Exact studies of equilibrium and nonequilibrium properties of correlated bosons in one-dimensional lattices

In this thesis, we study both equilibrium and nonequilibrium properties of hard-core bosons trapped in one-dimensional lattices. To perform many-body analyses of large systems, we utilize exact numerical approaches including an approach based on the Bose-Fermi mapping and the Lanczos method. We study noise correlations of hard-core bosons in homogeneous lattices, period-two superlattices, and disordered lattices, and focus on the scaling of such correlations with system size in the superfluid and insulating phases. We find that superfluid phases exhibit a leading linear scaling, while the leading terms in the scaling of the Mott-insulting and Bose-glass phases are constants. We also characterize the disorder-induced phase transition between a superfluid and a Bose-glass phase in an incommensurate lattice system by determining the critical exponents in the scaling of the momentum distribution and the noise correlations. We show that the phase transition is signaled by peaks in the first derivatives of the noise correlations with respect to the strength of quasiperiodic disorder, and the height of the peaks diverges with increasing system size. Furthermore, related to the nonequilibrium properties of isolated systems, we investigate the initial-state dependence of the outcome of relaxation dynamics following quantum quenches. Starting from a thermal state associated with a finite initial temperature, the entropy of the generalized Gibbs ensemble, introduced to describe integrable systems after relaxation, is found to be generally different from the entropy in thermal equilibrium. The disagreement is explained to stem from the distinction between the conserved quantities in the initial state and those in the thermal ensembles. On the other hand, if the initial state is selected to be an eigenstate of a nonintegrable (chaotic) model, a thermal-like "ergodic" sampling of the eigenstates of the integrable Hamiltonian is unveiled by computing the weighted energy density. We show that the distribution of the conserved quantities in the chaotic initial state coincides with the thermal ones in thermodynamic limit. Our results indicate that quenches starting from nonintegrable initial states will lead to thermalization even if the final system is integrable.