Nontrivial Flat Bands Research Articles

Mathematical results on the chiral models of twisted bilayer graphene (with an appendix by Mengxuan Yang and Zhongkai Tao)

The study of twisted bilayer graphene (TBG) is a hot topic in condensed matter physics with special focus on magic angles of twisting at which TBG acquires unusual properties. Mathematically, topologically non-trivial flat bands appear at those special angles. The chiral model of TBG pioneered by Tarnopolsky, Kruchkov, and Vishwanath (2019) has particularly nice mathematical properties and we survey, and in some cases, clarify, recent rigorous results which exploit them.

One-dimensional Dexter-type excitonic topological phase transition

The concept of the Su-Schrieffer-Heeger model, which was successfully introduced in topological photonics, can also be applied to the study of topological excitonics. Here we study the topological properties of a one-dimensional excitonic tight-binding model formed by two-level systems by taking into account the charge transfer and local excitations. The interactions between the two types of excitons give rise to a rich spectrum of physics, including the nontrivial topological phase in the uniform chain, unlike the conventional Su-Schrieffer-Heeger model, the topologically nontrivial flat bands, and, most importantly, the excitonic topological phase transition assisted by the Dexter electron exchange process. The excitonic topological phase leads to the development of “chiral superposition.” The topological edge states are robust because they are protected by the inversion symmetry. Based on our calculations, experiments for observing the edge states optically in a molecular chain are proposed. Published by the American Physical Society 2024

Multichannel coupling induced topological insulating phases with full multimerization

We propose and experimentally realize a class of quasi-one-dimensional topological lattices whose unit cells are constructed by coupled multiple identical resonators, with uniform hopping and inversion symmetry. In the presence of coupling-path-induced effective zero hopping within the unit cells, the systems are characterized by complete multimerization with degenerate −1 energy edge states for open boundary condition. Su–Schrieffer–Heeger subspaces with fully dimerized limits corresponding to pairs of nontrivial flat bands are derived from the Hilbert spaces. In particular, topological bound states in the continuum (BICs) are inherently present in even multimer chains, manifested by embedding the topological bound states into a continuous band assured by bulk-boundary correspondence. Moreover, we experimentally demonstrate the degenerate topological edge states and topological BICs in radio-frequency circuits.

Twistronics of Kekulé Graphene: Honeycomb and Kagome Flat Bands.

Kekulé-O order in graphene, which has recently been realized experimentally, induces Dirac electron masses on the order of m∼100 meV. We show that twisted bilayer graphene in which one or both layershave Kekulé-O order exhibits nontrivial flat electronic bands on honeycomb and kagome lattices. When only one layer has Kekulé-O order, there is a parameter regime for which the lowest four bands at charge neutrality form an isolated two-orbital honeycomb lattice model with two flat bands. The bandwidths are minimal at a magic twist angle θ≈0.7° and Dirac mass m≈100 meV. When both layers have Kekulé-O order, there is a large parameter regime around θ≈1° and m≳100 meV in which the lowest three valence and conduction bands at charge neutrality each realize isolated kagome lattice models with one flat band, while the next three valence and conduction bands are flat bands on triangular lattices. These flat band systems may provide a new platform for strongly correlated phases ofmatter.

General construction scheme for geometrically nontrivial flat band models

A singular flat band (SFB), a distinct class of the flat band, has been shown to exhibit various intriguing material properties characterized by the quantum distance. We present a general construction scheme for a tight-binding model hosting an SFB, where the quantum distance profile can be controlled. We first introduce how to build a compact localized state (CLS), endowing the flat band with a band-touching point and a specific value of the maximum quantum distance. Then, we develop a scheme designing a tight-binding Hamiltonian hosting an SFB starting from the obtained CLS, with the desired hopping range and symmetries. We propose several simple SFB models on the square and kagome lattices. Finally, we establish a bulk-boundary correspondence between the maximum quantum distance and the boundary modes for the open boundary condition, which can be used to detect the quantum distance via the electronic structure of the boundary states.

High Chern numbers in a perovskite-derived dice lattice (LaXO3)3/(LaAlO3)3(111) with X = Ti, Mn and Co

The dice lattice, containing a stack of three triangular lattices, has been proposed to exhibit nontrivial flat bands with nonzero Chern numbers, but unlike the honeycomb lattice it is much less studied. By employing density-functional theory (DFT) calculations with an on-site Coulomb repulsion term, we explore systematically the electronic and topological properties of (LaXO3)3/(LaAlO3)3(111) superlattices with X = Ti, Mn and Co, where a LaAlO3 trilayer spacer confines the LaXO3 (LXO) dice lattice. In the absence of spin-orbit coupling (SOC) with symmetry constrained to P3, the ferromagnetic (FM) phase of the LXO(111) trilayers exhibits a half-metallic band structure with multiple Dirac crossings and coupled electron-hole pockets around the Fermi energy. Symmetry lowering induces a significant rearrangement of bands and triggers a metal-to-insulator transition. Inclusion of SOC leads to a substantial anomalous Hall conductivity (AHC) around the Fermi energy reaching values up to sim -3e^2/h for X = Mn and Co in P3 symmetry and both in- and out-of-plane magnetization directions in the first case and along [001] in the latter. The dice lattice emerges as a promising playground to realise nontrivial topological phases with high Chern numbers.

Superconductivity, Charge Density Wave, and Supersolidity in Flat Bands with a Tunable Quantum Metric.

Predicting the fate of an interacting system in the limit where the electronic bandwidth is quenched is often highly nontrivial. The complex interplay between interactions and quantum fluctuations driven by the band geometry can drive competition between various ground states, such as charge density wave order and superconductivity. In this work, we study an electronic model of topologically trivial flat bands with a continuously tunable Fubini-Study metric in the presence of on-site attraction and nearest-neighbor repulsion, using numerically exact quantum MonteCarlo simulations. By varying the electron filling and the minimal spatial extent of the localized flat-band Wannier wave functions, we obtain a number of intertwined orders. These include a phase with coexisting charge density wave order and superconductivity, i.e., a supersolid. In spite of the nonperturbative nature of the problem, we identify an analytically tractable limit associated with a "small" spatial extent of the Wannier functions and derive a low-energy effective Hamiltonian that can well describe our numerical results. We also provide unambiguous evidence for the violation of any putative lower bound on the zero-temperature superfluid stiffness in geometrically nontrivial flat bands.

Topological Flatband Loop States in Fractal‐Like Photonic Lattices

AbstractNoncontractible loop states (NLSs) are a recently realized topological entity in flatband lattices, arising typically from the band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane‐touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, nontrivial NLSs and robust boundary modes in a system with such degeneracy are demonstrated. Based on a tailored photonic lattice constructed from the well‐known fractal Sierpinski gasket, the wavefunction singularities and the conditions for the existence of the NLSs are theoretically analyzed. It is shown that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real‐space topology. Experimentally, directly such flatband NLSs in a laser‐written Corbino‐shaped fractal‐like lattice are observed. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures, and band topology.

Structural Reconstruction Modulated Physical Properties of Titanium Oxide at the Monolayer Limit

To suppress surface dangling bonds, monolayer oxides derived from non-layered bulks usually undergo a pronounced structural reconstruction. It remains challenging to resolve these structural reconstructions and the induced distinct modulation of intrinsic properties. In this study, the structural reconstruction-modulated electronic, polaronic, and exitonic properties of a non-layered oxide at the monolayer limit are unraveled. Based on first-principles calculations and tight-binding simulations for a stable titanium dioxide (TiO2) monolayer, we show that its distinct surface Kagome sublattices host a topologically nontrivial flat band at the valence band edge. The strong electron–hole interaction in this monolayer oxide gives rise to a large exciton binding energy of around 2.49 eV. Interestingly, the monolayer TiO2 also exhibits strong electron–lattice coupling, which favors the formation of small electron polarons and thus greatly reduces its band gap energy into the visible light range. This work could be useful to understand the structural reconstruction-induced modulation of exotic physical and chemical properties for a broad range of non-layered oxides at the monolayer limit.

Observation of flat-band localization and topological edge states induced by effective strong interactions in electrical circuit networks

Flat-band topologies and localizations in noninteracting systems are extensively studied in different quantum and classical-wave systems. Recently, the exploration on the novel physics of flat-band localizations and topologies in interacting systems has aroused great interest. In particular, it is theoretically shown that the strong interaction could drive the formation of nontrivial topological flat bands; even dispersive trivial bands dominate the single-particle counterparts. However, the experimental observation of those interesting phenomena is still lacking. Here, we experimentally simulate the interaction-induced flat-band localizations and topological edge states in electrical circuit networks. We directly map the eigenstates of two correlated bosons in one-dimensional Aharonov-Bohm cages to modes of two-dimensional circuit lattices. In this case, by tuning the effective interaction strength through circuits' groundings, the two-boson flat bands and topological edge states are detected by measuring frequency-dependent impedance responses and voltage dynamics in the time domain. Our finding suggests a flexible platform to simulate the interaction-induced flat-band topology, and may possess potential applications in designing novel electronic devices.

Isospin polarized Chern insulator state of <i>C</i> = 4 in twisted double bilayer graphene

A flat band with nearly zero dispersion can be created by twisting the relative orientation of van der Waals materials, leading to a series of strongly correlated states, such as unconventional superconductivity, correlated insulating state, and orbital magnetism. The bandwidth and topological property of electronic band structure in a twisted double bilayer graphene are tunable by an external displacement field. This system can be an excellent quantum simulator to study the interplay between topological phase transition and strong electron correlation. Theoretical calculation shows that the <inline-formula><tex-math id="M4">\begin{document}$ {C}_{2x} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M4.png"/></alternatives></inline-formula> symmetry in twisted double bilayer graphene (TDBG) can be broken by an electric displacement field, leading the lowest conduction and valence band near charge neutrality to obtain a finite Chern number. The topological properties of the band and the symmetry breaking driven by the strong interaction make it possible to realize and regulate the old insulation state at low magnetic fields. Hence Chern insulator may emerge from this topological non-trivial flat band under strong electron interaction. Here, we observe Chern insulator state with Chern number 4 at filling factor <inline-formula><tex-math id="M5">\begin{document}$ \nu =1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M5.png"/></alternatives></inline-formula> under a small magnetic field on twisted double bilayer graphene with twist angle 1.48°. Moreover, the longitudinal resistance shows a peak under a parallel magnetic field and increases with temperature or field rising, which is similar to the Pomeranchuk effect in <sup>3</sup>He. This phenomenon indicates that Chern insulator at <inline-formula><tex-math id="M6">\begin{document}$ \nu =1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20230497_M6.png"/></alternatives></inline-formula> may originate from isospin polarization.

Novel electrical properties of moiré graphene systems

In this review, we discuss the electronic structures, topological properties, correlated states, nonlinear optical responses, as well as phonon and electron-phonon coupling effects of moiré graphene superlattices. First, we illustrate that topologically non-trivial flat bands and moiré orbital magnetism are ubiquitous in various twisted graphene systems. In particular, the topological flat bands of magic-angle twisted bilayer graphene can be explained from a zeroth pseudo-Landau-level picture, which can naturally explain the experimentally observed quantum anomalous Hall effect and some of the other correlated states. These topologically nontrivial flat bands may lead to nearly quantized piezoelectric response, which can be used to directly probe the valley Chern numbers in these moiré graphene systems. A simple and general chiral decomposition rule is reviewed and discussed, which can be used to predict the low-energy band dispersions of generic twisted multilayer graphene system and alternating twisted multilayer graphene system. This review further discusses nontrivial interaction effects of magic-angle TBG such as the correlated insulator states, density wave states, cascade transitions, and nematic states, and proposes nonlinear optical measurement as an experimental probe to distinguish the different “featureless” correlated states. The phonon properties and electron-phonon coupling effects are also briefly reviewed. The novel physics emerging from band-aligned graphene-insulator heterostructres is also discussed in this review. In the end, we make a summary and an outlook about the novel physical properties of moiré superlattices based on two-dimensional materials.

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.

Nonlinear Hall effects in strained twisted bilayer WSe2

Recently, it has been pointed out that the twisting of bilayer WSe2 would generate topologically non-trivial flat bands near the Fermi energy. In this work, we show that twisted bilayer WSe2 (tWSe2) with uniaxial strain exhibits a large nonlinear Hall (NLH) response due to the non-trivial Berry curvatures of the flat bands. Moreover, the NLH effect is greatly enhanced near the topological phase transition point which can be tuned by a vertical displacement field. Importantly, the nonlinear Hall signal changes sign across the topological phase transition point and provides a way to identify the topological phase transition and probe the topological properties of the flat bands. The strong enhancement and high tunability of the NLH effect near the topological phase transition point renders tWSe2 and related moire materials available platforms for rectification and second harmonic generations.

Non-Hermitian flat bands in rhombic microring resonator arrays.

We investigate the flat bands in a quasi-one-dimensional rhombic array composed of evanescently coupled microring resonators (MRRs) with non-Hermitian coupling. By changing the relative position of non-Hermitian coupling in each cell, we construct topologically trivial and nontrivial flat bands, where both the real and imaginary parts of energy bands become flat and coalesce into a single band. We show the nontrivial systems are able to support topological boundary modes isolated from the flat bulk bands although there is no band gap. The elusive topology of flat bands can be geometrically visualized by plotting the trajectories of their eigenvectors on Bloch sphere based on Majorana's stellar representation (MSR). Furthermore, we perform a full wave simulation and show the characteristics of flat bands, associated compact localized modes, and boundary modes are reflected from absorption spectra and field intensity profiles. The study may find potential applications in lasers, narrowband filters, and efficient light harvesting.

Floquet-engineered topological flat bands in irradiated twisted bilayer graphene

We propose a tunable optical setup to engineer topologically nontrivial flat bands in twisted bilayer graphene under circularly polarized light. Using both analytical and numerical calculations, we demonstrate that nearly flat bands can be engineered at small twist angles near the magic angles of the static system. The flatness and the gaps between these bands can be tuned optically by varying laser frequency and amplitude. We study the effects of interlayer hopping variations on Floquet flat bands and find that changes associated with lattice relaxation favor their formation. Furthermore, we find that, once formed, the flat bands carry nonzero Chern numbers. We show that at currently known values of parameters, such topological flat bands can be realized using circularly polarized UV laser light. Thus, our work opens the way to creating optically tunable, strongly correlated topological phases of electrons in moiré superlattices.Received 21 October 2019Revised 5 May 2020Accepted 2 November 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.043275Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChern insulatorsFractional quantum Hall effectPhysical SystemsFloquet systemsGrapheneTopological materialsCondensed Matter, Materials & Applied Physics

Building flat-band lattice models from Gram matrices

We propose a powerful and convenient method to systematically design flat-band lattice models, which overcomes the difficulties underlying the previous method. Especially, our method requires no elaborate calculations, applies to arbitrary spatial dimensions, and guarantees to result in a completely flat ground band. We use this method to generate several classes of lattice models, including models with both short- and long-range hoppings, both topologically trivial and non-trivial flat bands. Some of these models were previously known. Our method, however, provides crucial new insights. For example, we have reproduced and generalized the Kapit-Mueller model [Kapit and Mueller, Phys. Rev. Lett. \textbf{105}, 215303 (2010)] and demonstrated a universal scaling rule between the flat band degeneracy and the magnetic flux that was not noticed in previous studies. We show that the flat band of this model results from the (over-)completeness properties of coherent states.

Topological flat bands in twisted trilayer graphene

Twisted trilayer graphene (TLG) may be the simplest realistic system so far, which has flat bands with nontrivial topology. Here, we give a comprehensive calculation about its band structures and the band topology, i.e., valley Chern number of the nearly flat bands, with the continuum model. With realistic parameters, the magic angle of twisted TLG is about 1.12°, at which two nearly flat bands appears. Unlike the twisted bilayer graphene, a small twist angle can induce a tiny gap at all the Dirac points, which can be enlarged further by a perpendicular electric field. The valley Chern numbers of the two nearly flat bands in the twisted TLG depends on the twist angle θ and the perpendicular electric field E⊥. Considering its topological flat bands, the twisted TLG should be an ideal experimental platform to study the strongly correlated physics in topologically nontrivial flat band systems. And, due to its reduced symmetry, the correlated states in twisted TLG should be quite different from that in twisted bilayer graphene and twisted double bilayer graphene.

Novel phenomena in flatband photonic structures: from localized states to real-space topology

In recent years, flatband systems have aroused considerable interest in different branches of physics, from condensed-matter physics to engineered flatband structures such as in ultracold atoms, various metamaterials, electronic materials, and photonic waveguide arrays. Flatband localization, as an important phenomenon in solid state physics, is of broad interest in the exploration of many fundamental physics of many-body systems. We briefly review the recent experimental advances in light localization in engineered flatband lattices, with the emphasis on the optical induction technique of various photonic lattices and unconventional flatband states. The photonic lattices, established by various optical induction techniques, include quasi-one-dimensional diamond lattices and two-dimensional super-honeycomb, Lieb and Kagome lattices. Nontrivial flatband line states, independent of linear superpositions of conventional compact localized states, are demonstrated in photonic Lieb and super-honeycomb lattices, and they can be considered as an indirect illustration of the non-contractible loop states. Furthermore, we discuss alternative approaches to directly observing the non-contractible loop states in photonic Kagome lattices. These robust loop states are direct manifestation of real-space topology in such flatband systems. In this paper we do not intend to comprehensively account the vast flatband literature, but we briefly review the relevant work on photonic lattices mainly from our group. We hope that the mentioned concepts and techniques can be further explored and developed for subsequent applications in other structured photonic media such as photonic crystals, metamaterials, and other synthetic nanophotonic materials.

Negative flat band magnetism in a spin–orbit-coupled correlated kagome magnet

It has long been speculated that electronic flat band systems can be a fertile ground for hosting novel emergent phenomena including unconventional magnetism and superconductivity. Although flat bands are known to exist in a few systems such as heavy fermion materials and twisted bilayer graphene, their microscopic roles and underlying mechanisms in generating emergent behavior remain elusive. Here we use scanning tunneling microscopy to elucidate the atomically resolved electronic states and their magnetic response in the kagome magnet Co3Sn2S2. We observe a pronounced peak at the Fermi level, which is identified to arise from the kinetically frustrated kagome flat band. Increasing magnetic field up to +-8T, this state exhibits an anomalous magnetization-polarized Zeeman shift, dominated by an orbital moment in opposite to the field direction. Such negative magnetism can be understood as spin-orbit coupling induced quantum phase effects tied to non-trivial flat band systems. We image the flat band peak, resolve the associated negative magnetism, and provide its connection to the Berry curvature field, showing that Co3Sn2S2 is a rare example of kagome magnet where the low energy physics can be dominated by the spin-orbit coupled flat band. Our methodology of probing band-resolved ordering phenomena such as spin-orbit magnetism can also be applied in future experiments to elucidate other exotic phenomena including flat band superconductivity and anomalous quantum transport.