Artificial flat band systems: from lattice models to experiments

Abstract

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.

14 pages, 5 figures, to appear in Adv. Phys.: X

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• Superconducting Transitions in Flat Band Systems
• Compactly-supported Wannier functions and algebraic $K$-theory
• Observation of flat band for terahertz spoof plasmon in metallic kagomé lattice
• Spin order and phase transitions in chains of polariton condensates
• Polariton condensation in $S$- and $P$-flatbands in a two-dimensional Lieb lattice
• Superconductivity in repulsively interacting fermions on a diamond chain: flat-band induced pairing
• Flat bands in fractal-like geometry
• Compact localized states and flat bands from local symmetry partitioning
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• Quantum Anomalous Hall Effect in Flat Band Ferromagnet
• Desiging Artificial Lieb Lattice on Metal Surface
• Topological flat Wannier-Stark bands
• Nonlinear localized flatband modes with spin-orbit coupling
• Disorder-induced topological phase transitions on Lieb lattices
• Observation of Ground and Excited Flat Band States in Graphene Photonic Ribbons
• Quantum localized states in photonic flat-band lattices
• Lifetime of flatband states
• Spontaneous particle-hole symmetry breaking of correlated fermions on the Lieb lattice
• Topological Landau-Zener Bloch Oscillations in Photonic Floquet Lieb Lattices
• Single and double linear and nonlinear flatband chains: spectra and modes
• Interaction created effective flat bands in conducting polymers
• Emergent light crystal from frustration and pump engineering
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Cited by in corpus (303)

• Negative flat band magnetism in a spin-orbit coupled correlated kagome magnet
• Experimental observation of Aharonov-Bohm cages in photonic lattices
• Classification of flat bands according to the band-crossing singularity of Bloch wave functions
• Non-Bloch band collapse and chiral Zener tunneling
• Fantastic flat bands and where to find them: The CoSn-type compounds
• Tilted and type-III Dirac cones emerging from flat bands in photonic orbital graphene
• Preformed pairs in flat Bloch bands
• Electron-phonon coupling in the charge density wave state of CsV${}_3$Sb${}_5$
• Exciton-Photonics: From Fundamental Science to Applications
• Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics
• Flat Band Quantum Scar
• Magic Fermions: Carroll and Flat Bands
• Aharonov-Bohm Caging and Inverse Anderson transition in Ultracold Atoms
• Many-Body Flatband Localization
• Disordered flat bands on the kagome lattice
• Realization of robust boundary modes and non-contractible loop states in photonic Kagome lattices
• Electron states for gapped pseudospin-1 fermions in the field of charged impurity
• Universal $d=1$ flatband generator from compact localized states
• Quantum geometry and flat band Bose-Einstein condensation
• Engineering interfacial quantum states and electronic landscapes by molecular nanoarchitectures
• Flat-band localization in Creutz superradiance lattices
• Light-matter coupling and quantum geometry in moiré materials
• Nonlinear Polariton Fluids in a Flatband Reveal Discrete Gap Solitons
• Optical lattice platform for the SYK model
• Flatband generator in two dimensions
• Nonlinear dynamics of Aharonov-Bohm cages
• Flat bands by latent symmetry
• Flat band superconductivity in the square-octagon lattice
• Dirac points emerging from flat bands in Lieb-kagomé lattices
• Subradiant emission from regular atomic arrays: universal scaling of decay rates from the generalized Bloch theorem
• Quantum-statistical theory for laser-tuned transport and optical conductivities of dressed electrons in $α-\mc{T}_3$ materials
• Magnon crystallization in the kagome lattice antiferromagnet
• Photon-mediated localization in two-level qubit arrays
• Designer Flat Bands in Quasi-One-Dimensional Atomic Lattices
• Quantum Interactions in Topological R166 Kagome Magnet
• Winding vector: how to annihilate two Dirac points with the same charge
• Nonlinear caging in All-Bands-Flat Lattices
• Tuning the flat bands of the kagome metal CoSn with Fe, In, or Ni doping
• Engineered electronic states in atomically precise artificial lattices and graphene nanoribbons
• Photonic realization of a generic type of graphene edge states exhibiting topological flat band
• A Two-Dimensional Metal-Organic Framework on Superconducting NbSe${}_2$
• Quantum Caging in Interacting Many-Body All-Bands-Flat Lattices
• Origin of the flat band in heavily Cs doped graphene
• Anyonic-parity-time symmetry in complex-coupled lasers
• Synthesis and Local Probe Gating of a Monolayer Metal-Organic Framework
• Interaction-induced topological properties of two bosons in flat-band systems
• Fragile topology in line-graph lattices with two, three, or four gapped flat bands
• Flatband Line States in Photonic Super-Honeycomb Lattices
• Chiral quantum optics in photonic sawtooth lattices
• Nonlinear symmetry breaking of Aharonov-Bohm cages
• Flat band in two-dimensional non-Hermitian optical lattices
• Flat-band localization and interaction-induced delocalization of photons
• Generalized WKB theory for electron tunneling in gapped $\alpha -{\mathcal{T}}_3$ lattices
• RKKY interaction in a doped pseudospin-1 fermion system at finite temperature
• Stable Flatbands, Topology, and Superconductivity of Magic Honeycomb Networks
• Fractal-like photonic lattices and localized states arising from singular and nonsingular flatbands
• Exciton-polaritons in flatland: Controlling flatband properties in a Lieb lattice
• Critical-to-Insulator Transitions and Fractality Edges in Perturbed Flatbands
• Electron-Phonon Interactions in Flat Band Systems
• Designing flat-band tight-binding models with tunable multifold band touching points
• Two-body problem in a multiband lattice and the role of quantum geometry
• Extended flat-bands, entanglement and topological properties in a Creutz ladder
• Latent symmetry induced degeneracies
• Origin of flat-band superfluidity on the Mielke checkerboard lattice
• Domain wall induced spin-polarized flat bands in antiferromagnetic topological insulators
• Higher-order exceptional point and Landau-Zener Bloch oscillations in driven non-Hermitian photonic Lieb lattices
• Interaction induced doublons and embedded topological subspace in a complete flat-band system
• Quantum network transfer and storage with compact localized states induced by local symmetries
• Are Heavy Fermion Strange Metals Planckian?
• Flat-band-based multifractality in the all-band-flat diamond chain
• Non-thermalized Dynamics of Flat-Band Many-Body Localization
• Finite-temperature plasmons, damping and collective behavior for $\alpha -{\mathcal{T}}_3$ model
• Gap generation and flat band catalysis in dice model with local interaction
• Lifetime of flatband states
• Optical conductivity of semi-Dirac and pseudospin-1 models: Zitterbewegung approach
• Many-Body Effects and Optical Properties of Single- and Double Layer $\alpha$-${\mathcal{T}}_3$ Lattices
• Hidden mechanism for embedding the flat bands of Lieb, kagome, and checkerboard lattices in other structures
• Strictly localized states in the octagonal Ammann-Beenker quasicrystal
• Flat band induced by the interplay of synthetic magnetic flux and non-Hermiticity
• Flat band transport and Josephson effect through a finite-size sawtooth lattice
• Localization and criticality in antiblockaded 2D Rydberg atom arrays
• Geometry and superfluidity of the flat band in a non-Hermitian optical lattice
• Dynamical optical conductivity for gapped $\alpha -{\mathcal{T}}_3$ materials with a curved "flat" band
• Effect of magnetic field and chemical potential on the RKKY interaction in the $\alpha$-${\mathcal{T}}_3$ lattice
• Flat-Band Lasing in Silicon Waveguide-Integrated Metasurfaces
• Strain-induced pseudo-magnetic field in $\alpha -{\mathcal{T}}_3$ lattice
• Observation of flat-band and band transition in the synthetic space
• Magnetoplasmons for the $\alpha$-T${}_3$ model with filled Landau levels
• Thermodynamics of delta-chain with ferro- and antiferromagnetic interactions
• Isolated Photonic Flatband with the Effective Magnetic Flux in A Synthetic Space including the Frequency Dimension
• Flat band induced room temperature ferromagnetism in two dimensional systems
• Strain induced superconductor-insulator transition on Lieb lattice
• Interplay and competition between disorder and flat band in an interacting Creutz ladder
• Essential model parameters for nonreciprocal magnons in multisublattice systems
• Hidden symmetries in acoustic wave systems
• Decoding flat bands from compact localized states
• Anderson localization of two-dimensional massless pseudospin-1 Dirac particles in a correlated random one-dimensional scalar potential
• Designing pretty good state transfer via isospectral reductions
• Fractional quantum anomalous Hall effect in a singular flat band
• Topological States in Dimerized Quantum-Dot Chains Created by Atom Manipulation
• Graphene: Free electron scattering within an inverted honeycomb lattice
• Compactly Supported Wannier Functions and Strictly Local Projectors
• Influence of different disorder types on Aharonov-Bohm caging in the diamond chain
• On-surface Assembly of Au-Dicyanoanthracene Coordination Structures on Au(111)
• Suppression of non-equilibrium quasiparticle transport in flat band superconductors
• Non-Hermitian flatband generator in one dimension
• The Rhombi-Chain Bose-Hubbard Model: geometric frustration and interactions
• Metal-insulator transition in infinitesimally weakly disordered flatbands
• Collision of ${\varphi }^4$ kinks free of the Peierls-Nabarro barrier in the regime of strong discreteness
• Realistic flat-band model based on degenerate $p$-orbitals in two-dimensional ionic materials
• Spectroscopic Evidence for Dirac Nodal Surfaces and Nodal Rings in Superconductor NaAlSi
• Perfect localization on flat band binary one-dimensional photonic lattices
• Tailoring flat bands and topological phases in a multi-strand Creutz network
• Flat-Band Itinerant Antiferromagnetism in the Kagome Metal CoSn1-xInx
• Complex magnetic and spatial symmetry breaking from correlations in kagome flat bands
• Quench dynamics of quantum spin models with flat bands of excitations
• Wannier-Stark flatbands in Bravais lattices
• Many-body localization transition from flatband fine-tuning
• Excitation of localized condensates in the flat band of exciton-polariton Lieb lattice
• \textit{Ab initio} study on spin fluctuations of itinerant kagome magnet FeSn
• Exchange interaction, disorder, and stacking faults in rhombohedral graphene multilayers
• Flat bands and entanglement in the Kitaev ladder
• Topological inverse Anderson insulator
• Engineering topological phase transition and Aharonov-Bohm caging in a flux-staggered lattice
• Magnetization, d-wave superconductivity and non-Fermi liquid behavior in a crossover from dispersive to flat bands
• Electronic states of pseudospin-1 fermions in $\alpha -{\mathcal{T}}_3$ lattice ribbons in a magnetic field
• Localization properties in Lieb lattices and their extensions
• Drude weight and the many-body quantum metric in one-dimensional Bose systems
• Isolated flat bands in 2D lattices based on a novel path-exchange symmetry
• Spin-Imbalanced Fermi Superfluidity in a Lieb Lattice Hubbard Model
• Probing Flat Band Physics in Spin Ice Systems via Polarized Neutron Scattering
• Flat Band Induced Metal-Insulator Transitions for Weak Magnetic Flux and Spin-Orbit Disorder
• Melting of magnetization plateaus for kagome and square-kagome lattice antiferromagnets
• Highly degenerate photonic flat bands arising from complete graph configurations
• Cluster dynamics in two-dimensional lattice gases with inter-site interactions
• Line-graph-lattice crystal structures of stoichiometric materials
• Dynamical quantum phase transitions on cross-stitch flat band networks
• Observation of topologically distinct corner states in "bearded" photonic Kagome lattices
• Novel Electronic Structure of Nitrogen-Doped Lutetium Hydrides
• Stackings and effective models of bilayer dice lattices
• Optical conductivity of bilayer dice lattices
• Persistent homology analysis of a generalized Aubry-André-Harper model
• Evolution of ultra-flat band in van der Waals kagome semiconductor Pd3P2(S1-xSex)8
• Anti-$\mathcal{P}\mathcal{T}$ flatbands
• Kondo-lattice-mediated interactions in flat band systems
• Quantum electrodynamics in anisotropic and tilted Dirac photonic lattices
• A General Method to Design Acoustic Higher-Order Topological Insulators
• Dipole-dipole interactions mediated by a photonic flat band
• Going beyond the Chandrasekhar-Clogston limit in a flat-band superconductor
• Building Flat-Band Lattice Models from Gram Matrices
• Local Hilbert space fragmentation and weak thermalization in Bose-Hubbard diamond necklaces
• Unconventional delocalization in a family of 3D Lieb lattices
• Enhanced repulsively bound atom pairs in topological optical lattice ladders
• Topological chiral currents in the Gross-Neveu model extension
• 1D flat bands in phosphorene nanoribbons with pentagonal nature
• Temperature dependence of correlated electronic states in archetypal kagome metal CoSn
• Compact localised states in magnonic Lieb lattices
• Multiple flat bands and localized states in photonic super-Kagome lattices
• The GOE ensemble for quasiperiodic tilings without unfolding: $r$-value statistics
• Flat Bands and Compact Localised States: A Carrollian roadmap
• Signatures of many-body localization of quasiparticles in a flat band superconductor
• Conductance transition with interacting bosons in an Aharonov-Bohm cage
• Pairs, trimers and BCS-BEC crossover near a flat band: the sawtooth lattice
• Inseparable time-crystal geometries on the Möbius strip
• Classical ground states of spin lattices
• Scaling theory of wave confinement in classical and quantum periodic systems
• Two-flux tunable Aharonov-Bohm caging in a photonic lattice
• Wannier solitons in spin-orbit-coupled Bose-Einstein condensates in optical lattices with a flat-band
• Hubbard models and state preparation in an optical Lieb lattice
• Observation of Flat Frequency Bands at Open Edges and Antiphase Boundary Seams in Topological Mechanical Metamaterials
• Spectral Properties of Two Coupled Fibonacci Chains
• Supersymmetry on the lattice: Geometry, Topology, and Flat Bands
• Magnetoresistance signature of two-dimensional electronic states in Co${}_3$Sn${}_2$S${}_2$
• Infinite bound states and hydrogen atom-like energy spectrum induced by a flat band
• Polaron spectra and edge singularities for correlated flat bands
• Correlation functions and characteristic lengthscales in flat band superconductors
• Tunable Aharonov-Bohm cages through anti-$\mathcal{P}\mathcal{T}$-symmetric imaginary couplings
• Surge of power transmission in flat and nearly flat band lattices
• Non-ergodic one-magnon magnetization dynamics of the antiferromagnetic delta chain
• Effective Medium Model for Graphene Superlattices with Electrostatic and Magnetic Vector Potentials
• Critical States Generators from Perturbed Flatbands
• Nonequilibrium Landau-Zener Tunneling in Exciton-Polariton Condensates
• Omnidirectional excitation of surface waves and super-Klein tunneling at the interface between two different bi-isotropic media
• Optical properties of two dimensional Dirac Weyl materials with a flatband
• Robust incoherent perfect absorption
• Lieb lattices and pseudospin-1 dynamics under barrier- and well-like electrostatic interactions
• Flat bands and magnetism in $\mathbf{F}{\mathbf{e}}_4\mathbf{GeT}{\mathbf{e}}_2$ and $\mathbf{F}{\mathbf{e}}_5\mathbf{GeT}{\mathbf{e}}_2$ due to bipartite crystal lattices
• One-dimensional flat bands and Dirac cones in narrow zigzag dice lattice ribbons
• Ferromagnetism in armchair graphene nanoribbon heterostructures
• Confined states and topological phases in two-dimensional quasicrystalline $\pi$-flux model
• The Berezinskii-Kosterlitz-Thouless Transition and Anomalous Metallic Phase in a Hybrid Josephson Junction Array
• Topological Bogoliubov quasiparticles from Bose-Einstein condensate in a flat band system
• Superconductivity with Wannier-Stark Flat Bands
• Singular flat bands in the modified Haldane-Dice model
• RKKY interaction in one-dimensional flat-band lattices
• Excitation spectra of strongly interacting bosons in the flat-band Lieb lattice
• Strange metal phase of disordered magic-angle twisted bilayer graphene at low temperatures: from flatbands to weakly coupled Sachdev-Ye-Kitaev bundles
• Localized States in Local Isomorphism Classes of Pentagonal Quasicrystals
• Coulomb impurity on a Dice lattice: atomic collapse and bound states
• Flat-band engineering of quasi-one-dimensional systems via supersymmetric transformations
• Robustness of flat band superconductivity against disorder in a two-dimensional Lieb lattice model
• Superfluidity of dipolar excitons in a double layer of $\alpha -T_3$ with a mass term
• Doping induced multiferroicity and quantum anomalous Hall effect in $\alpha$-In${}_2$Se${}_3$ thin films
• Fubini-Study metric and topological properties of flat band electronic states: the case of an atomic chain with $s-p$ orbitals
• Compact Wannier Functions in One Dimension
• Complete escape from localization on a hierarchical lattice: A Koch fractal with all states extended
• Deformation of localized states and state transitions in systems of randomly hopping interacting fermions
• Certain exact many-body results for Hubbard model ground states testable in small quantum dot arrays
• Interplay of many-body interactions and quasiperiodic disorder in the all-band-flat diamond chain
• Engineering flux-controlled flat bands and topological states in a Stagome lattice
• A tale of two localizations: coexistence of flat bands and Anderson localization in a photonics-inspired amorphous system
• Trapping Hard-Core Bosons in Flatband Lattices
• Permanent oscillations and solitary wave behavior in flatband Heisenberg quantum spin systems
• Non-Hermitian synthetic lattices with light-matter coupling
• Stability of $(N+1)$-body fermion clusters in multiband Hubbard model
• Reflectionless pseudospin-1 Dirac systems via Darboux transformation and flat band solutions
• Dirac node engineering and flat bands in doped Dirac materials
• Electronic structure and transport in materials with flat bands: 2D materials and quasicrystals
• Differential entropy per particle as a probe of van Hove singularities and flat bands
• van Hove Singularity-Driven Emergence of Multiple Flat Bands in Kagome Superconductors
• Bound states and point interactions of the one-dimensional pseudospin-one Hamiltonian
• Mode conversion and resonant absorption in inhomogeneous materials with flat bands
• Renormalization approach to the analysis and design of Hermitian and non-Hermitian interfaces
• Two-dimensional flat-band solitons in superhoneycomb lattices
• The Fractal-Lattice Hubbard Model
• Optimization of metasurfaces for lasing with symmetry constraints on the modes
• Extended states in one-dimensional aperiodic lattices with linearly varying patches
• Spin-orbit interactions may relax the rigid conditions leading to flat bands
• How to realize compact and non-compact localized states in disorder-free hypercube networks
• Finite temperatures and flat bands: the Hubbard model on three-dimensional Lieb lattices
• An Asymmetric Spoof-Fluid-Spoof Acoustic Waveguide and its Application as a CO${}_2$ Sensor
• Lasing in para-Fermi class-B microring resonator arrays
• Transfer of solitons and half-vortex solitons via adiabatic passage
• Spin-selective Aharonov-Casher caging in a topological quantum network
• Ferromagnetism vs. Antiferromagnetism in Narrow-Band Systems: Competition Between Quantum Geometry and Band Dispersion
• Flux-mediated effective Su-Schrieffer-Heeger model in an impurity decorated diamond chain
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Anomalous Diffusion, Prethermalization, and Particle Binding in an Interacting Flat Band System
• Aharonov-Bohm caging of an electron in a quantum fractal
• Band structure and optical response of Kekulé-modulated $\alpha -{\mathcal{T}}_3$ model
• Many-Body Localization in Translational Invariant Diamond Ladders with Flat Bands
• Quantum engineering for compactly localized states in disordered Lieb lattices
• Topological phase transitions in superconductors with chiral symmetry
• Correlated metallic two-particle bound states in Wannier--Stark flatbands
• Identifying Instabilities with Quantum Geometry in Flat Band Systems
• Localized Fock Space Cages in Kinetically Constrained Models
• Twisted Type-II Rashba Homobilayer: A Platform for Tunable Topological Flat Bands
• Topological Triviality of Flat Hamiltonians
• Flat Band Generation through Interlayer Geometric Frustration in Intercalated Transition Metal Dichalcogenides
• Symmetry-protected flatband condition for Hamiltonians with local symmetry
• Flat bands in tight-binding lattices with anisotropic potentials
• Coexistence of ergodic and non-ergodic behavior and level spacing statistics in a one-dimensional model of a flat band superconductor
• Re-entrant topological order in strongly correlated nanowire due to Rashba spin-orbit coupling
• Current distribution and group velocities for electronic states on $\alpha -{\mathcal{T}}_3$ lattice ribbons in a magnetic field
• Observation of non-Markovian Radiative Phenomena in Structured Photonic Lattices
• Selective band interaction and long-range hopping in a structured environment with giant atoms
• Phase diagram of a coupled trimer system at half filling using the Hubbard model
• A non-Hermitian Su-Schrieffer-Heeger model with the energy levels of free parafermions
• Boundary Flat Bands with Topological Spin Textures Protected by Sub-chiral Symmetry
• Optical Black-hole Analog Created by Topological Phase Transition with a Long-lived Horizon
• In-band supratransmission in nonlinear flat band lattices
• Developing a semiclassical Wentzel-Kramers-Brillouin theory for $\alpha -{\mathcal{T}}_3$ model
• Topological phenomena demonstrated in photorefractive photonic lattices
• Synthetic $\pi$-flux system in 2D superconducting qubit array with tunable coupling
• Nonlinear gap modes and compactons in a lattice model for spin-orbit coupled exciton-polaritons in zigzag chains
• Dichotomy of flat bands in the van der Waals ferromagnet Fe${}_5$GeTe${}_2$
• Flat and almost flat bands in the quasi-one-dimensional Josephson junction array
• All-optical switching at the two-photon limit with interference-localized states
• Nonlinear compact localized modes in flux-dressed octagonal-diamond photonic lattice
• Quantum dots on the InAs(110) cleavage surface created by atom manipulation
• Localized waves in the nonlinear rhombic waveguide array
• Majorana edge states in s-wave kagome superconductors with Rashba interaction
• Non-Hermitian Aharonov-Bohm Cage
• Flat bands and topological phase transition in entangled Su-Schrieffer-Heeger chains
• Resonant cavity-QED with chiral flat bands
• Tunable zero modes and quantum interferences in flat-band topological insulators
• Heat percolation in many-body flatband localizing systems
• Rydberg Atoms in a Ladder Geometry: Quench Dynamics and Floquet Engineering
• Compact localized states of open scattering media
• Flat bands and band touching from real-space topology in hyperbolic lattices
• Quantum simulator of extended bipartite Hubbard model with broken sublattice symmetry: magnetism, correlations, and phase transitions
• Machine Learning Study on the Flat-Band States Constructed by Molecular-Orbital Representation with Randomness
• Non-local interactions and supersolidity of moiré excitons
• Engineering the Bogoliubov Modes through Geometry and Interaction: From Collective Edge Modes to Flat-band Excitations
• Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars
• Designer quantum matter in van der Waals heterostructures
• Superfluid weight in disordered flat-band superconductors as a competition between localization functionals
• Non-ergodic one-magnon magnetization dynamics of the kagome lattice antiferromagnet
• On flat bands in the $J_1$-$J_2$-$J_3$ XXZ sawtooth chain
• Compact localized currents in flat bands with broken time-reversal symmetry
• Topologically nontrivial flat bands and quantum Hall crossovers in square-octagon lattice materials
• Electric-field driven flat bands in the distorted sawtooth chain via the Katsura-Nagaosa-Balatsky mechanism
• Superconductivity and geometric superfluid weight of a tunable flat band system
• Chains of nanoparticles for flat-band emission and lasing
• Flux-induced strengthening of the magnetic couplings in a flat-band diamond chain
• Localization and Flat Bands in Bond-Inflated Lattices
• Direct Observation of the Zigzag Edge States of a Supramolecular Diatomic Kagome Lattice
• Zak Phase Dislocations in Trimer Lattices
• Phase and Photon Number Dependent NOON State Localization in Flat Band Lattices
• Influence of flat bands on RKKY interaction: perspective of Fano defects
• Proximity-induced flat bands and topological properties in a decorated diamond chain
• Flat band mediated photon-photon interactions in 2D waveguide QED networks
• Towards lattice-gas description of low-temperature properties above the Haldane and cluster-based Haldane ground states of a mixed spin-(1,1/2) Heisenberg octahedral chain