We provide an account of the static and dynamic properties of hard-core bosons in a one-dimensional lattice subject to a multichromatic quasiperiodic potential for which the single-particle spectrum has mobility edges. We use the mapping from strongly interacting bosons to weakly interacting fermions and provide exact numerical results for hard-core bosons in and out of equilibrium. In equilibrium, we find that the system behaves like a quasi-condensate (insulator) depending on whether the Fermi surface of the corresponding fermionic system lies in a spectral region where the single-particle states are delocalized (localized). We also study nonequilibrium expansion dynamics of initially trapped bosons and demonstrate that the extent of partial localization is determined by the single-particle spectrum.
We provide an account of the static and dynamic properties of hard-core bosons in a one-dimensional lattice subject to a multichromatic quasiperiodic potential for which the single-particle spectrum has mobility edges. We use the mapping from strongly interacting bosons to weakly interacting fermions and provide exact numerical results for hard-core bosons in and out of equilibrium. In equilibrium, we find that the system behaves like a quasi-condensate (insulator) depending on whether the Fermi surface of the corresponding fermionic system lies in a spectral region where the single-particle states are delocalized (localized). We also study nonequilibrium expansion dynamics of initially trapped bosons and demonstrate that the extent of partial localization is determined by the single-particle spectrum.
We explore the ground-state properties of cold atomic gases focusing on the cases of noninteracting fermions and hard-core (Tonks-Girardeau) bosons, trapped by the combination of two potentials (bichromatic lattice) with incommensurate periods. For such systems, two limiting cases have been thoroughly established. In the tight-binding limit, the single-particle states in the lowest occupied band show a localization transition, as the strength of the second potential is increased above a certain threshold. In the continuous limit, when the tight-binding approximation does not hold, a mobility edge is found, instead, whose position in energy depends upon the strength of the second potential. Here, we study how the crossover from the discrete to the continuum behavior occurs, and prove that signatures of the localization transition and mobility edge clearly appear in the generic many-body properties of the systems. Specifically, we evaluate the momentum distribution, which is a routinely measured quantity in experiments with cold atoms, and demonstrate that, even in the presence of strong boson-boson interactions (infinite in the Tonks-Girardeau limit), the single-particle mobility edge can be observed in the ground-state properties.
We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.
Noninteracting fermions in one dimension can undergo a localization-delocalization transition in the presence of a quasiperiodic potential as a function of that potential. In the presence of interactions, this transition transforms into a many-body localization (MBL) transition. Recent studies have suggested that this type of transition can also occur in models with quasiperiodic potentials that possess single-particle mobility edges. Two such models were studied by Modak and Mukerjee [Phys. Rev. Lett. 115, 230401 (2015)] but only one was found to exhibit an MBL transition in the presence of interactions while the other one did not. In this work we investigate the occurrence of MBL in the presence of weak interactions in five different models with single-particle mobility edges in one dimension with a view to obtaining a criterion for the same. We find that not all such models undergo a thermal-MBL phase transition in the presence of weak interactions. We propose a criterion to determine whether MBL is likely to occur in the presence of interaction based only on the properties of the noninteracting models. The relevant quantity $\ensuremath{\epsilon}$ is a measure of how localized the localized states are relative to how delocalized the delocalized states are in the noninteracting model. We also study various other features of the noninteracting models such as the divergence of the localization length at the mobility edge and the presence or absence of ``ergodicity'' and localization in their many-body eigenstates. However, we find that these features cannot be used to predict the occurrence of MBL upon the introduction of weak interactions.
Abstract We study the relaxation dynamics and statistical properties of hard-core bosons in a one-dimensional quasiperiodic lattice with a power-law potential following a quantum quench. Through numerical simulation of the time evolution of various one-body observables, including the on-site density, the momentum distribution, and natural orbital occupations, we find that these observables relax to non-thermal steady states. The temporal fluctuations around these steady values vanish in the thermodynamic limit. Moreover, the long-time statistics of all observables are precisely captured by a generalized Gibbs ensemble. This result confirms that generalized thermalization remains valid even in systems featuring mobility edges, where extended and localized single-particle states coexist. Our findings offer a quantitative basis for future experiments on generalized thermalization using quantum simulators with tunable mobility edges.
We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential h<hc, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for h>hc all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for h<hc, many-body states at the top (with E>E2) and the bottom of the spectrum (with E<E1) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at E1,2, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below E1 and above E2.
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.