Lieb Lattice Research Articles

Collapsing Behavior of the Ferrimagnetic Ground State of the S = 1/2 Heisenberg Antiferromagnet on the Lieb Lattice due to Frustration

Collapsing Behavior of the Ferrimagnetic Ground State of the <i>S</i> = 1/2 Heisenberg Antiferromagnet on the Lieb Lattice due to Frustration

Modulational instability in photonic Lieb lattice: a graph Laplacian approach

Modulational instability in photonic Lieb lattice: a graph Laplacian approach

Realization of a 2D Lieb Lattice in a Metal-Inorganic Framework with Partial Flat Bands and Topological Edge States.

Flat bands and Dirac cones in materials are the source of the exotic electronic and topological properties. The Lieb lattice is expected to host these electronic structures, arising from quantum destructive interference. Nevertheless, the experimental realization of a 2D Lieb lattice remained challenging to date due to its intrinsic structural instability. After computationally designing a Platinum-Phosphorus (Pt-P) Lieb lattice, it has successfully overcome its structural instability and synthesized on a gold substrate via molecular beam epitaxy. Low-temperature scanning tunneling microscopy and spectroscopy verify the Lieb lattice's morphology and electronic flat bands. Furthermore, topological Dirac edge states stemming from pronounced spin-orbit coupling induced by heavy Pt atoms are predicted. These findings convincingly open perspectives for creating metal-inorganic framework-based atomic lattices, offering prospects for strongly correlated phases interplayed with topology.

Individually tunable tunnelling coefficients in optical lattices using local periodic driving

Abstract Ultracold atoms in optical lattices have emerged as powerful quantum simulators of translationally invariant systems with many applications in e.g. strongly-correlated and topological systems. However, the ability to locally tune all Hamiltonian parameters remains an outstanding goal that would enable the simulation of a wider range of quantum phenomena. Motivated by recent advances in quantum gas microscopes and optical tweezers, we here show theoretically how local control over individual tunnelling links in an optical lattice can be achieved by incorporating local time-periodic potentials. We propose to periodically modulate the on-site energy of individual lattice sites and employ Floquet theory to demonstrate how this provides full individual control over the tunnelling amplitudes in one dimension. We provide various example configurations realising interesting topological models such as extended Su–Schrieffer–Heeger models that would be challenging to realise by other means. Extending to two dimensions, we demonstrate that local periodic driving in a Lieb lattice engineers a two-dimensional (2D) network with fully controllable tunnelling magnitudes. In a three-site plaquette, we show full simultaneous control over the relative tunnelling amplitudes and the gauge-invariant flux piercing the plaquette, providing a clear stepping stone to building a fully programmable 2D tight-binding model. We also explicitly demonstrate how utilise our technique to generate a magnetic field gradient in 2D. This local modulation scheme is applicable to many different lattice geometries.

Lattice generation utilizing fractional Talbot effect and high-order orbital angular momentum optical vortices

The concept of orbital angular momentum (OAM) has garnered significant attention from researchers since its discovery, with a steady uncovering of its properties over the past three decades. Initially overlooked, the role of OAM in the optical vortex lattice (OVL) has gained prominence due to recent scholarly efforts. This study delves into the modification of OAM by employing controlled topological charge to generate high-order OVL. Furthermore, it explores the Talbot effect within the context of multi-beam interference. The findings indicate that the interference of three beams of light results in the formation of both Kagome and Super-honeycomb lattices, whereas the interference of four beams leads to the emergence of the Lieb lattice. This research offers novel insights into light interference and light field regulation and its application.

Dynamical Conductivities for the Fermionic Lieb Lattice

Dynamical Conductivities for the Fermionic Lieb Lattice

Exact non-Hermitian mobility edges and robust flat bands in two-dimensional Lieb lattices with imaginary quasiperiodic potentials

The mobility edge (ME) is a critical energy delineates the boundary between extended and localized states within the energy spectrum, and it plays a crucial role in understanding the metal–insulator transition in disordered or quasiperiodic systems. While there have been extensive studies on MEs in one-dimensional non-Hermitian (NH) quasiperiodic lattices recently, the investigation of exact NH MEs in two-dimensional (2D) cases remains rare. In the present study, we introduce a 2D dissipative Lieb lattice (DLL) model with imaginary quasiperiodic potentials applied solely to the vertices of the Lieb lattice. By mapping this DLL model to the 2D NH Aubry–André–Harper model, we analytically derive the exact ME and find it associated with the absolute eigenenergies. We find that the eigenvalues of extended states are purely imaginary when the quasiperiodic potential is strong enough. Additionally, we demonstrate that the introduction of imaginary quasiperiodic potentials does not disrupt the flat bands inherent in the system. Finally, we propose a theoretical framework for realizing our model using the Lindblad master equation. Our results pave the way for further investigation of exact NH MEs and flat bands in 2D dissipative quasiperiodic systems.

Quantum engineering for compactly localized states in disordered Lieb lattices

Blending ordering within an uncorrelated disorder potential in families of 3D Lieb lattices preserves the macroscopic degeneracy of compact localized states and yields unconventional combinations of localized and delocalized phases—as shown in Liu et al. (Phys Rev B 106:214204, 2022). We proceed to reintroduce translation invariance in the system by further ordering the disorder, and discuss the spectral structure and eigenstates features of the resulting perturbed lattices. We restore order in steps by first (i) rendering the disorder binary—i.e., yielding a randomized checkerboard potential, then (ii) reordering the randomized checkerboard into an ordered one, and at last (iii) realigning all the checkerboard values yielding a constant potential shift, but only on a sub-lattice. Along this path, we test the influence of additional random impurities on the order restoration. We find that in each of these steps, about half of the dispersive states are projected upon the unperturbed sites hosting the degenerate compact states, while the remaining ones are localized in the perturbed sites with energy determined by the strength of checkerboard. This strategy, herewith implemented in the 3D Lieb lattice, highlights order restoration as experimental pathway to engineer spectral and states features in disordered lattice structures in the pursuit of quantum storage and memory applications.Graphic abstract

Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials

In this paper, we investigate the so-called electronic dressed states, a unified quasiparticle resulting from the interaction between electrons in a two-dimensional material with an off-resonance optical dressing field. If the frequency of this field is much larger than all characteristic energies in the system, such as the Fermi energy or bandgap(s), the electronic band structure is affected by radiation so that some important properties of the electron dispersions could be modified in a way desirable for practical applications. For example, circularly polarized light can be used to vary the bandgap of Dirac materials: it opens a gap in graphene and other metallic and semimetallic lattices, or it modifies the magnitude of an existing gap. This will either enhance or reduce a gap, depending on its initial value as well as properties of a host material. Here, we consider gapped dice and Lieb lattices as samples, and we put forward a full theoretical model to reveal how these electronic states are deformed by elliptically-polarized irradiation with a focus on the generation and modification of a bandgap.

Band engineering in two-dimensional porphyrin- and phthalocyanine-based covalent organic frameworks: insight from molecular design

Two-dimensional covalent organic frameworks (2D COFs) represent an emerging class of crystalline polymeric networks, characterized by their tunable architectures and porosity, synthetic adaptability, and interesting optical, magnetic, and electrical properties. The incorporation of porphyrin (Por) or phthalocyanine (Pc) core units into 2D COFs provides an ideal platform for exploring the relationship between the COF geometric structure and its electronic properties in the case of tetragonal symmetry. In this work, on the basis of tight-binding models and density functional theory calculations, we describe the generic types of electronic band structures that can arise in tetragonal COFs. Three tetragonal lattice symmetries are examined: the basic square lattice, the Lieb lattice, and the checkerboard lattice. The potential topological characteristics of each lattice are explored. The Por-/Pc-based COFs exhibit characteristic band dispersions that are directly linked to their lattice symmetries and the nature of the frontier molecular orbitals of their building units. We show that the band dispersions in these COFs can be tailored by choosing specific symmetries of the molecular building units and/or by modulating the relative energies of the core and linker units. These strategies can be extended to a wide array of COFs, offering an effective approach to engineering their electronic properties.

Non-perturbative dynamics of flat-band systems with correlated disorder

Abstract We develop a numerical method for the time evolution of Gaussian wave packets on flat-band lattices in the presence of correlated disorder. To achieve this, we introduce a method to generate random on-site energies with prescribed correlations. We verify this method with a one-dimensional (1D) cross-stitch model, and find good agreement with analytical results obtained from the disorderdressed evolution equations. This allows us to reproduce previous findings, that disorder can mobilize 1D flat-band states which would otherwise remain localized. As explained by the corresponding disorder-dressed evolution equations, such mobilization requires an asymmetric disorder-induced coupling to dispersive bands, a condition that is generically not fulfilled when the flat-band is resonant with the dispersive bands at a Dirac point-like crossing. We exemplify this with the 1D Lieb lattice. While analytical expressions are not available for the two-dimensional (2D) system due to its complexity, we extend the numerical method to the 2D α-T 3 model, and find that the initial flat-band wave packet preserves its localization when α = 0, regardless of disorder and intersections. However, when α ≠ 0, the wave packet shifts in real space. We interpret this as a Berry phase controlled, disorder-induced wave-packet mobilization. In addition, we present density functional theory calculations of candidate materials, specifically Hg1-xCdxTe. The flat-band emerges near the Γ point (k =0) in the Brillouin zone.

All‐Bands‐Flat Floquet Topological Photonic Insulators with Microring Lattices

Coupled microring lattices are versatile photonic systems that can be used to realize various topological phases of matter. In two‐dimensional (2D) microring lattices, the periodic and unidirectional circulation of light in each microring gives rise to a time‐like dimension, so that the lattice emulates a (2 + 1)D system with much richer topological behaviors than static 2D lattices. Accurate treatment of these systems requires a departure from the static tight‐binding model of coupled resonators and take into account the periodic coupling sequence of light in the lattice network. This article provides an overview of the theory and design of (2 + 1)D microring lattices for realizing Floquet topological photonic insulators (TPIs). Particular focus is placed on the microring Lieb lattice with perfect couplings, which emulates an anomalous Floquet insulator with all flat bands. Such a system exhibits some unique properties, including wide edge mode continuum exceeding a Floquet–Brillouin zone, super‐robustness to lattice disorder, Aharonov–Bohm (AB) caging and compact localized flat‐band states that can be used to realize high‐quality factor topological resonators. All‐bands‐flat Floquet–Lieb microring lattices provide a versatile platform for investigating topological physics as well as potential applications in realizing topologically‐protected photonic devices.

Isolated flat band in artificially designed Lieb lattice based on macrocycle supramolecular crystal

Isolated flat bands are known to host various strongly correlated phases due to the enhanced Coulomb interactions when the flat bands are gapped from dispersive bands. One way to achieve an isolated flat band is by breaking the on-site energy symmetry in a Lieb lattice. In this study, we demonstrate the design of such a Lieb lattice. The self-assembly of square-shaped macrocycle molecules on a Ag(111) surface forms a two-dimensional supramolecular crystal, comprising three types of nanopores with different sizes arranged in a Lieb lattice. The surface-state electrons of the Ag(111) substrate confined by these nanopores behave as quantum dots with specific energies depending on the pore size. Using scanning tunneling spectroscopy and plane-wave quantum simulation, we reveal that this artificial Lieb lattice exhibits an isolated flat band gapped at 0.16 eV from the nearest band. The supramolecular crystal is nearly defect-free and extends to sub-micrometer size, making it a practical platform for exploring the exotic properties of the isolated flat band.

Sublattice-selective percolation on bipartite planar lattices.

In conventional site percolation, all lattice sites are occupied with the same probability. For a bipartite lattice, sublattice-selective percolation instead involves two independent occupation probabilities, depending on the sublattice to which a given site belongs. Here, we determine the corresponding phase diagram for the two-dimensional square and Lieb lattices from quantifying the parameter regime where a percolating cluster persists for sublattice-selective percolation. For this purpose, we present an adapted Newman-Ziff algorithm. We also consider the critical exponents at the percolation transition, confirming previous Monte Carlo and renormalization-group findings that suggest sublattice-selective percolation belongs to the same universality class as conventional site percolation. To further strengthen this conclusion, we finally treat sublattice-selective percolation on the Bethe lattice (infinite Cayley tree) by an exact solution.

Super Klein tunneling in phononic Lieb lattices

Super Klein tunneling in phononic Lieb lattices

Merging toroidal dipole bound states in the continuum without up-down symmetry in Lieb lattice metasurfaces

Abstract The significance of bound states in the continuum (BICs) lies in their potential for theoretically infinite quality factors. However, their actual quality factors are limited by imperfections in fabrication, which lead to coupling with the radiation continuum. In this study, we present a novel approach to address this issue by introducing a merging BIC regime based on a Lieb lattice. By utilizing this approach, we effectively suppress the out-of-plane scattering loss, thereby enhancing the robustness of the structure against fabrication artifacts. Notably, unlike previous merging systems, our design does not rely on the up-down symmetry of metasurfaces. This characteristic grants more flexibility in applications that involve substrates and superstrates with different optical properties, such as microfluidic devices. Furthermore, we incorporate a lateral band gap mirror into the design to encapsulate the BIC structure. This mirror serves to suppress the in-plane radiation resulting from finite-size effects, leading to a remarkable ten-fold improvement in the quality factor. Consequently, our merged BIC metasurface, enclosed by the Lieb lattice photonic crystal mirror, achieves an exceptionally high-quality factor of 105 while maintaining a small footprint of 26.6 × 26.6 μm. Our findings establish an appealing platform that capitalizes on the topological nature of BICs within compact structures. This platform holds great promise for various applications, including optical trapping, optofluidics, and high-sensitivity biodetection, opening up new possibilities in these fields.

Flat band effects on the ground-state BCS-BEC crossover in atomic Fermi gases in a quasi-two-dimensional Lieb lattice

The ground-state superfluid behavior of ultracold atomic Fermi gases with a short-range attractive interaction in a quasi-two-dimensional Lieb lattice is studied using BCS mean-field theory, within the context of BCS-BEC crossover. We find that the flat band leads to nontrivial exotic effects. As the Fermi level enters the flat band, both the pairing gap and the in-plane superfluid density exhibit an unusual power law as a function of interaction, with strongly enhanced quantum geometric effects, in addition to a dramatic increase of compressibility as the interaction approaches the BCS limit. As the Fermi level crosses the van Hove singularities, the character of pairing changes from particle-like to hole-like or vice versa. We present the computed phase diagram, in which a pair density wave state emerges at high densities with relatively strong interaction strength.

Remarks on the Duffin-Kemmer-Petiau oscillator in an external magnetic field

This work re-examines the issue of spin-1 particles in a (2+1)-dimensional Duffin-Kemmer-Petiau oscillator (DKPO) in the presence of an external magnetic field. By following the appropriate procedure for the spin-1 sector of the Duffin-Kemmer-Petiau (DKP) theory, the previously used 6 × 6 representation in the literature is shown to be reducible to a 3 × 3 irreducible representation. This approach enabled us to find new aspects of the results recently disseminated in various studies, as well as other considerations overlooked and requiring revision. Finally, we present some applications of two-dimensional DKP theory in condensed-matter systems, particularly in Lieb lattices.

Topological band engineering and tunable thermal Hall effect in trimerized Lieb lattice ferromagnets

We theoretically study the topological properties of magnons and the relevant magnon thermal Hall effect in trimerized Lieb lattice ferromagnets in the presence of next-nearest-neighbor Dzyaloshinskii-Moriya interactions. By calculating the magnon band structures and their Chern numbers with the linear spin-wave theory, we show that the system can undergo a phase transition between a magnonic topological insulator phase and a magnonic trivial insulator phase. The main results are presented in the form of topological phase diagrams, where the Chern numbers or magnon thermal Hall conductivity are shown as a function of the two lattice trimerization parameters. We find a sharp change of the thermal Hall conductivity across the critical point of phase transformations associated with topological nontrivial edge states. The behaviors reflect that the existence of trimerizations breaks the C4 rotational symmetry of the Lieb lattice. Finally, we show that our theoretical predictions could be experimentally realized in high-temperature cuprate superconductors or organic magnetic materials. Published by the American Physical Society 2024

Effect of a flat band on a multiband two-dimensional Lieb lattice with intra- and interband interactions

In this study, we explore the effect of a single flat band in the electronic properties of a ferromagnetic two-dimensional Lieb lattice using the multiband Hubbard model with polarized carriers, spin-up and spin-down. We employ the self-consistent dynamical mean field theory and a Green functions cumulant expansion around the atomic limit to obtain the correlated densities of states while varying the intra- and interband interactions. Our findings demonstrate a renormalization of the correlated density of states in both the spin-up and spin-down carriers as we varied the intra- and interband interactions. We conclude that the presence of a flat band enables the system to maintain a metal state with itinerant ferromagnetism in the spin-up carrier.