Topological Flat Bands Research Articles

Observation of Flat Frequency Bands at Open Edges and Antiphase Boundary Seams in Topological Mechanical Metamaterials

Motivated by the recent theoretical studies on a two-dimensional (2D) chiral Hamiltonian based on the Su-Schrieffer-Heeger chains [L. Zhu, E. Prodan, and K. H. Ahn, Phys. Rev. B 99, 041117(R) (2019)PRBMDO2469-995010.1103/PhysRevB.99.041117], we experimentally and computationally demonstrate that topological flat frequency bands can occur at open edges of 2D planar metamaterials and at antiphase boundary seams of ring-shaped or tubular metamaterials. Specifically, using mechanical systems made of magnetically coupled spinners, we reveal that the presence of the edge or seam bands that are flat in the entire projected reciprocal space follows the predictions based on topological winding numbers. The edge-to-edge distance sensitively controls the flatness of the edge bands and the localization of excitations, consistent with the theoretical analysis. The analog of the fractional charge state is observed. Possible realizations of flat bands in a large class of metamaterials, including photonic crystals and electronic metamaterials, are discussed.

Superconductivity, pseudogap, and phase separation in topological flat bands

We study a two-dimensional model of an isolated narrow topological band at partial filling with local attractive interactions. Numerically exact quantum Monte Carlo calculations show that the ground state is a superconductor with a critical temperature that scales nearly linearly with the interaction strength. We also find a broad pseudogap regime at temperatures above the superconducting phase that exhibits strong pairing fluctuations and a tendency towards electronic phase separation; introducing a small nearest neighbor attraction suppresses superconductivity entirely and results in phase separation. We discuss the possible relevance of superconductivity in this unusual regime to the physics of flat band moir\'{e} materials, and as a route to designing higher temperature superconductors.

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.

Quantum phase transitions in a ν=12 bosonic fractional Chern insulator

Interacting quantum many-body systems often generate exotic orders, either topological or conventional competing orders. Understanding these phases and the phase transitions between them are common issues in condensed-matter physics. How to characterize these states in a convincing and convenient way remains ambiguous. Here, we report our numerical study on the interacting $\ensuremath{\nu}=\frac{1}{2}$ hard-core boson system with topological flat bands based on the infinite density matrix group algorithm. We explicitly show that the entanglement spectral flow can be used to diagnose the quantum phases emerging in the hard-core boson systems. In particular, the subsequent charge pumping exhibits a linear, sinelike, and indiscernible dependence by the flux threading in the fractional Chern insulator, supersolid, and solid state, respectively. The real-space boson occupancy in the thermodynamic limit is further calculated for a better understanding of the respective state. We compare the phase diagram obtained here with that obtained by the previously exact diagonalization method, and qualitative differences are observed. Therefore, the infinite density matrix group algorithm provides a convincing and convenient way to study the phase transition in such a strongly correlated topological system.

Magnon magic angles and tunable Hall conductivity in 2D twisted ferromagnetic bilayers

Twistronics is currently one of the most active research fields in condensed matter physics, following the discovery of correlated insulating and superconducting phases in twisted bilayer graphene (tBLG). Here, we present a magnonic analogue of tBLG. We study magnons in twisted ferromagnetic bilayers (tFBL) with collinear magnetic order, including exchange and weak Dzyaloshinskii-Moriya interactions (DMI). For negligible DMI, tFBL presents discrete magnon magic angles and flat moiré minibands analogous to tBLG. The DMI, however, changes the picture and renders the system much more exotic. The DMI in tFBL induces a rich topological magnon band structure for any twist angle. The twist angle turns to a control knob for the magnon valley Hall and Nernst conductivities. Gapped flat bands appear in a continuum of magic angles in tFBL with DMI. In the lower limit of the continuum, the band structure reconstructs to form several topological flat bands. The luxury of twist-angle control over band gaps, topological properties, number of flat bands, and valley Hall and Nernst conductivities renders tFBL a novel device from fundamental and applied perspectives.

Moiré magnons in twisted bilayer magnets with collinear order

We explore the moir\'e magnon bands in twisted bilayer magnets with next-nearest neighboring Dzyaloshinskii-Moriya interactions, assuming that the out-of-plane collinear magnetic order is preserved under weak interlayer coupling. By calculating the magnonic band structures and the topological Chern numbers for four representative cases, we find that (i) the valley moir\'e bands are extremely flat over a wide range of continuous twist angles; (ii) the topological Chern numbers of the lowest few flat bands vary significantly with the twist angle; and (iii) the lowest few topological flat bands in bilayer antiferromagnets entail nontrivial thermal spin transport in the transverse direction; These properties make twisted bilayer magnets an ideal platform to study the magnonic counterparts of moir\'e electrons, where the statistical distinction between magnons and electrons leads to fundamentally new physical behavior.

Enhanced repulsively bound atom pairs in topological optical lattice ladders

There is a growing interest in using cold-atom systems to explore the effects of strong interactions in topological band structures. Here we investigate interacting bosons in a Cruetz ladder, which is characterised by topological flat energy bands where it has been proposed that interactions can lead to the formation of bound atomic pairs giving rise to pair superfluidity. By investigating realistic experimental implementations, we understand how the lattice topology enhances the properties of bound pairs giving rise to relatively large effective pair-tunnelling in these systems which can lead to robust pair superfluidity, and we find lattice supersolid phases involving only pairs. We identify schemes for preparation of these phases via time-dependent parameter variation and look at ways to detect and characterise these systems in a lattice. This work provides a starting point for investigating the interplay between the effects of topology, interactions and pairing in more general systems, with potential future connections to quantum simulation of topological materials.

Topological flat bands in frustrated kagome lattice CoSn

Electronic flat bands in momentum space, arising from strong localization of electrons in real space, are an ideal stage to realize strongly-correlated phenomena. Theoretically, the flat bands can naturally arise in certain geometrically frustrated lattices, often with nontrivial topology if combined with spin-orbit coupling. Here, we report the observation of topological flat bands in frustrated kagome metal CoSn, using angle-resolved photoemission spectroscopy and band structure calculations. Throughout the entire Brillouin zone, the bandwidth of the flat band is suppressed by an order of magnitude compared to the Dirac bands originating from the same orbitals. The frustration-driven nature of the flat band is directly confirmed by the chiral d-orbital texture of the corresponding real-space Wannier functions. Spin-orbit coupling opens a large gap of 80 meV at the quadratic touching point between the Dirac and flat bands, endowing a nonzero Z2 invariant to the flat band. These findings demonstrate that kagome-derived flat bands are a promising platform for novel emergent phases of matter at the confluence of strong correlation and topology.

Excitonic Laughlin states in ideal topological insulator flat bands and their possible presence in moiré superlattice materials

We investigate few- and many-body states in half-filled maximally symmetric topological insulator flat bands realized by two degenerate Landau levels which experience opposite magnetic fields. This serves as a toy model of flat bands in moir\'e materials in which valleys have Chern numbers $C= \pm 1$. We argue that although the spontaneously polarized Ising Chern magnet is a natural ground state for repulsive Coulomb interactions, it can be in reasonable energetic competition with correlated states which can be viewed as Laughlin states of excitons when short distance corrections to the interaction are included. This is because charge neutral excitons in these bands behave effectively as charged particles in ordinary Landau levels. In particular, the Ising Chern magnet is no longer the ground state once the strength of a short range intra-valley repulsion is about $30\%$ larger than the inter-valley repulsion. Remarkably, these excitonic Laughlin states feature valley number fractionalization but no charge fractionalization and a quantized charge Hall conductivity identical to the Ising magnet, $\sigma_{xy}= \pm e^2/h$, and thus cannot be distinguished from it by ordinary charge transport measurements. The most compact excitonic Laughlin state that can be constructed in these bands is an analogue of $\nu=1/4$ bosonic Laughlin state and has no valley polarization even though it spontaneously breaks time reversal symmetry with a charge Hall conductivity $\sigma_{xy}= \pm e^2/h$.

Topological flat bands and correlated states in twisted monolayer-bilayer graphene

Monolayer graphene placed with a twist on top of AB-stacked bilayer graphene hosts topological flat bands in a wide range of twist angles. The dispersion of these bands and gaps between them can be efficiently controlled by a perpendicular electric field, which induces topological transitions accompanied by changes of the Chern numbers. In the regime where the applied electric field induces gaps between the flat bands, we find a relatively uniform distribution of the Berry curvature. Consequently, interaction-induced valley- and/or spin-polarized states at integer filling factors are energetically favorable. In particular, we predict a quantum anomalous Hall state at filling factor $\nu=1$ for a range of twist angles $1^\circ<\theta <1.4^\circ$. Furthermore, to characterize the response of the system to magnetic field, we computed the Hofstadter butterfly and the Wannier plot, which can be used to probe the dispersion and topology of the flat bands in this material.

Gate-tunable topological flat bands in twisted monolayer-bilayer graphene

We investigate the band structure of twisted monolayer-bilayer graphene (tMBG), or twisted graphene on bilayer graphene (tGBG), as a function of twist angles and perpendicular electric fields in search of optimum conditions for achieving isolated nearly flat bands. Narrow bandwidths comparable or smaller than the effective Coulomb energies satisfying $U_{\textrm{eff}} /W \gtrsim 1$ are expected for twist angles in the range of $0.3^{\circ} \sim 1.5^{\circ}$, more specifically in islands around $\theta \sim 0.5^{\circ}, \, 0.85^{\circ}, \,1.3^{\circ}$ for appropriate perpendicular electric field magnitudes and directions. The valley Chern numbers of the electron-hole asymmetric bands depend intrinsically on the details of the hopping terms in the bilayer graphene, and extrinsically on factors like electric fields or average staggered potentials in the graphene layer aligned with the contacting hexagonal boron nitride substrate. This tunability of the band isolation, bandwidth, and valley Chern numbers makes of tMBG a more versatile system than twisted bilayer graphene for finding nearly flat bands prone to strong correlations.

Fractional quantum Hall states for moiré superstructures in the Hofstadter regime

We study the transition of $\nu=1/3$ and $2/5$ fractional quantum Hall states of the honeycomb Hofstadter model as we tune to a two-orbital moir\'e superlattice Hamiltonian, motivated by the flat bands of twisted bilayer graphene in a perpendicular magnetic field. In doing so, we address the extent to which these states survive in moir\'e systems and analyze the nature of the transition. Through the use of a Peierls substitution, we determine the Landau-level splitting for the moir\'e Hamiltonian, and study the structure of the Chern bands for a range of magnetic flux per plaquette. We identify topological flat bands in the spectrum at low energies, with numerically tractable lattice geometries that can support the fractional quantum Hall effect. As we tune the model, we find that the orbital-polarized $\nu=1/3$ and $2/5$ states corresponding to the honeycomb Hofstadter model survive up to $\approx 30\%$ of typical moir\'e superlattice parameters, beyond which they transition into an insulating phase. We present evidence for this through density matrix renormalization group calculations on an infinite cylinder, by verifying the charge pumping, spectral flow, entanglement scaling, and conformal field theory edge counting. We conclude that fractional quantum Hall states from the Hofstadter model can persist up to hopping amplitudes of the same order as those typical for moir\'e superlattice Hamiltonians, which implies generally that fractional states for moir\'e superstructures can be discerned simply by analyzing the dominant terms in their effective Hamiltonians.

Quantum Hall effects of exciton condensate in topological flat bands

Tunable exciton condensates in two dimensional electron gas systems under strong magnetic field exhibits anomalous Hall transport owing to mutual Coulomb coupling, and have attracted a lot of research activity. Here, we explore another framework using topological flat band models in the absence of Landau levels, for realizing the many-body exciton phases of two-component fermions under strong intercomponent interactions. By developing new diagnosis based on the state-of-the-art density-matrix renormalization group and exact diagonalization, we show the theoretical discovery of the emergence of Halperin (111) quantum Hall effect at a total filling factor $\nu=1$ in the lowest Chern band under strong Hubbard repulsion, which is classified by the unique ground state with bulk charge insulation and spin superfluidity, The topological nature is further characterized by one edge branch of chiral propagating Luttinger modes with level counting $1,1,2,3,5,7$ in consistent with the conformal field theory description. Moreover, with nearest-neighbor repulsions, we propose the Halperin (333) fractional quantum Hall effect at a total filling factor $\nu=1/3$ in the lowest Chern band.

Non-Abelian fractional Chern insulator in disk geometry

Non-Abelian (NA) fractional topological states with quasi-particles obeying NA braiding statistics have attracted intensive attentions for both its fundamental nature and the prospect for topological quantum computation. To date, there are many models proposed to realize the NA fractional topological states, such as the well-known Moore-Read quantum Hall states and the Non-Abelian fractional Chern insulators (NA-FCIs). Here, we investigate the NA-FCI in disk geometry with three-body hard-core bosons loaded into a topological flat band. This stable $\nu = 1$ bosonic NA-FCI is characterized by the edge excitations and the ground-state angular momentum. Based on the generalized Pauli principle and the Jack polynomials, we successfully construct a trial wave function for the NA-FCI. Moreover, a $\nu = 1/2$ Abelian FCI state emerges with the increase of the on-site interaction and it can be identified with the help of the trial wave function as well. Our findings not only lead to an optimal wave function for the NA-FCI, but also directly provide an effective approach for future researches on paired topological states.

Magnetic properties on Chern insulator of checkerboard lattice with topological flat band

By means of the mean-field method and the random phase approximation, we study the magnetic properties of the correlated Chern insulator on a checkerboard lattice with topological flat band. The antiferromagnetic (AF) order is found to be more stable than the ferromagnetic (FM) order at half filling. While at quarter filling, the system becomes a FM-Chern insulator induced by the FM order. The phase diagram is more complex for other fillings. FM order is more stable than AF order for small doping due to the flatness of band structure, while FM and AF orders compete at large doping.

Topological models in rotationally symmetric quasicrystals

We investigate the physics of quasicrystalline models in the presence of a uniform magnetic field, focusing on the presence and construction of topological states. This is done by using the Hofstadter model but with the sites and couplings denoted by the vertex model of the quasicrystal, giving the Hofstadter vertex model. We specifically consider two-dimensional quasicrystals made from tilings of two tiles with incommensurate areas, focusing on the five-fold Penrose and the eight-fold Ammann-Beenker tilings. This introduces two competing scales; the uniform magnetic field and the incommensurate scale of the cells of the tiling. Due to these competing scales the periodicity of the Hofstadter butterfly is destroyed. We observe the presence of topological edge states on the boundary of the system via the Bott index that exhibit two way transport along the edge. For the eight-fold tiling we also observe internal edge-like states with non-zero Bott index, which exhibit two way transport along this internal edge. The presence of these internal edge states is a new characteristic of quasicrystalline models in magnetic fields. We then move on to considering interacting systems. This is challenging, in part because exact diagonalization on a few tens of sites is not expected to be enough to accurately capture the physics of the quasicrystalline system, and in part because it is not clear how to construct topological flatbands having a large number of states. We show that these problems can be circumvented by building the models analytically, and in this way we construct models with Laughlin type fractional quantum Hall ground states.

Topological flat bands without magic angles in massive twisted bilayer graphenes

Twisted bilayer graphene (TBG) hosts nearly flat bands with narrow bandwidths of a few meV at certain {\em magic} twist angles. Here we show that in twisted gapped Dirac material bilayers, or massive twisted bilayer graphenes (MTBG), isolated nearly flat bands below a threshold bandwidth $W_c$ are expected for continuous small twist angles up to a critical $\theta_c$ depending on the flatness of the original bands and the interlayer coupling strength. Narrow bandwidths of $W \lesssim $30 meV are expected for $\theta \lesssim 3^{\circ} $ for twisted Dirac materials with intrinsic gaps of $\sim 2$ eV that finds realization in monolayers of gapped transition metal dichalcogenides (TMDC), silicon carbide (SiC) among others, and even narrower bandwidths in hexagonal boron nitride (BN) whose gaps are $\sim 5$ eV, while twisted graphene systems with smaller gaps of a few tens of meV, e.g. due to alignment with hexagonal boron nitride, show vestiges of the magic angles behavior in the bandwidth evolution. The phase diagram of finite valley Chern numbers of the isolated moire bands expands with increasing difference between the sublattice selective interlayer tunneling parameters. The valley contrasting circular dichroism for interband optical transitions is constructive near $0^{\circ}$ and destructive near $60^{\circ}$ alignments and can be tuned through electric field and gate driven polarization of the mini-valleys. Combining massive Dirac materials with various intrinsic gaps, Fermi velocities, interlayer tunneling strengths suggests optimistic prospects of increasing $\theta_c$ and achieving correlated states with large $U/W$ effective interaction versus bandwidth ratios.

Photonic materials in circuit quantum electrodynamics

Photonic synthetic materials provide an opportunity to explore the role of microscopic quantum phenomena in determining macroscopic material properties. There are, however, fundamental obstacles to overcome — in vacuum, photons not only lack mass, but also do not naturally interact with one another. Here, we review how the superconducting quantum circuit platform has been harnessed in the last decade to make some of the first materials from light. We describe the structures that are used to imbue individual microwave photons with matter-like properties such as mass, the nonlinear elements that mediate interactions between these photons, and quantum dynamic/thermodynamic approaches that can be used to assemble and stabilize strongly correlated states of many photons. We then describe state-of-the-art techniques to generate synthetic magnetic fields, engineer topological and non-topological flat bands and explore the physics of quantum materials in non-Euclidean geometries — directions that we view as some of the most exciting for this burgeoning field. Finally, we discuss upcoming prospects, and in particular opportunities to probe novel aspects of quantum thermalization and detect quasi-particles with exotic anyonic statistics, as well as potential applications in quantum information science. This Review Article surveys the physics of many-body quantum states formed by microwave photons in circuit quantum electrodynamics environments.

Exact topological flat bands from continuum Landau levels

We construct and characterize tight binding Hamiltonians which contain a completely flat topological band made of continuum lowest Landau level wavefunctions sampled on a lattice. We find an infinite family of such Hamiltonians, with simple analytic descriptions. These provide a valuable tool for constructing exactly solvable models. We also implement a numerical algorithm for finding the most local Hamiltonian with a flat Landau level. We find intriguing structures in the spatial dependence of the matrix elements for this optimized model. The models we construct serve as foundations for numerical and experimental studies of topological systems, both non-interacting and interacting.

Topological properties and orbital magnetism in twisted graphene systems

We review and discuss the electronic structures, topological properties and orbital magnetism in twisted bilayer (TBG) and multilayer graphene systems. Moiré pattern is formed in twisted bilayer graphene due to the mutual twist of the two graphene layers. The moiré potential induced by the twist can generate opposite pseudo magnetic fields in the Moiré supercell, which are coupled with the Dirac fermions and generate two sets of pseudo Landau levels with opposite Chern numbers &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$\pm1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. The two flat bands for each valley each spin of TBG are equivalent to the two zeroth pseudo Landau levels with opposite Chern numbers and opposite sublattice polarizations. Such a pseudo-Landau-level representation has significant implications on the quantum anomalous Hall states observed at integer fillings of the flat bands in TBG at the magic angle. The origin of the magic angle can also be naturally explained by using the pseudo-Landau-level picture. We further discuss twisted multilayer graphene systems, and show that topological flat bands generally exist in the twisted multilayer graphene systems. These topological flat bands have nonzero valley Chern numbers, which can be described by a succinct formula under certain approxmations. These topological flat bands in twisted bilayer and multilayer graphene systems are associated with orbital magnetism. A valley polarized state in the twist graphene system is an orbital magnetic state with nontrivial current-loop pattern in the moiré supercell. The experimentally observed correlated insulating states at &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$\pm 1/2$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; fillings and at charge neutrality point of magic-angle TBG can be valley polarized states, which are associated with compensating current loops and induce staggered orbital magnetizations on the moiré length scale. If &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$C_{2z}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20200506_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; symmetry is broken due to the alignment of hexagonal boron nitride substrate, then a valley-polarized ground state would be a moiré orbital ferromagnetic state, which exhibits not only (quantum) anomalous Hall effect, but also novel magneto-optical and nonlinear optical responses.