The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: I. Basic Results

The problem of finding the exact energies and configurations for the Frenkel-Kontorova model consisting of particles in one dimension connected to their nearest-neighbors by springs and placed in a periodic potential consisting of segments from parabolas of identical (positive) curvature but arbitrary height and spacing, is reduced to that of minimizing a certain convex function defined on a finite simplex.Comment: 12 RevTeX pages, using AMS-Fonts (amssym.tex,amssym.def), 6 Postscript figures, accepted by Phys. Rev.

Bosons in one-dimensional incommensurate superlattices

We investigate numerically the zero-temperature physics of the one-dimensional Bose-Hubbard model in an incommensurate cosine potential, recently realized in experiments with cold bosons in optical superlattices L. Fallani et al., Phys. Rev. Lett. 98, 130404, (2007)]. An incommensurate cosine potential has intermediate properties between a truly periodic and a fully random potential, displaying a characteristic length scale (the quasi-period) which is shown to set a finite lower bound to the excitation energy of the system at special incommensurate fillings. This leads to the emergence of gapped incommensurate band-insulator (IBI) phases along with gapless Bose-glass (BG) phases for strong quasi-periodic potential, both for hardcore and softcore bosons. Enriching the spatial features of the potential by the addition of a second incommensurate component appears to remove the IBI regions, stabilizing a continuous BG phase over an extended parameter range. Moreover we discuss the validity of the local-density approximation in presence of a parabolic trap, clarifying the notion of a local BG phase in a trapped system; we investigate the behavior of first- and second-order coherence upon increasing the strength of the quasi-periodic potential; and we discuss the ab-initio derivation of the Bose-Hubbard Hamiltonian with quasi-periodic potential starting from the microscopic Hamiltonian of bosons in an incommensurate superlattice.Comment: 22 pages, 28 figure

Exploring the grand-canonical phase diagram of interacting bosons in optical lattices by trap squeezing

In this paper we theoretically discuss how quantum simulators based on trapped cold bosons in optical lattices can explore the grand-canonical phase diagram of homogeneous lattice boson models, via control of the trapping potential independently of all other experimental parameters (trap squeezing). Based on quantum Monte Carlo, we establish the general scaling relation linking the global chemical potential to the Hamiltonian parameters of the Bose-Hubbard model in a parabolic trap, describing cold bosons in optical lattices; we find that this scaling relation is well captured by a modified Thomas-Fermi scaling behavior - corrected for quantum fluctuations - in the case of high enough density and/or weak enough interactions, and by a mean-field Gutzwiller Ansatz over a much larger parameter range. The above scaling relation allows to control experimentally the chemical potential, independently of all other Hamiltonian parameters, via trap squeezing; given that the global chemical potential coincides with the local chemical potential in the trap center, measurements of the central density as a function of the chemical potential gives access to the information on the bulk compressibility of the Bose-Hubbard model. Supplemented with time-of-flight measurements of the coherence properties, the measurement of compressibility enables one to discern among the various possible phases realized by bosons in an optical lattice with or without external (periodic or random) potentials -- e.g. superfluid, Mott insulator, band insulator, and Bose glass. We theoretically demonstrate the trap-squeezing investigation of the above phases in the case of bosons in a one-dimensional optical lattice, and in a one-dimensional incommensurate superlattice.Comment: 27 pages, 26 figures. v2: added references and further discussion of the local-density approximation

Localization in one-dimensional incommensurate lattices beyond the Aubry-Andr\'e model

Localization properties of particles in one-dimensional incommensurate lattices without interaction are investigated with models beyond the tight-binding Aubry-Andr\'e (AA) model. Based on a tight-binding t_1 - t_2 model with finite next-nearest-neighbor hopping t_2, we find the localization properties qualitatively different from those of the AA model, signaled by the appearance of mobility edges. We then further go beyond the tight-binding assumption and directly study the system based on the more fundamental single-particle Schr\"odinger equation. With this approach, we also observe the presence of mobility edges and localization properties dependent on incommensuration.Comment: 5 pages, 6 figure

Localization in one dimensional lattices with non-nearest-neighbor hopping: Generalized Anderson and Aubry-Andr\'e models

We study the quantum localization phenomena of noninteracting particles in one-dimensional lattices based on tight-binding models with various forms of hopping terms beyond the nearest neighbor, which are generalizations of the famous Aubry-Andr\'e and noninteracting Anderson model. For the case with deterministic disordered potential induced by a secondary incommensurate lattice (i.e. the Aubry-Andr\'e model), we identify a class of self dual models, for which the boundary between localized and extended eigenstates are determined analytically by employing a generalized Aubry-Andr\'e transformation. We also numerically investigate the localization properties of non-dual models with next-nearest-neighbor hopping, Gaussian, and power-law decay hopping terms. We find that even for these non-dual models, the numerically obtained mobility edges can be well approximated by the analytically obtained condition for localization transition in the self dual models, as long as the decay of the hopping rate with respect to distance is sufficiently fast. For the disordered potential with genuinely random character, we examine scenarios with next-nearest-neighbor hopping, exponential, Gaussian, and power-law decay hopping terms numerically. We find that the higher order hopping terms can remove the symmetry in the localization length about the energy band center compared to the Anderson model. Furthermore, our results demonstrate that for the power-law decay case, there exists a critical exponent below which mobility edges can be found. Our theoretical results could, in principle, be directly tested in shallow atomic optical lattice systems enabling non-nearest-neighbor hopping.Comment: 18 pages, 24 figures updated with additional reference

Order in extremal trajectories

Given a chaotic dynamical system and a time interval in which some quantity takes an unusually large average value, what can we say of the trajectory that yields this deviation? As an example, we study the trajectories of the archetypical chaotic system, the baker's map. We show that, out of all irregular trajectories, a large-deviation requirement selects (isolated) orbits that are periodic or quasiperiodic. We discuss what the relevance of this calculation may be for dynamical systems and for glasses

Oscillatory Instabilities of Standing Waves in One-Dimensional Nonlinear Lattices

In one-dimensional anharmonic lattices, we construct nonlinear standing waves (SWs) reducing to harmonic SWs at small amplitude. For SWs with spatial periodicity incommensurate with the lattice period, a transition by breaking of analyticity versus wave amplitude is observed. As a consequence of the discreteness, oscillatory linear instabilities, persisting for arbitrarily small amplitude in infinite lattices, appear for all wave numbers Q not equal to zero or \pi. Incommensurate analytic SWs with |Q|>\pi/2 may however appear as 'quasi-stable', as their instability growth rate is of higher order.Comment: 4 pages, 6 figures, to appear in Phys. Rev. Let

Absence of Wavepacket Diffusion in Disordered Nonlinear Systems

We study the spreading of an initially localized wavepacket in two nonlinear chains (discrete nonlinear Schroedinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverge as a function of time. We find that the participation number of a wavepacket does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times the dynamical state consists of a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wavepacket, a limit profile for the norm/energy distribution with infinite second moment should exist in all cases which rules out the possibility of slow energy diffusion (subdiffusion). This limit profile could be a quasiperiodic solution (KAM torus)