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Localized modes in linear and nonlinear octagonal-diamond lattices with two flat bands

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Abstract

We consider a two-dimensional octagonal-diamond network with a fine-tuned diagonal coupling inside the diamond-shaped unit cell. Its linear spectrum exhibits coexistence of two dispersive bands (DBs) and two flat bands (FBs), touching one of the DBs embedded between them. Analogous to the kagome lattice, one of the FBs will constitute the ground state of the system for a proper sign choice of the Hamiltonian. The system is characterized by two different flat-band fundamental octagonal compactons, originating from the destructive interference of fully geometric nature. In the presence of a nonlinear amplitude (on-site) perturbation, the single-octagon linear modes continue into one-parameter families of nonlinear compact modes with the same amplitude and phase structure. However, numerical stability analysis indicates that all strictly compact nonlinear modes are unstable, either purely exponentially or with oscillatory instabilities, for weak and intermediate nonlinearities and sufficiently large system sizes. Stabilization may appear in certain ranges for finite systems and, for the compacton originating from the band at the spectral edge, also in a regime of very large focusing nonlinearities. In contrast to the kagome lattice, the latter compacton family will become unstable already for arbitrarily weak defocusing nonlinearity for large enough systems. We show analytically the existence of a critical system size consisting of 12 octagon rings, such that the ground state for weak defocusing nonlinearity is a stable single compacton for smaller systems, and a continuation of a nontrivial, noncompact linear combination of single compacton modes for larger systems. Investigating generally the different nonlinear localized (noncompact) mode families in the semi-infinite gap bounded by this FB, we find that, for increasing (defocusing) nonlinearity the stable ground state will continuously develop into an exponentially localized mode with two main peaks in antiphase. At a critical nonlinearity strength a symmetry-breaking pitchfork bifurcation appears, so that the stable ground state is single peaked for larger defocusing nonlinearities. We also investigate numerically the mobility of localized modes in this regime and find that the considered modes are generally immobile both with respect to axial and diagonal phase-gradient perturbations.

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... This paper is continuation of our previous study of the flux-free nonlinear ODL, which was characterized by the band triplet consisting of two FBs and one dispersive band (DB), or four isolated DB in homogeneous or dimerized variant, respectively, in the linear limit [20]. We investigated the properties of the nonlinear compact localized modes and considered the corresponding system ground state in the presence of nonlinearity. ...
... As illustrated in figure 2, two of them are fully degenerate (dispersionless) FBs with energy: β = 0 and −2. The first one touches the embedded DB at the boundaries of the BZ, and the second one touches the same DB at the center of BZ [20]. Corresponding band crossings are conical and parabolic, respectively, similar to the Lieb [46,47] and Kagome lattices [46]. ...
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Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.
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We report on the engineering of a nondispersive (flat) energy band in a geometrically frustrated lattice of micropillar optical cavities. By taking advantage of the non-Hermitian nature of our system, we achieve bosonic condensation of exciton polaritons into the flat band. Because of the infinite effective mass in such a band, the condensate is highly sensitive to disorder and fragments into localized modes reflecting the elementary eigenstates produced by geometric frustration. This realization offers a novel approach to studying coherent phases of light and matter under the controlled interplay of frustration, interactions, and dissipation.
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We investigate the magnetic and metal(M)-insulator(I) phase diagram of the 1/5-depleted square-lattice Hubbard model at 1/4-filling by the mean-field approximation. There exist three magnetic phases of para(P)-, antiferro(AF)-, and ferro-magnetic(F) types, each realized for the large intra-square hopping t1, inter-square hopping t2, and Coulomb interaction U, respectively. Within each magnetic phase, the M-I transition of Lifshitz type emerges and, finally, six kind of phases are identified in U-t1/t2 plane. When t1 = t2, we find that the Dirac cone and nearly flat-band around \Gamma point form the SU(3) multiplet. The SU(3) effective theory well-describes the phase transitions between PI, PM, and AF phases. The PI and AFI phases are characterized by different Berry phases as in polyacetylene or graphene.
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We provide an overview of recent experimental and theoretical developments in the area of optical discrete solitons. By nature, discrete solitons represent self-trapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of self-localized states has been successfully observed in both one- and two-dimensional nonlinear waveguide arrays. In recent years such photonic lattices have been implemented or induced in a variety of material systems, including those with cubic (Kerr), quadratic, photorefractive, and liquid-crystal nonlinearities. In all cases the underlying periodicity or discreteness leads to altogether new families of optical solitons that have no counterpart whatsoever in continuous systems. We first review the linear properties of photonic lattices that are key in the understanding of discrete solitons. The physics and dynamics of the fundamental discrete and gap solitons are then analyzed along with those of many other exotic classes — e.g. twisted, vector and multi-band, cavity, spatio-temporal, random-phase, vortex, and non-local lattice solitons, just to mention a few. The possibility of all-optically routing optical discrete solitons in 2D and 3D periodic environments using soliton collisions is also presented. Finally, soliton formation in optical quasi-crystals and at the boundaries of waveguide array structures are discussed.
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Exact solutions to a nonlinear Schrödinger lattice with a saturable nonlinearity are reported. For finite lattices we find two different standing-wave-like solutions, and for an infinite lattice we find a localized soliton-like solution. The existence requirements and stability of these solutions are discussed, and we find that our solutions are linearly stable in most cases. We also show that the effective Peierls–Nabarro barrier potential is nonzero thereby indicating that this discrete model is quite likely nonintegrable.
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We realize a two-dimensional kagome lattice for ultracold atoms by overlaying two commensurate triangular optical lattices generated by light at the wavelengths of 532 and 1064 nm. Stabilizing and tuning the relative position of the two lattices, we explore different lattice geometries including a kagome, a one-dimensional stripe, and a decorated triangular lattice. We characterize these geometries using Kapitza-Dirac diffraction and by analyzing the Bloch-state composition of a superfluid released suddenly from the lattice. The Bloch-state analysis also allows us to determine the ground-state distribution within the superlattice unit cell. The lattices implemented in this work offer a near-ideal realization of a paradigmatic model of many-body quantum physics, which can serve as a platform for future studies of geometric frustration.
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Nonlinear classical Hamiltonian lattices exhibit generic solutions - discrete breathers. They are time-periodic and (typically exponentially) localized in space. The lattices have discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. We will introduce the concept of these localized excitations and review their basic properties including dynamical and structural stability. We then focus on advances in the theory of discrete breathers in three directions - scattering of waves by these excitations, persistence of discrete breathers in long transient processes and thermal equilibrium, and their quantization. The second part of this review is devoted to a detailed discussion of recent experimental observations and studies of discrete breathers, including theoretical modelling of these experimental situations on the basis of the general theory of discrete breathers. in particular we will focus on their detection in Josephson junction networks, arrays of coupled nonlinear optical waveguides, Bose-Einstein condensates loaded on optical lattices, antiferromagnetic layered structures, PtCl based single crystals and driven micromechanical cantilever arrays. (C) 2008 Elsevier B.V. All rights reserved.
Article
When the two arcs of the continuous phonon spectrum of the Floquet matrix of a discrete breather overlap on the unit circle, the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable. More precisely, when the model parameters vary, the breather in the finite system exhibits a large number of collisions between the Floquet eigenvalues belonging to the phonon spectrum. These collisions correspond to complex cascades of instability thresholds followed near after by re-entrant stability thresholds. We interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the ac-tion. Then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges, but simultaneously the maximum amplitude of the instabilities vanishes, so that the breather in the infinite system recovers its linear stability (as long as all its other localized modes remain stable). This is the situa-tion which is required in Cretegny et al. [Physica D 119 (1998) 73–87] for having inelastic phonon scattering with two channels. We also analyze the size effects when a Floquet eigenvalue associated with a localized mode collides with the Floquet continuous phonon spectrum with different Krein signature. In contrast to the previous case, the infinite system is unstable after the collision. © 1998 Elsevier Science B.V.
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We describe a system of coupled bosonic wells by means of the Bose-Hubbard model. Its dynamics is investigated via the time-dependent variational principle and coherent states. Some exact solutions of the dynamics, which have a vortex-like structure, are explicitly found, and used together with semiclassical requantization to describe the dynamics in the Mott insulator regime.
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The linear propagation properties of laser written hexagonal waveguide arrays in fused silica have been studied for the first time. We determine the behavior of the coupling constants for different waveguide separations and for different wavelengths. The high accuracy of these arrays demonstrates the ability to fabricate complex and elongated devices based on evanescently coupled waveguide structures using the fs direct writing approach.
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We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein–Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schrödinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with non-trivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes.
Article
We consider two types of strongly localized modes in discrete nonlinear lattices. Taking the lattice nonlinear Schrödinger (NLS) equation as a particular but rather fundamental example, we show that (1) the discreteness effects may be understood in the ``standard'' discrete NLS model as arising from an effective periodic potential similar to the Peierls-Nabarro (PN) barrier potential for kinks in the Frenkel-Kontorova model; (2) this PN potential vanishes in the completely integrable Ablowitz-Ladik variant of the NLS equation; and hence (3) the PN potential arises from the nonintegrability of the discrete physical models and determines the stability properties of the stationary localized modes.
Article
We reveal that even weak inherent discreteness of a nonlinear model can lead to instabilities of the localized modes it supports. We present the first example of an oscillatory instability of dark solitons, and analyse how it may occur for dark solitons of the discrete nonlinear Schrodinger and generalized Ablowitz-Ladik equations. Comment: 11 pages, 4 figures, to be published in Physical Review Letters
Article
We demonstrate that there exist stationary states of Bose-Einstein condensates in an optical lattice that do not satisfy the usual Bloch periodicity condition. Using the discrete model appropriate to the tight-binding limit we determine energy bands for period-doubled states in a one-dimensional lattice. In a complementary approach we calculate the band structure from the Gross-Pitaevskii equation, considering both states of the usual Bloch form and states which have the Bloch form for a period equal to twice that of the optical lattice. We show that the onset of dynamical instability of states of the usual Bloch form coincides with the occurrence of period-doubled states with the same energy. The period-doubled states are shown to be related to periodic trains of solitons. Comment: 4 pages, 3 figures, change of content