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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]. ...

... On the other hand, in the area of focusing nonlinearity g < − 1 the CLS 1 -like pattern is preserved for long propagation distances after balancing its energy with the neighborhood, as can be seen on the state overlap and participation number profiles in figures 5 and 6. Regarding the nonlinear CLS 0 equivalent originating from the singular FB, which is the member of robust FB-DB-FB triplet strongly separated from the second DB [20], the nonlinearity-induced mixing affects the mode overlap for |g| > 1 in the area of defocusing nonlinearity, as can be confirmed by the profile of the ρ(g) for CLS 0 , figure 5. Regarding P, with increasing the propagation length we observed the changes in its value over the whole region |g| > 1 but with maximal relative discrepancy of the order of 1%. The increase (g < − 1) or decrease (g > 1) of P can be associated to the increase or decrease of the number of 'excited' lattice sites, respectively. ...

Tuning the values of artificial flux in the two-dimensional octagonal-diamond lattice drives topological phase transitions, including between singular and non-singular flatbands. We study the dynamical properties of nonlinear compact localized modes that can be continued from linear flatband modes. We show how the stability of the compact localized modes can be tuned by the nonlinearity strength or the applied artificial flux. Our model can be realized using ring resonator lattices or nonlinear waveguide arrays.

... Flatbands [20][21][22][23][24], a type of special energy band structures in condensed physics and photonics, are well known for their zero dispersion in at least one energy band of their spectra. Various flatband lattices have been theoretically and experimentally constructed, such as one-dimensional flatband lattices [22], Lieb lattices [23], and Kagome lattices [24] and the square-root topological rhombic lattice [25]. ...

... Various flatband lattices have been theoretically and experimentally constructed, such as one-dimensional flatband lattices [22], Lieb lattices [23], and Kagome lattices [24] and the square-root topological rhombic lattice [25]. The most typical common feature of these systems is that their eigenmodes are localized in real space; namely, they all support the existence of compact localized states (CLS) [20][21][22][23][24]. Although various intriguing phenomena have been inspired by the flatband features, such as topological superconductor [26], topological flatband insulator [27], and Anderson localization [28]. ...

We study a quasi-one-dimensional non-reciprocal Hermitian hourglass photonic lattice that can accomplish multiple functions. Under the effect of non-reciprocal coupling, this lattice can produce an energy isolation effect, two kinds of flatbands, and energy band inversion. The excitation and propagation of a single energy band and multiple energy bands can be realized; in the flatband condition, the system has compact localized states, and the flatbands can be excited by a straightforward method. Our findings advance the theory of energy band regulation in artificial photonic lattices.

... Twodimensional thresholdless solitons bifurcating from linear compact modes have been found in kagome arrays [20]. Flat-band solitons and nonlinear localized flat-band modes were also reported in sawtooth-like lattices [21], diamond chain lattices [22], [23], Stub lattices [24], [25], Lieb lattices [26]- [29], and octagonal-diamond lattices [30]. A summary of recent theoretical and experimental advances in the area of multidimensional localized structures in optical media, including the formation of different types of lattice solitons in discrete and continuous physical models can be found in Ref. [31]. ...

Flat-band periodic materials are characterized by a linear spectrum containing at least one band where the propagation constant remains nearly constant irrespective of the Bloch momentum across the Brillouin zone. These materials provide a unique platform for investigating phenomena related to light localization. Meantime, the interaction between flat-band physics and nonlinearity in continuous systems remains largely unexplored, particularly in continuous systems where the band flatness deviates slightly from zero, in contrast to simplified discrete systems with exactly flat bands. Here, we use a continuous superhoneycomb lattice featuring a flat band in its spectrum to theoretically and numerically introduce a range of stable flat-band solitons. These solutions encompass fundamental, dipole, multi-peak, and even vortex solitons. Numerical analysis demonstrates that these solitons are stable in a broad range of powers. They do not bifurcate from the flat band and can be analyzed using Wannier function expansion leading to their designation as Wannier solitons. These solitons showcase novel possibilities for light localization and transmission within nonlinear flat-band systems.

... Two-dimensional thresholdless solitons bifurcating from linear compact modes have been found in kagome arrays [20]. Flat-band solitons and nonlinear localized flat-band modes were also reported in sawtooth-like lattices [21], diamond chain lattices [22,23], Stub lattices [24,25], Lieb lattices [26][27][28][29], and octagonal-diamond lattices [30]. A summary of recent theoretical and experimental advances in the area of multidimensional localized structures in optical media, including the formation of different types of lattice soli-arXiv:2407.11811v1 ...

Flat-band periodic materials are characterized by a linear spectrum containing at least one band where the propagation constant remains nearly constant irrespective of the Bloch momentum across the Brillouin zone. These materials provide a unique platform for investigating phenomena related to light localization. Meantime, the interaction between flat-band physics and nonlinearity in continuous systems remains largely unexplored, particularly in continuous systems where the band flatness deviates slightly from zero, in contrast to simplified discrete systems with exactly flat bands. Here, we use a continuous superhoneycomb lattice featuring a flat band in its spectrum to theoretically and numerically introduce a range of stable flatband solitons. These solutions encompass fundamental, dipole, multi-peak, and even vortex solitons. Numerical analysis demonstrates that these solitons are stable in a broad range of powers. They do not bifurcate from the flat band and can be analyzed using Wannier function expansion leading to their designation as Wannier solitons. These solitons showcase novel possibilities for light localization and transmission within nonlinear flat-band systems.

... Therefore it was no surprise that the introduction of local Kerr nonlinearity in notable flatband networks yielded diverse examples of compact breather solutions [33][34][35][36][37][38]. Their existence have been partly explained by a continuation criterium from linear CLS of flatband networks to compact breathers introduced in Ref. 39 for CLS whose nonzero amplitudes are all equal. ...

Nonlinear networks can host spatially compact time periodic solutions called compact breathers. Such solutions can exist accidentally (i.e. for specific nonlinear strength values) or parametrically (i.e. for any nonlinear strength). In this work we introduce an efficient generator scheme for one-dimensional nonlinear lattices which support either types of compact breathers spanned over a given number U of lattice's unit cells and any number of sites v per cell - scheme which can be straightforwardly extended to higher dimensions. This scheme in particular allows to show the existence and explicitly construct examples of parametric compact breathers with inhomogeneous spatial profiles -- extending previous results which indicated that only homogeneous parametric compact breathers exist. We provide explicit d=1 lattices with different v supporting compact breather solutions for U=1,2.

... Stability conditions for fundamental and compact spatially localized states have also been studied from a more theoretical perspective for a nonlinear diamond/rhombic lattice [60,61]; however, no coherent mobility was reported. Very recently, we explored octagonal-diamond lattices [62], a system presenting two flat bands. We were not able to observe mobility, but oscillation around the input position. ...

During the last decades, researchers of different scientific areas have investigated several systems and materials to suggest new ways of transporting and localizing light. These problems are probably main goals in any search for new configurations and new emerging properties, independently of the degree of complexity of suggested methods. Fortunately, fabrication techniques in photonics have consolidated during the last decades, allowing the experimental implementation of different theoretical ideas which were neither tested nor validated. Specifically, we will focus on recent advances in the implementation of Flat Band (FB) photonic systems. FB periodical structures have at least two bands in their linear spectrum, with one of them completely flat. This implies the emergence of linear photonic states which are completely localized in space and that can be located in different regions across the lattice. This localization occurs as a result of destructive interference, what naturally depends on the particular lattice geometry. In addition, flat band systems also posses dispersive states which make possible the observation of ballistic transport as well. Therefore, FB photonic lattices constitute an unique platform for studying localization and transport, without requiring the inclusion of any sophisticated interaction/effect, rather a smart and simple geometry.

We study the mobility of localized solutions in a nonlinear Lieb photonic lattice. We characterize different families of nonlinear solutions looking for different regions of parameters to observe coherent transport across the system. In particular, we analytically derive a family of compact discrete solitons, which originate at the flat band of this lattice at zero power. For low level of power, we found a transparent region where two well-localized nonlinear modes, which are close in Hamiltonian, show a very good mobility. We numerically observe a perfect transport across the system with negligible radiation, where two compact solutions adiabatically transform one into the other. Although the ring mode stabilizes for larger power, it is not parametrically connected to any other stationary solution and, therefore, it is not allowed to move across the system for high power.

We consider the effect of disorder on the tight-binding Hamiltonians with a flat band and derive a common mathematical formulation of the average density of states and inverse participation ratio applicable for a wide range of them. The system information in the formulation appears through a single parameter which plays an important role in search of the critical points for disorder driven transitions in flat bands [P. Shukla, arXiv:1807.02436]. In a weak disorder regime, the formulation indicates an insensitivity of the statistical measures to disorder strength, thus confirming the numerical results obtained by our as well as previous studies.

A flatband representing a highly degenerate and dispersionless manifold state of electrons may offer unique opportunities for the emergence of exotic quantum phases. To date, definitive experimental demonstrations of flatbands remain to be accomplished in realistic materials. Here, we present the first experimental observation of a striking flatband near the Fermi level in the layered Fe3Sn2 crystal consisting of two Fe kagome lattices separated by a Sn spacing layer. The band flatness is attributed to the local destructive interferences of Bloch wave functions within the kagome lattices, as confirmed through theoretical calculations and modelings. We also establish high-temperature ferromagnetic ordering in the system and interpret the observed collective phenomenon as a consequence of the synergetic effect of electron correlation and the peculiar lattice geometry. Specifically, local spin moments formed by intramolecular exchange interaction are ferromagnetically coupled through a unique network of the hexagonal units in the kagome lattice. Our findings have important implications to exploit emergent flat-band physics in special lattice geometries.

Flatbands are receiving increasing theoretical and experimental attention in the field of photonics, in particular in the field of photonic lattices. Flatband photonic lattices consist of arrays of coupled waveguides or resonators where the peculiar lattice geometry results in at least one completely flat or dispersionless band in its photonic band structure. Although bearing a strong resemblance to structural slow light, this independent research direction is instead inspired by analogies with “frustrated” condensed matter systems. In this Perspective, we critically analyze the research carried out to date, discuss how this exotic physics may lead to novel photonic device applications, and chart promising future directions in theory and experiment.

Linear wave equations on flat band networks host compact localized eigenstates (CLS). Nonlinear wave equations on translationally invariant flat band networks can host compact discrete breathers - time periodic and spatially compact localized solutions. Such solutions can appear as one-parameter families of continued linear compact eigenstates, or as discrete sets on families of non-compact discrete breathers, or even on purely dispersive networks with fine-tuned nonlinear dispersion. In all cases, their existence relies on destructive interference. We use CLS amplitude distribution properties and orthogonality conditions to derive existence criteria and stability properties for compact discrete breathers as continued CLS.

Certain lattice wave systems in translationally invariant settings have one or more spectral bands that are strictly flat or independent of momentum in the tight binding approximation, arising from either internal symmetries or fine-tuned coupling. These flat bands display remarkable strongly-interacting phases of matter. Originally considered as a theoretical convenience useful for obtaining exact analytical solutions of ferromagnetism, flat bands have now been observed in a variety of settings, ranging from electronic systems to ultracold atomic gases and photonic devices. Here we review the design and implementation of flat bands and chart future directions of this exciting field.

We experimentally study a Stub photonic lattice and excite their localized linear states originated from an isolated Flat Band at the center of the linear spectrum. By exciting these modes in different regions of the lattice, we observe that they do not diffract across the system and remain well trapped after propagating along the crystal. By using their wave nature, we are able to combine - in phase and out of phase - two neighbor states into a coherent superposition. These observations allow us to propose a novel setup for performing three different all-optical logical operations such as OR, AND, and XOR, positioning Flat Band systems as key setups to perform all-optical operations at any level of power.

We report on the experimental realization of a quasi-one-dimensional photonic graphene ribbon supporting four flat-bands. We study the dynamics of fundamental and dipolar modes, which are analogous to the s and p orbitals, respectively. In the experiment, both modes (orbitals) are effectively decoupled from each other, implying two sets of six bands, where two of them are completely flat (dispersionless). Using an image generator setup, we excite the s and p flat-band modes and demonstrate their non-diffracting propagation for the first time. Our results open an exciting route towards photonic emulation of higher orbital dynamics.

We propose and experimentally realize a new kind of bound states in the continuum (BICs) in a class of systems constructed by coupling multiple identical one-dimensional chains, each with inversion symmetry. In such systems, a specific separation of the Hilbert space into a topological and a nontopological subspace exists. Bulk-boundary correspondence in the topological subspace guarantees the existence of a localized interface state which can lie in the continuum of extended states in the nontopological subspace, forming a BIC. Such a topological BIC is observed experimentally in a system consisting of coupled acoustic resonators.

We develop a simple and general method to construct arbitrary Flat Band lattices. We identify the basic ingredients behind zero-dispersion bands and develop a method to construct extended lattices based on a consecutive repetition of a given mini-array. The number of degenerated localized states is defined by the number of connected mini-arrays times the number of modes preserving the symmetry at a given connector site. In this way, we create one or more (depending on the lattice geometry) complete degenerated Flat Bands for quasi-one and two-dimensional systems. We probe our method by studying several examples, and discuss the effect of additional interactions like anisotropy or nonlinearity. At the end, we test our method by studying numerically a ribbon lattice using a continuous description.

We investigate, theoretically and experimentally, a photonic realization of a Sawtooth lattice. This special lattice exhibits two spectral bands, with one of them experiencing a complete collapse to a highly degenerate flat band for a special set of inter-site coupling constants. We report the observation of different transport regimes, including strong transport inhibition due to the appearance of the non-diffractive flat band. Moreover, we excite localized Shockley surface states residing in the gap between the two linear bands.

We report the first experimental demonstration of localized flat-band states in optically induced Kagome photonic lattices. Such lattices exhibit a unique band structure with the lowest band being completely flat (diffractionless) in the tight-binding approximation. By taking the advantage of linear superposition of the flat-band eigenmodes of the Kagome lattices, we demonstrate a high-fidelity transmission of complex patterns in such two-dimensional pyrochlore-like photonic structures. Our numerical simulations find good agreement with experimental observations, upholding the belief that flat-band lattices can support distortion-free image transmission.

We present a simple, yet effective, approach for optical induction of Lieb photonic lattices, which typically rely on the femtosecond laser writing technique. Such lattices are established by judiciously overlapping two sublattices (an “egg-crate” lattice and a square lattice) with different periodicities through a self-defocusing photorefractive medium. Furthermore, taking advantage of the superposition of localized flat-band states inherent in the Lieb lattices, we demonstrate distortion-free image transmission in such two-dimensional perovskite-like photonic structures. Our experimental observations find good agreement with numerical simulations.

The localized mode propagation in binary nonlinear kagome ribbons is investigated with the premise to ensure controlled light propagation through photonic lattice media. Particularity of the linear system characterized by the dispersionless flat band in the spectrum is the opening of new minigaps due to the "binarism." Together with the presence of nonlinearity, this determines the guiding mode types and properties. Nonlinearity destabilizes the staggered rings found to be nondiffracting in the linear system, but can give rise to dynamically stable ringlike solutions of several types: unstaggered rings, low-power staggered rings, hour-glass-like solutions, and vortex rings with high power. The type of solutions, i.e., the energy and angular momentum circulation through the nonlinear lattice, can be controlled by suitable initial excitation of the ribbon. In addition, by controlling the system "binarism" various localized modes can be generated and guided through the system, owing to the opening of the minigaps in the spectrum. All these findings offer diverse technical possibilities, especially with respect to the high-speed optical communications and high-power lasers.

We experimentally demonstrate the photonic realization of a dispersionless flat band in a quasi-one-dimensional photonic lattice fabricated by ultrafast laser inscription. In the nearest neighbor tight binding approximation, the lattice supports two dispersive and one nondispersive (flat) band. We experimentally excite superpositions of flat-band eigenmodes at the input of the photonic lattice and show the diffractionless propagation of the input states due to their infinite effective mass. In the future, the use of photonic rhombic lattices, together with the successful implementation of a synthetic gauge field, will enable the observation of Aharonov–Bohm photonic caging.

We discuss the properties of nonlinear localized modes in sawtooth lattices,
in the framework of a discrete nonlinear Schr\"odinger model with general
on-site nonlinearity. Analytic conditions for existence of exact compact
three-site solutions are obtained, and explicitly illustrated for the cases of
power-law (cubic) and saturable nonlinearities. These nonlinear compact modes
appear as continuations of linear compact modes belonging to a flat dispersion
band. While for the linear system a compact mode exists only for one specific
ratio of the two different coupling constants, nonlinearity may lead to
compactification of otherwise non-compact localized modes for a range of
coupling ratios, at some specific power. For saturable lattices, the
compactification power can be tuned by also varying the nonlinear parameter.
Introducing different onsite energies and anisotropic couplings yield further
possibilities for compactness tuning. The properties of strongly localized
modes are investigated numerically for cubic and saturable nonlinearities, and
in particular their stability over large parameter regimes is shown. Since the
linear flat band is isolated, its compact modes may be continued into compact
nonlinear modes both for focusing and defocusing nonlinearities. Results are
discussed in relation to recent realizations of sawtooth photonic lattices.

We present the first experimental demonstration of a new type of localized state in the continuum, namely, compacton-like linear states in flat-band lattices. To this end, we employ photonic Lieb lattices, which exhibit three tight-binding bands, with one being perfectly flat. Discrete predictions are confirmed by realistic continuous numerical simulations as well as by direct experiments. Our results could be of great importance for fundamental physics as well as for various applications where light needs to be conducted in a diffractionless and localized manner over long distances.

We show experimentally how a non-diffracting state can be excited in a
photonic Lieb lattice. This lattice supports three energy bands, including a
perfectly flat middle band, which corresponds to an infinite effective mass
with zero dispersion. We show that a suitable optical input state can be
prepared so as to only excite the flat band. We analyse, both experimentally
and theoretically, the evolution of such photonic flat-band states, and show
their remarkable robustness, even in the presence of disorder.

Quantum magnetic phase transition in square-octagon lattice was investigated by cellular dynamical mean field theory combining with continuous time quantum Monte Carlo algorithm. Based on the systematic calculation on the density of states, the double occupancy and the Fermi surface evolution of square-octagon lattice, we presented the phase diagrams of this splendid many particle system. The competition between the temperature and the on-site repulsive interaction in the isotropic square-octagon lattice has shown that both antiferromagnetic and paramagnetic order can be found not only in the metal phase, but also in the insulating phase. Antiferromagnetic metal phase disappeared in the phase diagram that consists of the anisotropic parameter λ and the on-site repulsive interaction U while the other phases still can be detected at T = 0.17. The results found in this work may contribute to understand well the properties of some consuming systems that have square-octagon structure, quasi square-octagon structure, such as ZnO.

We analyze the transport of light in the bulk and at the edge of photonic
Lieb lattices, whose unique feature is the existence of a flat band
representing stationary states in the middle of the band structure that can
form localized bulk states. We find that transport in bulk Lieb lattices is
significantly affected by the particular excitation site within the unit cell,
due to overlap with the flat band states. Additionally, we demonstrate the
existence of new edge states in anisotropic Lieb lattices. These states arise
due to a virtual defect at the lattice edges and are not described by the
standard tight-binding model.

Macroscopically degenerate flat bands (FB) in periodic lattices host compact
localized states which appear due to destructive interference and local
symmetry. Interference provides a deep connection between the existence of flat
band states (FBS) and the appearance of Fano resonances for wave propagation.
We introduce generic transformations detangling FBS and dispersive states into
lattices of Fano defects. Inverting the transformation, we generate a continuum
of FB models. Our procedure allows us to systematically treat perturbations
such as disorder and explain the emergence of energy-dependent localization
length scaling in terms of Fano resonances.

We explore the fundamental question of the critical nonlinearity value needed to dynamically localize energy in discrete nonlinear cubic (Kerr) lattices. We focus on the effective frequency and participation ratio of the profile to determine the transition into localization in one-, two-, and three-dimensional lattices. A simple and general criterion is developed, for the case of an initially localized excitation, to define the transition region in parameter space ("dynamical tongue") from a delocalized to a localized profile. We introduce a method for computing the dynamically excited frequencies, which helps us validate our stationary ansatz approach and the effective frequency concept. A general analytical estimate of the critical nonlinearity is obtained, with an extra parameter to be determined. We find this parameter to be almost constant for two-dimensional systems and prove its validity by applying it successfully to two-dimensional binary lattices.

We consider a model for a two-dimensional Kagome-lattice with defocusing
nonlinearity, and show that families of localized discrete solitons may
bifurcate from localized linear modes of the flat band with zero power
threshold. Each family of such fundamental nonlinear modes corresponds to a
unique configuration in the strong-nonlinearity limit. By choosing well-tuned
dynamical perturbations, small-amplitude, strongly localized solutions from
different families may be switched into each other, as well as moved between
different lattice positions. In a window of small power, the lowest-energy
state is a symmetry-broken localized state, which may appear spontaneously.

For more than a decade, intrinsic localized modes have been theoretical constructs. Only recently have they been observed in physical systems as distinct as charge-transfer solids, Josephson junctions, photonic structures, and micromechanical oscillator arrays.

A simple system of ordinary differential equations is introduced which has applications to the dynamics of small molecules, molecular crystals, self-trapping in amorphous semiconductors, and globular proteins. Analytical, numerical and perturbation methods are used to study the properties of stationary solutions. General solution trajectories can be either sinusoidal, periodic, quasiperiodic or chaotic.

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schrödinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to the existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy.

We theoretically investigate a tight binding model of fermions hopping on the square-octagon lattice which consists of a square lattice with plaquette corners themselves decorated by squares. Upon the inclusion of second neighbor spin-orbit coupling or non-Abelian gauge fields, time-reversal symmetric topological Z_2 band insulators are realized. Additional insulating and gapless phases are also realized via the non-Abelian gauge fields. Some of the phase transitions involve topological changes to the Fermi surface. The stability of the topological phases to various symmetry breaking terms is investigated via the entanglement spectrum. Our results enlarge the number of known exactly solvable models of Z_2 band insulators, and are potentially relevant to the realization and identification of topological phases in both the solid state and cold atomic gases. Comment: 12 pages, 9 figures

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 0,pi. Incommensurate analytic SWs with |Q|>pi/2 may however appear as "quasistable," as their instability growth rate is of higher order.

We suggest an effective method for controlling nonlinear switching in arrays of weakly coupled optical waveguides. We demonstrate digitized switching of a narrow input beam for as many as 11 waveguides in the engineered waveguide arrays.

Dynamical properties of discrete solitons in nonlinear Schrödinger lattices with saturable nonlinearity are studied in the framework of the one-dimensional discrete Vinetskii-Kukhtarev model. Two stationary strongly localized modes, centered on site (A) and between two neighboring sites (B), are obtained. The associated Peierls-Nabarro potential is bounded and has multiple zeros indicating strong implications on the stability and dynamics of the localized modes. Besides a stable propagation of mode A, a stable propagation of mode B is also possible. The enhanced ability of the large power solitons to move across the lattice is pointed out and numerically verified.

The capability to temporarily arrest the propagation of optical signals is one of the main challenges hampering the ever more widespread use of light in rapid long-distance transmission as well as all-optical on-chip signal processing or computations. To this end, flat-band structures are of particular interest, since their hallmark compact eigenstates not only allow for the localization of wave packets, but importantly, also protect their transverse profile from deterioration without the need for additional diffraction management. In this work, we experimentally demonstrate that, far from being a nuisance to be compensated, judiciously tailored loss distributions can, in fact, be the key ingredient in synthesizing such flat bands in non-Hermitian environments. We probe their emergence in the vicinity of an exceptional point and directly observe the associated compact localized modes that can be excited at arbitrary positions of the periodic lattice.

Phase frustration in periodic lattices is responsible for the formation of dispersionless flatbands. The absence of any kinetic energy scale makes flatband physics critically sensitive to perturbations and interactions. We report on the experimental investigation of the nonlinear response of cavity polaritons in the gapped flatband of a one-dimensional Lieb lattice. We observe the formation of gap solitons with quantized size and abrupt edges, a signature of the frozen propagation of switching fronts. This type of gap soliton belongs to the class of truncated Bloch waves, and has only been observed in closed systems up to now. Here, the driven-dissipative character of the system gives rise to a complex multistability of the flatband nonlinear domains. These results open up an interesting perspective regarding more complex 2D lattices and the generation of correlated photon phases.

We present the appearance of nearly flat-band states with nonzero Chern numbers in a two-dimensional “diamond-octagon” lattice model comprising two kinds of elementary plaquette geometries, diamond and octagon, respectively. We show that the origin of such nontrivial topological nearly flat bands can be described by a short-ranged tight-binding Hamiltonian. By considering an additional diagonal hopping parameter in the diamond plaquettes along with an externally fine-tuned magnetic flux, it leads to the emergence of such nearly flat-band states with nonzero Chern numbers for our simple lattice model. Such topologically nontrivial nearly flat bands can be very useful to realize the fractional topological phenomena in lattice models when the interaction is taken into consideration. In addition, we also show that perfect band flattening for certain energy bands, leading to compact localized states can be accomplished by fine-tuning the parameters of the Hamiltonian of the system. We compute the density of states and the wave-function amplitude distribution at different lattice sites to corroborate the formation of such perfectly flat-band states in the energy spectrum. Considering the structural homology between a diamond-octagon lattice and a kagome lattice, we strongly believe that one can experimentally realize a diamond-octagon lattice using ultracold quantum gases in an optical lattice setting. A possible application of our lattice model could be to design a photonic lattice using single-mode laser-induced photonic waveguides and study the corresponding photonic flat bands.

We study two models of correlated bond and site disorder on the kagome lattice considering both translationally invariant and completely disordered systems. The models are shown to exhibit a perfectly flat ground-state band in the presence of disorder for which we provide exact analytic solutions. Whereas in one model the flat band remains gapped and touches the dispersive band, the other model has a finite gap, demonstrating that the band touching is not protected by topology alone. Our model also displays fully saturated ferromagnetic ground states in the presence of repulsive interactions, an example of disordered flat band ferromagnetism.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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