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arXiv:1404.6315 · Journal of the Physical Society of Japan

Topological Properties of Ultracold Bosons in One-Dimensional Quasiperiodic Optical Lattice

Fuyuki Matsuda, Masaki Tezuka, Norio Kawakami

Published Jul 24, 2014• 23 citations
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Abstract

We analyze topological properties of the one-dimensional Bose-Hubbard model with a quasiperiodic superlattice potential. This system can be realized in interacting ultracold bosons in optical lattice in the presence of an incommensurate superlattice potential. We first analyze the quasiperiodic superlattice made by the cosine function, which we call Harper-like Bose-Hubbard model. We compute the Chern number and observe a gap-closing behavior as the interaction strength $U$ is changed. Also, we discuss the bulk-edge correspondence in our system. Furthermore, we explore the phase diagram as a function of $U$ and a continuous deformation parameter $β$ between the Harper-like model and another important quasiperiodic lattice, the Fibonacci model. We numerically confirm that the incommensurate charge density wave (ICDW) phase is topologically non-trivial and it is topologically equivalent in the whole ICDW region.

DOI
10.7566/jpsj.83.083707

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We analyze topological properties of the one-dimensional Bose-Hubbard model with a quasiperiodic superlattice potential. This system can be realized in interacting ultracold bosons in optical lattice in the presence of an incommensurate superlattice potential. We first analyze the quasiperiodic superlattice made by the cosine function, which we call Harper-like Bose-Hubbard model. We compute the Chern number and observe a gap-closing behavior as the interaction strength UUU is changed. Also, we discuss the bulk-edge correspondence in our system. Furthermore, we explore the phase diagram as a function of UUU and a continuous deformation parameter βββ between the Harper-like model and another important quasiperiodic lattice, the Fibonacci model. We numerically confirm that the incommensurate charge density wave (ICDW) phase is topologically non-trivial and it is topologically equivalent in the whole ICDW region.
Recently, the relation between one-dimensional (1D) quasicrystals and 2D topological insulators has been demonstrated theoretically and also experimentally by using optical waveguides.It has been shown that 1D quasicrystals can be classified in terms of topology for 2D integer quantum Hall systems.Such quasicrystals can be also realized in ultracold atoms loaded in optical superlattices.In this study, we consider a 1D Bose-Hubbard model in a quasiperiodic superlattice.It is known that a gap in the excitation spectrum is induced by the interaction, and the resulting Mott insulating phase is characterized by a nonzero Chern number.The system is called a topological Mott insulator.In the non-interacting case, topological equivalence between the Fibonacci quasicrystal and the Harper model is already known.However, it has not been clarified whether such topological equivalence exists in topological Mott insulators.We show numerically that the topological equivalence exists in a wide range of quasiperiodic Bose-Hubbard model.
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 quasiperiod) 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 quasiperiodic potential for both hard-core and soft-core 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 the 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 quasiperiodic potential; and we discuss the ab initio derivation of the Bose-Hubbard Hamiltonian with quasiperiodic potential starting from the microscopic Hamiltonian of bosons in an incommensurate superlattice.
Ultracold atoms in one-dimensional (1D) bichromatic optical lattices constitute a surprisingly simple system for the study of topological insulators. We show that bosons in 1D bichromatic lattices present as a general feature the existence, at equal fractional filling, of Mott-insulator phases with different topological character. These different phases are a direct consequence of the bosonic and interacting nature of the particles and the topological nature of the Bloch bands. We demonstrate that the associated hidden topological transitions may occur both as a function of the superlattice strength and due to intersite interactions. We also discuss the topological character of incommensurate density-wave phases in quasiperiodic superlattices.
We study the topological properties of an extended Bose-Hubbard model with cyclically modulated hopping and on-site potential parameters, which can be realized with ultracold bosonic atoms in a one-dimensional optical superlattice. We show that the interacting bosonic chain at half filling and in the deep Mott insulating regime can simulate bosonic Chern insulators with a topological phase diagram similar to that of the Haldane model of noninteracting fermions. Furthermore, we explore the topological properties of the ground state by calculating the many-body Chern number, the quasiparticle energy spectrum with gapless edge modes, the topological pumping of the interacting bosons, and the topological phase transition from normal (trivial) to topological Mott insulators. We also present the global phase diagram of the many-body ground state, which contains a superfluid phase and two Mott insulating phases with trivial (a zero Chern number) and nontrivial topologies (a nonzero Chern number), respectively.
We compute the phase diagram of the one-dimensional Bose-Hubbard model with a quasiperiodic potential by means of the density-matrix renormalization group technique. This model describes the physics of cold atoms loaded in an optical lattice in the presence of a superlattice potential whose wavelength is incommensurate with the main lattice wavelength. After discussing the conditions under which the model can be realized experimentally, the study of the density vs the chemical potential curves for a nontrapped system unveils the existence of gapped phases at incommensurate densities interpreted as incommensurate charge-density-wave phases. Furthermore, a localization transition is known to occur above a critical value of the potential depth V2{V}_{2}V2​ in the case of free and hard-core bosons. We extend these results to soft-core bosons for which the phase diagrams at fixed densities display new features compared with the phase diagrams known for random box distribution disorder. In particular, a direct transition from the superfluid phase to the Mott-insulating phase is found at finite V2{V}_{2}V2​. Evidence for reentrances of the superfluid phase upon increasing interactions is presented. We finally comment on different ways to probe the emergent quantum phases and most importantly, the existence of a critical value for the localization transition. The latter feature can be investigated by looking at the expansion of the cloud after releasing the trap.
We study topological properties of the Bose-Hubbard model with repulsive interactions in a one-dimensional optical superlattice. We find that the Mott insulator states of the single-component (two-component) Bose-Hubbard model under fractional fillings are topological insulators characterized by a nonzero charge (or spin) Chern number with nontrivial edge states. For ultracold atomic experiments, we show that the topological Chern number can be detected through measuring the density profiles of the bosonic atoms in a harmonic trap.