occupancy following a quench in the on-site interaction U, and assess whether its long-time value is captured by a canonical thermal ensemble. We employ time-dependent variational Monte Carlo methods combined with transformer-based Neural-Network Quantum States to accurately describe the nonequilibrium dynamics of fermions, especially for the behavior at long times, thereby accessing regimes that were previously inaccessible to numerical simulations. Our results reveal two dynamical behaviors: for weak to intermediate interactions, the double occupancy rapidly approaches the thermal prediction, consistent with ergodic evolution; beyond a critical interaction UC, the dynamics remains distinct from the thermal expectation on the timescales investigated, revealing signatures of a prethermal plateau delaying fast relaxation. These results establish numerical simulation as a powerful tool to probe nonequilibrium quantum phenomena in correlated fermionic matter.
We investigate signatures of non-ergodic behavior in the real-time evolution of a one-dimensional Bose-Hubbard model, where the initial state is a doubly occupied density-wave state. We show that the occupation dynamics at strong interactions is dominated by doublon-holon exchange which leads to a domain wall excitation and propagation. The latter manifests as a negated staggered pattern in the density-density correlations. While the single-particle and the pair correlation functions show highly localized correlations that decay rapidly away from the nearest neighbor. We show that the time scale of the domain-wall excitations depends on the inverse of the interaction strength and therefore dictates the slow relaxation dynamics. In the presence of a parabolic trap, the occupation dynamics at the edges become frozen and further suppresses the propagation of correlations. This suppression happens even for trap strengths weaker than the tunneling rate. We also show that the model can be mapped to an antiferromagnetic transverse-field Ising model in the limit of strong interactions and that the correlation-propagation velocity in the original model is well captured by the group velocity of the spin-wave excitation in the effective spin model.
We investigate the formation of charge and spin ordering by starting from a noninteracting state and studying how it evolves in time under a Hamiltonian with finite electronic interactions. We consider the one-dimensional, half-filled extended Hubbard model, which we solve within time-dependent density matrix renormalization group. By employing linear finite-time quenches in on-site and nearest-neighbor interactions, we find the existence of impulse, intermediate, and adiabatic regimes of time evolution. For the quenches we analyze, we observe that the adiabatic regime is reached with distinct ramping timescales depending on whether the charge density wave (CDW) or the spin density wave (SDW) is formed. The former needs to be slower than the latter to prevent entangled excited states from being accessed during the quench. More interestingly, in the intermediate regime, we observe an enhancement of the entanglement entropy with respect to its initial value, which precedes the formation of the CDW ordering; a similar enhancement is not seen in the quench toward SDW. Our findings also show that the breaking of the system integrability, by turning on the nearest-neighbor interactions, does not give rise to significant changes in the nonequilibrium behavior within the adiabatic approximation.
Two fermions occupying the same site of a lattice model with strongly repulsive Hubbard-type interaction U form a doublon, a long-living excitation, the decay of which is suppressed because of energy conservation. By means of an exact-diagonalization approach based on the Krylov-space technique, we study the dynamics of a single doublon, of two doublons, and of a doublon in the presence of two additional fermions prepared locally in the initial state of the extended Hubbard model. The time dependence of the expectation value of the double occupancy at the different sites of a large one-dimensional lattice is analyzed by perturbative arguments. In this way the spatiotemporal evolution of the doublon can be understood. The initial decay takes place on a short time scale 1/U, and the long-time average of the decayed fraction of the total double occupancy scales as 1/U2. We demonstrate how the dynamics of a doublon in the initial state is related to the spectrum of two-fermion excitations obtained from linear-response theory, we work out the difference between doublons composed of fermions vs doublons composed of bosons, and we show that despite the increase of phase space for inelastic decay processes, the stability of a doublon is enhanced by the presence of additional fermions on an intermediate time scale.
We compute exactly the von Neumann entanglement entropy of the eta-pairing states - a large set of exact excited eigenstates of the Hubbard Hamiltonian. For the singlet eta-pairing states the entropy scales with the logarithm of the spatial dimension of the (smaller) partition. For the eta-pairing states with finite spin magnetization density, the leading term can scale as the volume or as the area-times-log, depending on the momentum space occupation of the Fermions with flipped spins. We also compute the corrections to the leading scaling. In order to study the eigenstate thermalization hypothesis (ETH), we also compute the entanglement Rényi entropies of such states and compare them with the corresponding entropies of thermal density matrix in various ensembles. Such states, which we find violate strong ETH, may provide a useful platform for a detailed study of the time-dependence of the onset of thermalization due to perturbations which violate the total pseudospin conservation.