A system of trapped ions under the action of off-resonant standing-waves can be used to simulate a variety of quantum spin models. In this work, we describe theoretically quantum phases that can be observed in the simplest realization of this idea: quantum Ising and XY models. Our numerical calculations with the Density Matrix Renormalization Group method show that experiments with ion traps should allow one to access general properties of quantum critical systems. On the other hand, ion trap quantum spin models show a few novel features due to the peculiarities of induced effective spin-spin interactions which lead to interesting effects like long-range quantum correlations and the coexistence of different spin phases.

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^{a}. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a>0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a<1) and short-range hops (a>1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.

One-dimensional quasi-periodic systems with power-law hopping, 1/r a , differ from both the standard Aubry-Azbel-Harper (AAH) model and from power-law systems with uncorrelated disorder. Whereas in the AAH model all single-particle states undergo a transition from ergodic to localized at a critical quasi-disorder strength, short-range power-law hops with a > 1 can result in mobility edges. We find that there is no localization for long-range hops with a ≤ 1, in contrast to the case of uncorrelated disorder. Systems with long-range hops rather present ergodic-to-multifractal edges and a phase transition from ergodic to multifractal (extended but non-ergodic) states. Both mobility and ergodic-to-multifractal edges may be clearly revealed in experiments on expansion dynamics.

We show that a time-dependent magnetic field inducing a periodically modulated scattering length may lead to interesting novel scenarios for cold gases in optical lattices, characterized by a nonlinear hopping depending on the number difference at neighboring sites. We discuss the rich physics introduced by this hopping, including pair superfluidity, exactly defect-free Mott-insulator states for finite hopping, and pure holon and doublon superfluids. We also address experimental detection, showing that the introduced nonlinear hopping may lead in harmonically trapped gases to abrupt drops in the density profile marking the interface between different superfluid regions.PACS numbers: 37.10. Jk, 67.85.Hj, 73.43.Nq Ultracold atoms in optical lattices formed by laser beams provide an excellent environment for studying lattice models of general relevance in condensed-matter physics, and in particular, variations of the celebrated Hubbard model [1,2]. Cold lattice gases allow for an unprecedented degree of control of various experimental parameters, even in real time. In particular, interparticle interactions can be changed by means of Feshbach resonances [3]. Moreover, recent milestone achievements allow for site-resolved detection, permitting the study of in situ densities [4,5], and more involved measurements, as that of nonlocal parity order [6].The modulation of the lattice parameters in real time opens interesting possibilities of control and quantum engineering. In particular, a periodic lattice shaking translates by means of Floquet's theorem [7,8] into a modified hopping constant [9], which may even reverse its sign as shown in experiments [10,11]. This technique has been employed to drive the Mott-insulator (MI) to superfluid (SF) transition [12], and to simulate frustrated classical magnetism [13]. Recent experiments have explored as well the fascinating perspectives offered by periodically driven lattices in strongly correlated gases [14,15].The effective Hubbard-like models describing ultracold lattice gases are typically characterized by a hopping independent of the number of particles at the sites. This is, however, not necessarily the case. Multiband physics [16][17][18] and dipolar interactions for sufficiently large dipole moments [19] may lead to occupation-dependent hopping. A major consequence of nonlinear hopping is the possibility to observe pair superfluidity (PSF) [19,20], which resembles pairing in SF Fermi gases, although for bosons superfluidity exists as well without pairing.In this Letter, we consider a cold lattice gas in the presence of a periodically modulated magnetic field. In the vicinity of a Feshbach resonance, this field induces modulated interparticle interactions [21]. Interestingly, Ref.[22] has shown that periodic modulations of the interaction strength may lead to a many-body coherent destruction of tunneling in two-mode Bose-Einstein condensates. As shown below, the generalization of this effect to lattice gases leads under proper conditions to an effective Hubbard-like ...

We determine the phase diagram and the momentum distribution for a one-dimensional Bose gas with\ud repulsive short-range interactions in the presence of a two-color lattice potential, with an incommensurate ratio\ud among the respective wavelengths, by using a combined numerical density matrix renormalization group and\ud analytical bosonization analysis. The system displays a delocalized superfluid phase at small values of the\ud intensity of the secondary lattice V2 and a localized Bose-glasslike phase at larger intensity V2. We analyze\ud the localization transition as a function of the height V2 beyond the known limits of free and hard-core bosons.\ud We find that weak repulsive interactions disfavor the localized phase, i.e., they increase the critical value of V2\ud at which localization occurs. In the case of integer filling of the primary lattice, the phase diagram at fixed\ud density displays, in addition to a transition from a superfluid to a Bose glass phase, a transition to a Mottinsulating\ud state for not too large V2 and large repulsion. We also analyze the emergence of a Bose-glass phase\ud by looking at the evolution of the Mott-insulator lobes when increasing V2. The Mott lobes shrink and\ud disappear above a critical value of V2. Finally, we characterize the superfluid phase by the momentum distribution,\ud and show that it displays a power-law decay at small momenta typical of Luttinger liquids, with an\ud exponent depending on the combined effect of the interactions and of the secondary lattice. In addition, we\ud observe two side peaks that are due to the diffraction of the Bose gas by the second lattice. This latter feature\ud could be observed in current experiments as characteristics of pseudo-random Bose systems