Compact Localized States Research Articles

Flat Bands in Network Superstructures of Atomic Chains

We investigate the origin of the ubiquitous existence of flat bands in the network superstructures of atomic chains, where one-dimensional (1D) atomic chains array periodically. While there can be many ways to connect those chains, we consider two representative ways of linking them, the dot-type and triangle-type links. Then, we construct a variety of superstructures, such as the square, rectangular, and honeycomb network superstructures with dot-type links and the honeycomb superstructure with triangle-type links. These links provide the wavefunctions with an opportunity to have destructive interference, which stabilizes the compact localized state (CLS). In the network superstructures, there exist multiple flat bands proportional to the number of atoms of each chain, and the corresponding eigenenergies can be found from the stability condition of the compact localized state. Finally, we demonstrate that the finite bandwidth of the nearly flat bands of the network superstructures arising from the next-nearest-neighbor hopping processes can be suppressed by increasing the length of the chains consisting of the superstructures.

Topological Flat Bands in Self-Complementary Plasmonic Metasurfaces.

Photonics can be confined in real space with dispersion vanishing in the momentum space due to destructive interference. In this Letter, we report the experimental realization of flat bands with nontrivial topology in a self-complementary plasmonic metasurface. The band diagram and compact localized states are measured. In these nontrivial band gaps, we observe the topological edge states by near-field measurements. Furthermore, we propose a digitalized metasurface by loading controllable diodes with C_{3} symmetry in every unit cell. By pumping a digital signal into the metasurface, we investigate the interaction between incident waves and the dynamic metasurface. Experimental results indicate that compact localized states in the nontrivial flat band could enhance the wave-matter interactions to convert more incident waves to time-modulated harmonic photonics. Although our experiments are conducted in the microwave regime, extending the related concepts into the optical plasmonic systems is feasible. Our findings pave an avenue toward planar integrated photonic devices with nontrivial flat bands and exotic transmission phenomena.

Delocalization and re-entrant localization of flat-band states in non-Hermitian disordered lattice models with flat bands

Abstract We present a numerical study of Anderson localization in disordered non-Hermitian lattice models with flat bands. Specifically, we consider 1D stub and 2D kagome lattices that have a random scalar potential and a uniform imaginary vector potential and calculate the spectra of the complex energy, the participation ratio, and the winding number as a function of the strength of the imaginary vector potential, h. The flat-band states are found to show a double transition from localized to delocalized and back to localized states with h, in contrast to the dispersive-band states going through a single delocalization transition. When h is sufficiently small, all flat-band states are localized. As h increases above a certain critical value h1, some pairs of flat-band states become delocalized. The participation ratio associated with them increases substantially and their winding numbers become nonzero. As h increases further, more and more flat-band states get delocalized until the fraction of the delocalized states reaches a maximum. For larger h values, a re-entrant localization takes place and, at another critical value h2, all flat-band states return to compact localized states with very small participation ratios and zero winding numbers. This re-entrant localization transition, which is due to the interplay among disorder, non-hermiticity, and the flat band, is a phenomenon occurring in many models having an imaginary vector potential and a flat band simultaneously. We explore the spatial characteristics of the flat-band states by calculating the local density distribution.

Flat band based multifractality in the all-band-flat diamond chain

We study the effect of quasiperiodic Aubry-Andr\'e disorder on the energy spectrum and eigenstates of a one-dimensional all-bands-flat (ABF) diamond chain. The ABF diamond chain possesses three dispersionless flat bands with all the eigenstates compactly localized on two unit cells in the zero disorder limit. The fate of the compact localized states in the presence of the disorder depends on the symmetry of the applied potential. We consider two cases here: a symmetric one, where the same disorder is applied to the top and bottom sites of a unit cell and an antisymmetric one, where the disorder applied to the top and bottom sites are of equal magnitude but with opposite signs. Remarkably, the symmetrically perturbed lattice preserves compact localization, although the degeneracy is lifted. When the lattice is perturbed antisymmetrically, not only is the degeneracy is lifted but compact localization is also destroyed. Fascinatingly, all eigenstates exhibit a multifractal nature below a critical strength of the applied potential. A central band of eigenstates continue to display an extended yet non-ergodic behaviour for arbitrarily large strengths of the potential. All other eigenstates exhibit the familiar Anderson localization above the critical potential strength. We show how the antisymmetric disordered model can be mapped to a $\frac{\pi}{4}$ rotated square lattice with nearest and selective next-nearest neighbour hopping and a staggered magnetic field - such models have been shown to exhibit multifractality. Surprisingly, the antisymmetric disorder (with an even number of unit cells) preserves chiral symmetry - we show this by explicitly writing down the chiral operator.

Localized dynamics arising from multiple flat bands in a decorated photonic Lieb lattice

Photonic lattices have emerged as an ideal testbed for localizing light in space. Among others, the most promising approach is based on flat band systems and their related nondiffracting compact localized states. So far, only compact localized states arising from a single flat band have been found. Such states typically appear static, thus not allowing adaptive or evolutionary features of light localization. Here, we report on the first experimental realization of an oscillating compact localized state arising from multiple flat bands. We observe an oscillatory intensity beating during propagation in a two-dimensional photonic decorated Lieb lattice. The photonic system is realized by direct femtosecond laser writing and hosts most importantly multiple flat bands at different eigenenergies in its band structure. Our results open new avenues for evolution dynamics in the up to now static phenomenon of light localization in periodic waveguide structures and extend the current understanding of light localization in flat band systems.

Designer flat bands: Topology and enhancement of superconductivity

We construct quasi one-dimensional topological and non-topological three-band lattices with tunable band gap and winding number of the flat band. Using mean field (MF) and exact density matrix renormalization group (DMRG) calculations, we show explicitly how the band gap affects pairing and superconductivity (SC) in a Hubbard model with attractive interactions. We show excellent agreement between MF and DMRG. When a phase twist is applied on the system, a phase difference appears between pairing order parameters on different sublattices, and this plays a very important role in the SC density. The SC weight, $D_s$, on the gapped topological, $W\neq0$, flat band increases linearly with interaction strength, $U$, for low values, and with a slope that depends on the details of the compact localized state at $U=0$. As $U\to 0$ for the gapped non-topological flat band ($W=0$), $D_s$ decays with a power law faster than quadratic but slower than exponential. This indicates that isolated non-topological flat bands are less beneficial to SC. In the gapless case (flat band touching the band above it), we find at low $U$ (both for $W=0$ and $W\neq 0$) that $D_s\propto U^\varphi$ with $\varphi<1$ contrary to the $U{\rm ln}\, ({\rm const.}/U)$ behavior reported in the literature. In other words, $D_s$ increases faster than linearly for low $U$ thus favoring SC at weak interaction more than the gapped case. For systems with touching bands, we observe that the one-body correlation length, $\xi$, diverges as a power law as $U\rightarrow0$, while for the isolated flat band $\xi(U\to 0)$ is a constant smaller than one lattice spacing. Both behaviors are distinct from the exponential divergence of $\xi$ in the dispersive case. Our results re-establish that the BCS mean field and quantum metric alone are insufficient to characterize SC at weak coupling.

Reconfigurable photon localization by coherent drive and dissipation in photonic lattices

The engineering of localized modes in photonic structures is one of the main targets of modern photonics. An efficient strategy to design these modes is to use the interplay of constructive and destructive interference in periodic photonic lattices. This mechanism is at the origin of the defect modes in photonic bandgaps, bound states in the continuum, and compact localized states in flat bands. Here, we show that in lattices of lossy resonators, the addition of external optical drives with a controlled phase enlarges the possibilities of manipulating interference effects and allows for the design of novel types of localized modes. Using a honeycomb lattice of coupled micropillars resonantly driven with several laser spots at energies within its photonic bands, we demonstrate the localization of light in at-will geometries down to a single site. These localized modes are fully reconfigurable and have the potentiality of enhancing nonlinear effects and of controlling light–matter interactions with single site resolution.

Localized States Emerging from Singular and Nonsingular Flat Bands in a Frustrated Fractal‐Like Photonic Lattice

AbstractSingular and nonsingular flat bands in a Sierpinski fractal‐like photonic lattice are reported. It is demonstrated that the lowest two bands, being isolated and degenerate due to geometrical frustration, are nonsingular and thus can be spanned by a complete set of compact localized states. These states are experimentally proven to propagate diffractionless in the photonic lattice. The results reveal the interplay between geometrical frustration, degenerate flat bands, and compact localized states in a single photonic lattice, and pave the way to photonic spin liquid ground states.

Deformation of localized states and state transitions in systems of randomly hopping interacting fermions

We numerically study the random-hopping fermions (the Cruetz ladder) with repulsion and investigate how the interactions deform localized eigenstates by means of the one particle-density matrix (OPDM). The ground state exhibits resurgence of localization from the compact localized state to strong-repulsion-induced localization. On the other hand, excited states in the middle of the spectrum tend to extend by the repulsion. The transition property obtained by numerical calculations of the OPDM is deeply understood by studying a solvable model in which local integrals of motion (LIOMs) are obtained explicitly. The present work clarifies the utility of the OPDM and also how compact-support LIOMs in non-interacting limit are deformed by the repulsion.

Designing flat-band tight-binding models with tunable multifold band touching points

Being dispersionless, flat bands on periodic lattices are solely characterized by their macroscopically degenerate eigenstates: compact localized states (CLSs) in real space and Bloch states in reciprocal space. Based on this property, this work presents a straightforward method to build flat-band tight-binding models with short-range hoppings \emph{on any periodic lattice}. The method consists in starting from a CLS and engineering families of Bloch Hamiltonians as quadratic (or linear) functions of the associated Bloch state. The resulting tight-binding models not only exhibit a flat band, but also multifold quadratic (or linear) band touching points (BTPs) whose number, location, and degeneracy can be controlled to a large extent. Quadratic flat-band models are ubiquitous: they can be built from any arbitrary CLS, on any lattice, in any dimension and with any number $N\geq2$ of bands. Linear flat-band models are rarer: they require $N\geq3$ and can only be built from CLSs that fulfill certain compatibility relations with the underlying lattice. Most flat-band models from the literature can be classified according to this scheme: Mielke's and Tasaki's models belong to the quadratic class, while the Lieb, dice and breathing Kagome models belong to the linear class. Many novel flat-band models are introduced, among which an $N=4$ bilayer honeycomb model with fourfold quadratic BTPs, an $N=5$ dice model with fivefold linear BTPs, and an $N=3$ Kagome model with BTPs that can be smoothly tuned from linear to quadratic.

Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands

We experimentally realize fractal-like photonic lattices by use of the cw-laser-writing technique, thereby observing distinct compact localized states (CLSs) associated with different flatbands in the same lattice setting. Such triangle-shaped lattices, akin to the first generation Sierpinski lattices, possess a band structure where singular non-degenerate and nonsingular degenerate flatbands coexist. By proper phase modulation of an input excitation beam, we demonstrate not only the simplest CLSs but also their superimposition into other complex mode structures. Our experimental and numerical results are corroborated by theoretical analysis. Furthermore, we show by numerical simulation a dynamical oscillation of the flatband states due to beating of the CLSs that have different eigenenergies. These results may provide inspiration for exploring fundamental phenomena arising from the interplay of fractal structure, flatband singularity, and real-space topology.

Kagome Flatbands for Coherent Exciton-Polariton Lasing

Kagome lattices supporting Dirac cone and flatband dispersions are well known as a highly frustrated, two-dimensional lattice system. Particularly the flatbands therein are attracting continuous interest based on their link to topological order, correlations and frustration. In this work, we realize coupled microcavity implementations of Kagome lattices hosting exciton-polariton quantum fluids of light. We demonstrate precise control over the dispersiveness of the flatband as well as selective condensation of exciton-polaritons into the flatband. Subsequently, we focus on the spatial and temporal coherence properties of the laser-like emission from these polariton condensates that are closely connected to the flatband nature of the system. Notably, we find a drastic increase in coherence time due to the localization of flatband condensates. Our work illustrates the outstanding suitability of the exciton-polariton system for detailed studies of flatband states as a platform for microlaser arrays in compact localized states, including strong interactions, topology and non-linearity.

General construction of flat bands with and without band crossings based on wave function singularity

In this work, we develop a systematic method of constructing flat-band models with and without band crossings. Our construction scheme utilizes the symmetry and spatial shape of a compact localized state (CLS) and also the singularity of the flat-band wave function obtained by a Fourier transform of the CLS (FT-CLS). In order to construct a flat-band model systematically using these ingredients, we first choose a CLS with a specific symmetry representation in a given lattice. Then, the singularity of FT-CLS indicates whether the resulting flat band exhibits a band crossing point or not. A tight-binding Hamiltonian with the flat band corresponding to the FT-CLS is obtained by introducing a set of basis molecular orbitals, which are orthogonal to the FT-CLS. Our construction scheme can be systematically applied to any lattice so that it provides a powerful theoretical framework to study exotic properties of both gapped and gapless flat bands arising from their wave function singularities.

Multiple quantum scar states and emergent slow thermalization in a flat-band system

Quantum many-body scars (QMBS) appear in a flat-band model with interactions on the saw-tooth lattice. The flat-band model includes a compact support localized eigenstates, called compact localized state (CLS). Some characteristic many-body states can be constructed from the CLSs at a low-filling on the flat-band. These many-body states are degenerate. Starting with such degenerate states we concretely show how to construct multiple QMBSs with different eigenenergies embedded in the entire spectrum. If the degeneracy is lifted by introducing hopping modulation or weak perturbations, these states lifted by these ways can be viewed as multiple QMBSs. In this work, we focus on the study of the perturbation-induced QMBS. Perturbed states, which are connected to the exact QMBSs in the unperturbed limit, indicate common properties of conventional QMBS systems, that is, a subspace with sub-volume or area law scaling entanglement entropy, which indicates the violation of the strong eigenstate thermalization hypothesis (ETH). Also for a specific initial state, slow-thermalization dynamics appears. We numerically demonstrate these subjects. The flat-band model with interactions is a characteristic example in non-integrable systems with the violation of the strong ETH and the QMBS.

Flat band, spin-1 Dirac cone, and Hofstadter diagram in the fermionic square kagome model

We study characteristic band structures of the fermions on a square kagome lattice, one of the two-dimensional lattices hosting a corner-sharing network of triangles. We show that the band structures of the nearest-neighbor tight-binding model exhibit many characteristic features, including a flat band which is ubiquitous among frustrated lattices. On the flat band, we elucidate its origin by using the molecular-orbital representation and also find localized exact eigenstates called compact localized states. In addition to the flat band, we also find two spin-1 Dirac cones with different energies. These spin-1 Dirac cones are not described by the simplest effective Dirac Hamiltonian because the middle band is bended and the energy spectrum is particle-hole asymmetric. We also investigated the Hofstadter problem on a square kagome lattice in the presence of an external field and find that the profile of the Chern numbers around the modified spin-1 Dirac cones coincides with the conventional one.

Strictly localized states in the octagonal Ammann-Beenker quasicrystal

Ammann-Beenker lattice is a two-dimensional quasicrystal with eightfold symmetry, which can be described as a projection of a cut from a four-dimensional simple cubic lattice. We consider the vertex tight-binding model on this lattice and investigate the strictly localized states at the center of the spectrum. We use a numerical method based on the generation of finite lattices around a given perpendicular space point and QR decomposition of the Hamiltonian to count the strictly localized states. We apply this method to count the frequency of localized states in lattices of up to 100 000 sites. We obtain an orthogonal set of compact localized states by diagonalizing the position operator projected onto the manifold spanned by the zero-energy states. We identify 20 localized state types and calculate their exact frequencies through their perpendicular space images. Unlike the Penrose lattice, all the localized state types are eightfold symmetric around an eight edge vertex, and all vertex types can support localized states. The total frequency of these 20 types gives a lower bound of ${f}_{\mathrm{LS}}=30\phantom{\rule{0.16em}{0ex}}796\ensuremath{-}21\phantom{\rule{0.16em}{0ex}}776\sqrt{2}\ensuremath{\simeq}0.085\phantom{\rule{0.16em}{0ex}}47$ for the fraction of strictly localized states in the spectrum. This value is in agreement with the numerical calculation and very close to the recently conjectured exact fraction of localized states ${f}_{\mathrm{Ex}}=3/2\ensuremath{-}\sqrt{2}\ensuremath{\simeq}0.085\phantom{\rule{0.16em}{0ex}}79$.

Flat-band generator in two dimensions

Dispersionless bands -- \emph{flatbands} -- provide an excellent testbed for novel physical phases due to the fine-tuned character of flatband tight-binding Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flatbands support compact localized states, which allowed to develop systematic flatband generators in $d=1$ dimension in Phys. Rev. B {\bf 95} 115135 (2017) and Phys. Rev. B {\bf 99} 125129 (2019). Here we extend this generator approach to $d=2$ dimensions. The \emph{shape} of a compact localized state turns into an important additional flatband classifier. This allows us to obtain analytical solutions for classes of $d=2$ flatband networks and to re-classify and re-obtain known ones, such as the checkerboard, kagome, Lieb and Tasaki lattices. Our generator can be straightforwardly generalized to three lattice dimensions as well.

Nonthermalized dynamics of flat-band many-body localization

We find that a flat-band fermion system with interactions and without disorders exhibits non-thermalized ergodicity-breaking dynamics, an analog of many-body localization (MBL). In the previous works, we observed flat-band many-body localization (FMBL) in the Creutz ladder model. The origin of FMBL is a compact localized state governed by local integrals of motion (LIOMs), which are to be obtained explicitly. In this work, we clarify the dynamical aspects of FMBL. We first study dynamics of two-particles, and find that the states are not substantially modified by weak interactions, but the periodic time evolution of entanglement entropy emerges as a result of a specific mechanism inherent in the system. On the other hand, as the strength of the interactions is increased, the modification of the states takes place with inducing instability of the LIOMs. Furthermore, many-body dynamics of the system at finite fillings is numerically investigated by time-evolving block decimation (TEBD) method. For a suitable choice of the filling, non-thermal and low entangled dynamics appears. This behavior is a typical example of the disorder-free FMBL.

Flat band quantum scar

We show that a quantum scar state, an atypical eigenstate breaking eigenstate thermalization hypothesis embedded in a many-body energy spectrum, can be constructed in flat band systems. The key idea of our construction is to make use of orthogonal compact localized states. We concretely discuss our construction scheme, taking a saw-tooth flat lattice system as an example, and numerically demonstrate the presence of a quantum scar state. Examples of higher-dimensional systems are also addressed. Our construction method of quantum scar has broad applications to various flat band systems.

From compact localized states to many-body scars in the random quantum comb

In this work we investigate the effects of configurational disorder on the eigenstates and dynamical properties of a tight-binding model on a quasi-one-dimensional comb lattice, consisting of a backbone decorated with linear offshoots of randomly distributed lengths. We show that all eigenstates are exponentially localized along the backbone of the comb. Moreover, we demonstrate the presence of an extensive number of compact localized states with precisely zero localization length. We provide an analytical understanding of these states and show that they survive in the presence of density-density interactions along the backbone of the system where, for sufficiently low but finite particle densities, they form many-body scar states. Finally, we discuss the implications of these compact localized states on the dynamical properties of systems with configurational disorder, and the corresponding appearance of long-lived transient behaviour in the time evolution of physically relevant product states.