Flat-band Physics Research Articles

Probing Flat Band Physics in Spin Ice Systems via Polarized Neutron Scattering.

In this Letter, we illustrate how polarized neutron scattering can be used to isolate the spin-spin correlations of modes forming flat bands in a frustrated magnetic system hosting a classical spin liquid phase. In particular, we explain why the nearest-neighbor spin ice model, whose interaction matrix has two flat bands, produces a dispersionless (i.e., "flat") response in the non-spin-flip (NSF) polarized neutron scattering channel and demonstrate that NSF scattering is a highly sensitive probe of correlations induced by weak perturbations that lift the flat band degeneracy. We use this to explain the experimentally measured dispersive (i.e., nonflat) NSF channel of the dipolar spin ice compound Ho_{2}Ti_{2}O_{7}.

Flat-Band-Induced Anomalous Anisotropic Charge Transport and Orbital Magnetism in Kagome Metal CoSn.

For solids, the dispersionless flat band has long been recognized as an ideal platform for achieving intriguing quantum phases. However, experimental progress in revealing flat-band physics has so far been achieved mainly in artificially engineered systems represented as magic-angle twisted bilayer graphene. Here, we demonstrate the emergence of flat-band-dominated anomalous transport and magnetic behaviors in CoSn, a paramagnetic kagome-lattice compound. By combination of angle-resolved photoemission spectroscopy measurements and first-principles calculations, we reveal the existence of a kagome-lattice-derived flat band right around the Fermi level. Strikingly, the resistivity within the kagome lattice plane is more than one order of magnitude larger than the interplane one, in sharp contrast with conventional (quasi-) two-dimensional layered materials. Moreover, the magnetic susceptibility under the out-of-plane magnetic field is found to be much smaller as compared with the in-plane case, which is revealed to be arising from the introduction of a unique orbital diamagnetism. Systematic analyses reveal that these anomalous and giant anisotropies can be reasonably attributed to the unique properties of flat-band electrons, including large effective mass and self-localization of wave functions. Our results broaden the already fascinating flat-band physics, and demonstrate the feasibility of exploring them in natural solid-state materials in addition to artificial ones.

Electric field driven flat bands: Enhanced magnetoelectric and electrocaloric effects in frustrated quantum magnets

The $J_1$-$J_2$ quantum spin sawtooth chain is a paradigmatic one-dimensional frustrated quantum spin system exhibiting unconventional ground-state and finite-temperature properties. In particular, it exhibits a flat energy band of one-magnon excitations accompanied by an enhanced magnetocaloric effect for two singular ratios of the basal interactions $J_1$ and the zigzag interactions $J_2$. In our paper, we demonstrate that one can drive the spin system into a flat-band scenario by applying an appropriate electric field, thus overcoming the restriction of fine-tuned exchange couplings $J_1$ and $J_2$ and allowing one to tune more materials towards flat-band physics, that is to show a macroscopic magnetization jump when crossing the magnetic saturation field, a residual entropy at zero temperature as well as an enhanced magnetocaloric effect. While the magnetic field acts on the spin system via the ordinary Zeeman term, the coupling of an applied electric field with the spins is given by the sophisticated Katsura-Nagaosa-Balatsky (KNB) mechanism, where the electric field effectively acts as a Dzyaloshinskii-Moriya spin-spin interaction. The resulting novel features are corresponding reciprocal effects: We find a magnetization jump driven by the electric field as well as a jump of the electric polarization driven by the magnetic field, i.e.\ the system exhibits an extraordinarily strong magnetoelectric effect. Moreover, in analogy to the enhanced magnetocaloric effect the system shows an enhanced electrocaloric effect.

Kondo lattice mediated interactions in flat-band systems

Electronic flat bands represent a paradigmatic platform to realize strongly correlated matter due to their associated divergent density of states. In common instances, including electron-electron interactions leads to magnetic instabilities for repulsive interactions and superconductivity for attractive interactions. Nevertheless, interactions of Kondo nature in flat band systems have remained relatively unexplored. Here we address the emergence of interacting states mediated by Kondo lattice coupled to a flat band system. Combining dynamical mean-field theory and tensor networks methods to solve flat band Kondo lattice models in one and two dimensions, we show the emergence of a robust underscreened regime leading to a magnetically ordered state in the flat band. Our results put forward flat band Kondo lattice models as a platform to explore the genuine interplay between flat band physics and many-body Kondo screening.

Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in the strong coupling limit

When two identical two-dimensional (2D) periodic lattices are stacked in parallel after rotating one layer by a certain angle relative to the other layer, the resulting bilayer system can lose lattice periodicity completely and become a 2D quasicrystal. Twisted bilayer graphene with 30-degree rotation is a representative example. We show that such quasicrystalline bilayer systems generally develop macroscopically degenerate localized zero-energy states (ZESs) in strong coupling limit where the interlayer couplings are overwhelmingly larger than the intralayer couplings. The emergent chiral symmetry in strong coupling limit and aperiodicity of bilayer quasicrystals guarantee the existence of the ZESs. The macroscopically degenerate ZESs are analogous to the flat bands of periodic systems, in that both are composed of localized eigenstates, which give divergent density of states. For monolayers, we consider the triangular, square, and honeycomb lattices, comprised of homogenous tiling of three possible planar regular polygons: the equilateral triangle, square, and regular hexagon. We construct a compact theoretical framework, which we call the quasiband model, that describes the low energy properties of bilayer quasicrystals and counts the number of ZESs using a subset of Bloch states of monolayers. We also propose a simple geometric scheme in real space which can show the spatial localization of ZESs and count their number. Our work clearly demonstrates that bilayer quasicrystals in strong coupling limit are an ideal playground to study the intriguing interplay of flat band physics and the aperiodicity of quasicrystals.

Ferromagnetic percolation transition in a multiorbital flat band assisted by Hund's coupling

By connecting Hund's physics with flat band physics, we establish an exact result for studying ferromagnetism in a multiorbital system. We consider a two-layer model consisting of a ${p}_{x}, {p}_{y}$-orbital honeycomb lattice layer and an $f$-orbital triangular lattice layer with sites aligned with the centers of the honeycomb plaquettes. The system features a flat band that admits a percolation representation for an appropriate chemical potential difference between the two layers. In this representation, the ground-state space is spanned by maximum-spin clusters of localized single-particle states, and averaging over the ground states yields a correlated percolation problem with weights due to the spin degeneracy of the clusters. A paramagnetic-ferromagnetic transition occurs as the band approaches half filling and the ground states become dominated by states with a large maximum-spin cluster, as shown by Monte Carlo simulation.

Isolated flat bands in a lattice of interlocking circles

Flat-band physics has attracted much attention in recently years because of its interesting properties and important applications. Some typical lattices have been proposed to generate flat bands, such as Kagome and Lieb lattices. The flat bands in these lattices contact with other bands rather than isolated. However, an ideal flat band should be isolated, because isolation is a prerequisite for a number of important applications. Here, we propose a new lattice that can produce isolated flat bands. The lattice is named as interlocking-circles lattice because its pattern seems like interlocking circles. Moreover, the new lattice is realized in graphene by hydrogenation. In the hydrogenated graphene, there are two nontrivial isolated flat bands appearing around the Fermi level. Upon hole or electron doping, the flat bands split into spin-polarized bands and then result in a ferromagnetic graphene. Our work not only proposes a new type of lattice but also opens a way to find systems with isolated flat bands.

Cluster dynamics in two-dimensional lattice gases with intersite interactions

Sufficiently strong inter-site interactions in extended-Hubbard and XXZ spin models result in dynamically-bound clusters at neighboring sites. We show that the dynamics of these clusters in two-dimensional lattices is remarkably different and richer than that of repulsively-bound on-site clusters in gases without inter-site interactions. Whereas on-site pairs move in the same lattice as individual particles, nearest-neighbor dimers perform an interacting quantum walk in a different lattice geometry, leading to a peculiar dynamics characterized by more than one time scale. The latter is general for any lattice geometry, but it is especially relevant in triangular and diamond lattices, where dimers move resonantly in an effective kagome and Lieb lattice, respectively. As a result, dimers experience partial quasi-localization due to an effective flat band, and may move slower than longer clusters. This surprising link between anomalously slow quantum walk dynamics in these models and flat-band physics may be readily observed in experiments with lanthanide atoms.

Observation of Flat Bands in Gated Semiconductor Artificial Graphene.

Flat bands near M points in the Brillouin zone are key features of honeycomb symmetry in artificial graphene (AG) where electrons may condense into novel correlated phases. Here we report the observation of van Hove singularity doublet of AG in GaAs quantum well transistors, which presents the evidence of flat bands in semiconductor AG. Two emerging peaks in photoluminescence spectra tuned by backgate voltages probe the singularity doublet of AG flat bands and demonstrate their accessibility to the Fermi level. As the Fermi level crosses the doublet, the spectra display dramatic stability against electron density, indicating interplays between electron-electron interactions and honeycomb symmetry. Our results provide a new flexible platform to explore intriguing flat band physics.

A continuous family of fully frustrated Heisenberg models on the Kagome lattice

We find that the antiferromagnetic Heisenberg model on the Kagome lattice with nearest-neighboring exchange coupling (NN-KAFH) belongs to a continuous family of fully frustrated Heisenberg model on the Kagome lattice, which has no preferred classical ordering pattern. The model within this family consists of the first-, second- and the third-neighboring exchange coupling J 1, J 2, and J 3, with . We find that when , the lowest band of , namely, the Fourier transform of the exchange coupling, is totally non-dispersive. Exact diagonalization calculation indicates that the ground state of the spin- NN-KAFH is locally stable under the perturbation of J 2 and J 3 when and only when . Interestingly, we find that the same flat band physics is also playing an important role in the RVB description of the spin liquid state on the Kagome lattice. In particular, we show that the extensively studied U(1) Dirac spin liquid state on the Kagome lattice can actually be generated from a continuous family of gauge inequivalent RVB mean-field ansatz, which host very different mean-field spinon dispersion.

Flat bands in twisted bilayers of polar two-dimensional semiconductors

We investigate the Bloch flat bands in twisted bilayers from nonpolar to polar two-dimensional semiconductors using first-principles calculations and density functional based tight-binding simulations. First, to delineate the underlying mechanism of the formation of the flat bands, we rely on a tight-binding model of modified graphene where a bias between the A-B sublattice of the hexagonal lattice is introduced. By analyzing the evolution of the valence and conduction band edges of the bilayer of the modified graphene with different stacking patterns, a mechanism attributed to the splitting of the defect-like band edge states induced by different stacking patterns is revealed. The magic angle mechanism is no longer needed. Next, guided by the revealed mechanism, we predict the formation of flat bands in twisted bilayers of a series of two-dimensional systems from nonpolar to polar semiconductors. Our finding has important implications for exploring the flat band physics in low dimensions.

Flat bands and Z2 topological phases in a non-Abelian kagome lattice

We introduce a non-Abelian kagome lattice model that has both time-reversal and inversion symmetries and study the flat band physics and topological phases of this model. Due to the coexistence of both time-reversal and inversion symmetries, the energy bands consist of three doubly degenerate bands whose energy and conditions for the presence of flat bands could be obtained analytically, allowing us to tune the flat band with respect to the other two dispersive bands from the top to the middle and then to the bottom of the three bands. We further study the gapped phases of the model and show that they belong to the same phase as the band gaps only close at discrete points of the parameter space, making any two gapped phases adiabatically connected to each other without closing the band gap. Using the Pfaffian approach based on the time-reversal symmetry and parity characterization from the inversion symmetry, we calculate the bulk topological invariants and demonstrate that the unique gapped phases belong to the $Z_2$ quantum spin Hall phase, which is further confirmed by the edge state calculations.

Variational study of the ground state and spin dynamics of the spin- 12 kagome antiferromagnetic Heisenberg model and its implication for herbertsmithite ZnCu3(OH)6Cl2

We find that the best RVB state of the spin-$\frac{1}{2}$ Kagome antiferromagnetic Heisenberg model(spin-$\frac{1}{2}$ KAFH) is described by a $Z_{2}$ gapped mean field ansatz, which hosts a mean field spinon dispersion very different from that of the widely studied $U(1)$ Dirac spin liquid state. However, we find that the physical spin fluctuation spectrum calculated from the Gutzwiller projected RPA(GRPA) theory above such an RVB state is actually gapless and is almost identical to that above the $U(1)$ Dirac spin liquid state. We find that such a peculiar behavior can be attributed to the unique flat band physics on the Kagome lattice, which makes the mapping between the mean field ansatz and the RVB state non-injective. We find that the spin fluctuation spectrum of the spin-$\frac{1}{2}$ KAFH is not at all featureless, but is characterized by a prominent spectral peak at about $0.25J$ around the $\mathbf{M}$ point, which is immersed in a broad continuum extending to $2.7J$. Based on these results, we argue that the spectral peak below 2 meV in the inelastic neutron scattering(INS) spectrum of Hebertsmithite ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$, which has been attributed to the contribution of Cu$^{2+}$ impurity spins occupying the Zn$^{2+}$ site, should rather be understood as the intrinsic contribution from the Kagome layer. We propose to verify such a picture by measuring the Knight shift on the Cu site, rather than the O site, which is almost blind to the spin fluctuation at the $\mathbf{M}$ point as a result of the strong antiferromagnetic correlation between nearest neighboring spins on the Kagome lattice.

Effect of bilayer stacking on the atomic and electronic structure of twisted double bilayer graphene

Twisted double bilayer graphene has recently emerged as an interesting moir\'e material that exhibits strong correlation phenomena that are tunable by an applied electric field. Here we study the atomic and electronic properties of three different graphene double bilayers: double bilayers composed of two AB stacked bilayers (AB/AB), double bilayers composed of two AA stacked bilayers (AA/AA) as well as heterosystems composed of one AB and one AA bilayer (AB/AA). The atomic structure is determined using classical force fields. We find that the inner layers of the double bilayer exhibit significant in-plane and out-of-plane relaxations, similar to twisted bilayer graphene. The relaxations of the outer layers depend on the stacking: atoms in AB bilayers follow the relaxations of the inner layers, while atoms in AA bilayers attempt to avoid higher-energy AA stacking. For the relaxed structures, we calculate the electronic band structures using the tight-binding method. All double bilayers exhibit flat bands at small twist angles, but the shape of the bands depends sensitively on the stacking of the outer layers. To gain further insight, we study the evolution of the band structure as the outer layers are rigidly moved away from the inner layers, while preserving their atomic relaxations. This reveals that the hybridization with the outer layers results in an additional flattening of the inner-layer flat band manifold. Our results establish AA/AA and AB/AA twisted double bilayers as interesting moir\'e materials with different flat band physics compared to the widely studied AB/AB system.

Interplay of quantum phase transition and flat band in hybrid lattices

We investigate the interplay of superradiant phase transition (SPT) and energy band physics in an extended Dicke-Hubbard lattice whose unit cell consists of a Dicke model coupled to an atomless cavity. We found in such a periodic lattice the critical point that occurs in a single Dicke model becomes a critical region that is periodically changing with the wavenumber $k$. In the weak-coupling normal phase of the system we observed a flat band and its corresponding localization that can be controlled by the ground-state SPT. Our work builds the connection between flat band physics and SPT, which may fundamentally broaden the regimes of many-body theory and quantum optics.

Lithium intercalation in MoS2 bilayers and implications for moiré flat bands

Intercalation of lithium atoms between layers of 2D materials can alter their atomic and electronic structure. We investigate effects of Li intercalation in twisted bilayers of the transition metal dichalcogenide MoS$_2$ through first-principles calculations, tight-binding parameterization based on the Wannier transformation, and analysis of moir\'e band structures through an effective continuum model. The energetic stability of different intercalation sites for Li between layers of MoS$_2$ are classified according to the local coordination type and the number of vertically aligned Mo atoms, suggesting that the Li atoms will cluster in certain regions of the moir\'e superlattice. The proximity of a Li atom has a dramatic influence on the interlayer interaction between sulfur atoms, deepening the moir\'e potential well and leading to better isolation of the flat bands in the energy spectrum. These results point to the usefulness for the use of chemical intercalation as a powerful means for controlling moir\'e flat-band physics in 2D semiconductors.

Flat bands in the CoSn-type compounds

Quantum interference on the kagome lattice generates electronic bands with narrow bandwidth, called flat bands. Crystal structures incorporating this lattice can host strong electron correlations with non-standard ingredients, but only if these bands lie at the Fermi level. In the six compounds with the CoSn structure type (FeGe, FeSn, CoSn, NiIn, RhPb, and PtTl) the transition metals form a kagome lattice. The two iron variants are robust antiferromagnets so we focus on the latter four and investigate their thermodynamic and transport properties. We consider these results and calculated band structures to locate and characterize the flat bands in these materials. We propose that CoSn and RhPb deserve the community's attention for exploring flat band physics.

Isolated Photonic Flatband with the Effective Magnetic Flux in a Synthetic Space Including the Frequency Dimension

AbstractExploration of the flatband in photonics is fundamentally important, aiming to control the localization of light for potential applications in optical communications. Herein, the flatband physics is studied in a synthetic space including the frequency axis of light. A ring‐resonator array is used to construct a synthetic Lieb‐type lattice, where the modulation phase distribution supports a locally non‐zero effective magnetic flux pointing into the synthetic space. It is found that the flatband is isolated from other dispersive bands and the light can still be localized even with perturbation from the group velocity dispersion of the waveguide. This work points out a route towards manipulating the localization performance of a wavepacket of light along both the spatial dimension and the frequency dimension, which holds the potential implication for controlling the storage of optical information in dispersive materials.

Flat-band physics in the spin-1/2 sawtooth chain

We consider the strongly anisotropic spin-1/2 XXZ model on the sawtooth-chain lattice with ferromagnetic longitudinal interaction Jzz = ΔJ and aniferromagnetic transversal interaction Jxx = Jyy = J > 0. At Δ = −1∕2 the lowest one-magnon excitation band is dispersionless (flat) leading to a massively degenerate set of ground states. Interestingly, this model admits a three-coloring representation of the ground-state manifold [H.J. Changlani et al., Phys. Rev. Lett. 120, 117202 (2018)]. We characterize this ground-state manifold and elaborate the low-temperature thermodynamics of the system. We illustrate the manifestation of the flat-band physics of the anisotropic model by comparison with two isotropic flat-band Heisenberg sawtooth chains. Our analytical consideration is complemented by exact diagonalization and finite-temperature Lanczos method calculations.Graphical abstract

Magnetic and electronic properties of spin-orbit coupled Dirac electrons on a (001) thin film of double-perovskite Sr2FeMoO6

We present an interacting model for the electronic and magnetic behavior of a strained $(001)$ atomic layer of Sr$_2$FeMoO$_6,$ which shows room-temperature ferrimagnetism and magnetoresistance with potential spintronics application in the bulk. We find that the strong spin-orbit coupling in the molybdenum 4$d$ shell gives rise to a robust ferrimagnetic state with an emergent spin-polarized electronic structure consisting of flat bands and four massive or massless Dirac dispersions. Based on the spin-wave theory, we demonstrate that the magnetic order remains intact for a wide range of doping, leading to the possibility of exploring flat band physics, such as Wigner crystallization in electron-doped Sr$_{2-x}$La$_{x}$FeMoO$_6.$