Non-Abelian gauge potentials in graphene bilayers

Nonlinear optics of graphitic carbon allotropes: from 0D to 3D

The dimensionality of materials fundamentally influences their electronic and optical properties, presenting a complex interplay with nonlinear optical (NLO) characteristics that remains largely unexplored. In this review, we focus on the influence of dimensionality on the NLO properties of graphitic allotropes, ranging from 0D fullerenes, 1D carbon nanotubes, and 2D graphene, to 3D graphite, all of which share a consistent sp2 hybridized chemical bonding structure. We examine the distinct physical and NLO properties across these dimensions, underscoring the profound impact of dimensionality. Notably, dimension-specific physical phenomena, such as Luttinger liquid in 1D and Landau quantization in 2D, play a significant role in shaping NLO phenomena. Finally, we explore the promising potential of NLO properties in systems with mixed dimensionalities, setting the stage for future breakthroughs and innovative applications.

Supersymmetry dictated topology in periodic gauge fields and realization in strained and twisted 2D materials

Supersymmetry (SUSY) of a Hamiltonian dictates double degeneracy between a pair of superpartners (SPs) transformed by supercharge, except at zero energy where modes remain unpaired in many cases. Here we explore a SUSY of complete isospectrum between SPs-with paired zero modes-realized by 2D electrons in zero-flux periodic gauge fields, which can describe twisted or periodically strained 2D materials. We find their low-energy sector containing zero (or threshold) modes must be topologically non-trivial, by proving that Chern numbers of the two SPs have a finite difference dictated by the number of zero modes and energy dispersion in their vicinity. In 30° twisted bilayer (double bilayer) transition metal dichalcogenides subject to periodic strain, we find one SP is topologically trivial in its lowest miniband, while the twin SP of identical dispersion has a Chern number of 1 (2), in stark contrast to time-reversal partners that have to be simultaneously trivial or nontrivial. For systems whose physical Hamiltonian corresponds to the square root of a SUSY Hamiltonian, such as twisted or strained bilayer graphene, we reveal that topological properties of the two SUSY SPs are transferred respectively to the conduction and valence bands, including the contrasted topology in the low-energy sector and identical topology in the high-energy sector. This offers a unified perspective for understanding topological properties in many flat-band systems described by such square-root models. Both types of SUSY systems provide unique opportunities for exploring correlated and topological phases of matter.

Dynamic Acoustic Beamshaping with Coupling-Immune Moiré Metasurfaces

Moiré effects arising from mutually twisted metasurfaces have showcased remarkable wave manipulation capabilities, unveiling tantalizing emerging phenomena such as acoustic moiré flat bands and topological phase transitions. However, the pursuit of strong near-field coupling in layers has necessitated acoustic moiré metasurfaces to be tightly stacked at narrow distances in the subwavelength range. Here, moiré effects beyond near-field interlayer coupling in acoustics are reported and the concept of coupling-immune moiré metasurfaces is proposed. Remote acoustic moiré effects decoupled from the interlayer distance are theoretically, numerically, and experimentally demonstrated. Tunable out-of-plane acoustic beam scanning is successfully achieved by dynamically controlling twist angles. The engineered coupling-immune properties are further extended to multilayered acoustic moiré metasurfaces and manipulation of acoustic vortices. Good robustness against external disturbances is also observed for the fabricated coupling-immune acoustic moiré metasurfaces. The presented work unlocks the potential of twisted moiré devices for out-of-plane acoustic beam shaping, enabling practical applications in remote dynamic detection, and multiplexed underwater acoustic communication.

Chirality Probe of Twisted Bilayer Graphene in the Linear Transport Regime

We propose minimal transport experiments in the coherent regime that can probe the chirality of twisted moiré structures. We show that only with a third contact and in the presence of an in-plane magnetic field (or another time-reversal symmetry breaking effect) a chiral system may display nonreciprocal transport in the linear regime. We then propose to use the third lead as a voltage probe and show that opposite enantiomers give rise to different voltage drops on the third lead. Additionally, in the scenario of layer-discriminating contacts, the third lead can serve as a current probe capable of detecting different handedness even in the absence of a magnetic field. In a complementary configuration, applying opposite voltages on the two layers of the third lead gives rise to a chiral (super)current in the absence of a source-drain voltage whose direction is determined by its chirality.

Twisted photonic Weyl meta-crystals and aperiodic Fermi arc scattering

As a milestone in the exploration of topological physics, Fermi arcs bridging Weyl points have been extensively studied. Weyl points, as are Fermi arcs, are believed to be only stable when preserving translation symmetry. However, no experimental observation of aperiodic Fermi arcs has been reported so far. Here, we continuously twist a bi-block Weyl meta-crystal and experimentally observe the twisted Fermi arc reconstruction. Although both the Weyl meta-crystals individually preserve translational symmetry, continuous twisting operation leads to the aperiodic hybridization and scattering of Fermi arcs on the interface, which is found to be determined by the singular total reflection around Weyl points. Our work unveils the aperiodic scattering of Fermi arcs and opens the door to continuously manipulating Fermi arcs.

Strain-Induced Quasi-1D Channels in Twisted Moiré Lattices

We study the effects of strain in moiré systems composed of honeycomb lattices. We elucidate the formation of almost perfect one-dimensional moiré patterns in twisted bilayer systems. The formation of such patterns is a consequence of an interplay between twist and strain which gives rise to a collapse of the reciprocal space unit cell. As a criterion for such collapse we find a simple relation between the two quantities and the material specific Poisson ratio. The induced one-dimensional behavior is characterized by two, usually incommensurate, periodicities. Our results offer explanations for the complex patterns of one-dimensional channels observed in low angle twisted bilayer graphene systems and twisted bilayer dicalcogenides. Our findings can be applied to any hexagonal twisted moiré pattern and can be easily extended to other geometries.

Synergistic correlated states and nontrivial topology in coupled graphene-insulator heterostructures

Graphene has aroused great attention due to the intriguing properties associated with its low-energy Dirac Hamiltonian. When graphene is coupled with a correlated insulating substrate, electronic states that cannot be revealed in either individual layer may emerge in a synergistic manner. Here, we theoretically study the correlated and topological states in Coulomb-coupled and gate-tunable graphene-insulator heterostructures. By electrostatically aligning the electronic bands, charge carriers transferred between graphene and the insulator can yield a long-wavelength electronic crystal at the interface, exerting a superlattice Coulomb potential on graphene and generating topologically nontrivial subbands. This coupling can further boost electron-electron interaction effects in graphene, leading to a spontaneous bandgap formation at the Dirac point and interaction-enhanced Fermi velocity. Reciprocally, the electronic crystal at the interface is substantially stabilized with the help of cooperative interlayer Coulomb coupling. We propose a number of substrate candidates for graphene to experimentally demonstrate these effects.

Untwisting Moiré Physics: Almost Ideal Bands and Fractional Chern Insulators in Periodically Strained Monolayer Graphene

Moiré systems have emerged in recent years as a rich platform to study strong correlations. Here, we will propose a simple, experimentally feasible setup based on periodically strained graphene that reproduces several key aspects of twisted moiré heterostructures-but without introducing a twist. We consider a monolayer graphene sheet subject to a C_{2}-breaking periodic strain-induced pseudomagnetic field with period L_{M}≫a, along with a scalar potential of the same period. This system has almost ideal flat bands with valley-resolved Chern number ±1, where the deviation from ideal band geometry is analytically controlled and exponentially small in the dimensionless ratio (L_{M}/l_{B})^{2}, where l_{B} is the magnetic length corresponding to the maximum value of the pseudomagnetic field. Moreover, the scalar potential can tune the bandwidth far below the Coulomb scale, making this a very promising platform for strongly interacting topological phases. Using a combination of strong-coupling theory and self-consistent Hartree-Fock, we find quantum anomalous Hall states at integer fillings. At fractional filling, exact diagonaliztion reveals a fractional Chern insulator at parameters in the experimentally feasible range. Overall, we find that this system has larger interaction-induced gaps, smaller quasiparticle dispersion, and enhanced tunability compared to twisted graphene systems, even in their ideal limit.

Unconventional Flat Chern Bands and 2e Charges in Skyrmionic Moiré Superlattices

The interplay of topological characteristics in real space and reciprocal space can lead to the emergence of unconventional topological phases. In this Letter, we implement a novel mechanism for generating higher-Chern flat bands on the basis of twisted bilayer graphene (TBG) coupled to topological magnetic structures in the form of the skyrmion lattice. In particular, we discover a scenario for generating |C| = 2 dispersionless electronic bands when the skyrmion periodicity and the moiré periodicity match. Following the Wilczek argument, the statistics of the charge-carrying excitations in this case is bosonic, characterized by electronic charge Q = 2e, which is even in units of electron charge e. The skyrmion coupling strength triggering the topological phase transition is realistic, with its lower bound estimated as 4 meV. The Hofstadter butterfly spectrum results in an unexpected quantum Hall conductance sequence ±2e2h,±4e2h,... for TBG with the skyrmion order.

Chiral Decomposition of Twisted Graphene Multilayers with Arbitrary Stacking

We formulate the chiral decomposition rules that govern the electronic structure of a broad family of twisted N + M multilayer graphene configurations that combine arbitrary stacking order and a mutual twist. We show that at the magic angle in the chiral limit the low-energy bands of such systems are composed of chiral pseudospin doublets that are energetically entangled with two flat bands per valley induced by the moiré superlattice potential. The analytic construction is supported by explicit numerical calculations based on realistic parametrization. We further show that vertical displacement fields can open energy gaps between the pseudospin doublets and the two flat bands, such that the flat bands may carry nonzero valley Chern numbers. These results provide guidelines for the rational design of topological and correlated states in generic twisted graphene multilayers.

Optimal twist angle for a graphene-like bilayer

The first optimal-or 'magic'-angle leading to the nullity of the Dirac/Fermi velocity for twisted bilayer graphene is re-evaluated in the Bistritzer-MacDonald set-up (Bistritzer and MacDonald 2011Proc. Natl Acad. Sci.10812233-7). From the details of that calculation we study the resulting alterations when the properties of the two layers are not exactly the same. A moiré combination of lattices without relative rotation but with different spacing lengths may also lead to a vanishing Dirac velocity. Hopping amplitudes can vary as well, and curvature is one of the possible causes for their change. In the case of small curvature values and situations dominated by hopping energy scales, the optimal angle becomes wider than in the 'flat' case.

Local atomic-morphology-resolved edge states in twisted bilayer graphene nanoribbons

We study the properties of edge states for a selected (10,1)[(4,3)] twisted bilayer graphene (TBG) nanoribbon with minimal edges but a majority of zigzag edges. By using the tight-binding and Green's function methods, we find a remarkable rule of a local electronic transfer for the edge states. As the energy away from the Fermi level, the transfer is in the order of convex AB-, concave AB-, concave AA- and convex AA-stacked regions of the ribbon curve edges. We illustrate that this rule comes from the difference in interlayer couplings among the four types of local geometries at edges. Further, an in-plane transverse electric field can rearrange the edge bands and enlarge the energy regimes, leading to the lowest energy states modified from AB-stacked edge states to AA-stacked ones. The realignment of the edge bands results from the interplay between the interlayer coupling and the potential difference induced by the transverse electric field, which results in different bonding and antibonding edge states, i.e. the edge bands. In contrast, the total energy regime of the edge bands remain nearly unchanged under a relative strong off-plane perpendicular electric field, and the typical AA-stacked edge states are still maintained even the rotational symmetry of two layers is broken. Until a sufficiently strong value, the TBG nanoribbon tends to behave as two noninteracting monolayer ribbons except for a band distortion in low-energy regime. The conductance spectra reflects the edge bands well. We also discussed the influence of edge defects in the TBG nanoribbon on transport properties. It is found that the contributed conductance of each type of edge states shows different degrees of suppression for a monatomic vacancy in the corresponding region of edges.

Moiré metasurfaces for dynamic beamforming

Recent advances in digitally programmable metamaterials have accelerated the development of reconfigurable intelligent surfaces (RIS). However, the excessive use of active components (e.g., pin diodes and varactor diodes) leads to high costs, especially for those operating at millimeter-wave frequencies, impeding their large-scale deployments in RIS. Here, we introduce an entirely different approach-moiré metasurfaces-to implement dynamic beamforming through mutual twists of two closely stacked metasurfaces. The superposition of two high-spatial-frequency patterns produces a low-spatial-frequency moiré pattern through the moiré effect, which provides the surface impedance profiles to generate desired radiation patterns. We demonstrate experimentally that the direction of the radiated beams can continuously sweep over the entire reflection space along predesigned trajectories by simply adjusting the twist angle and the overall orientation. Our work opens previously unexplored directions for synthesizing far-field scattering through the direct contact of mutually twisted metallic patterns with different plane symmetry groups.

Tunable Sample-Wide Electronic Kagome Lattice in Low-Angle Twisted Bilayer Graphene

Overlaying two graphene layers with a small twist angle θ can create a moiré superlattice to realize exotic phenomena that are entirely absent in a graphene monolayer. A representative example is the predicted formation of localized pseudo-Landau levels (PLLs) with kagome lattice in tiny-angle twisted bilayer graphene (TBG) with θ<0.3° when the graphene layers are subjected to different electrostatic potentials. However, this was shown only for the model of rigidly rotated TBG, which is not realized in reality due to an interfacial structural reconstruction. It is believed that the interfacial structural reconstruction strongly inhibits the formation of the PLLs. Here, we systematically study electronic properties of the TBG with 0.075°≤θ<1.2° and demonstrate, unexpectedly, that the PLLs are quite robust for all the studied TBG. The structural reconstruction suppresses the formation of the emergent kagome lattice in the tiny-angle TBG. However, for the TBG around the magic angle, the sample-wide electronic kagome lattices with tunable lattice constants are directly imaged by using a scanning tunneling microscope. Our observations open a new direction to explore exotic correlated phases in moiré systems.

Intrinsic superflat bands in general twisted bilayer systems

Twisted bilayer systems with discrete magic angles, such as twisted bilayer graphene featuring moiré superlattices, provide a versatile platform for exploring novel physical properties. Here, we discover a class of superflat bands in general twisted bilayer systems beyond the low-energy physics of magic-angle twisted counterparts. By considering continuous lattice dislocation, we obtain intrinsic localized states, which are spectrally isolated at lowest and highest energies and spatially centered around the AA stacked region, governed by the macroscopic effective energy potential well. Such localized states exhibit negligible inter-cell coupling and support the formation of superflat bands in a wide and continuous parameter space, which can be mimicked using a twisted bilayer nanophotonic system. Our finding suggests that general twisted bilayer systems can realize continuously tunable superflat bands and the corresponding localized states for various photonic, phononic, and mechanical waves.

Moiré-Driven Topological Transitions and Extreme Anisotropy in Elastic Metasurfaces

The twist angle between a pair of stacked 2D materials has been recently shown to control remarkable phenomena, including the emergence of flat-band superconductivity in twisted graphene bilayers, of higher-order topological phases in twisted moiré superlattices, and of topological polaritons in twisted hyperbolic metasurfaces. These discoveries, at the foundations of the emergent field of twistronics, have so far been mostly limited to explorations in atomically thin condensed matter and photonic systems, with limitations on the degree of control over geometry and twist angle, and inherent challenges in the fabrication of carefully engineered stacked multilayers. Here, this work extends twistronics to widely reconfigurable macroscopic elastic metasurfaces consisting of LEGO pillar resonators. This work demonstrates highly tailored anisotropy over a single-layer metasurface driven by variations in the twist angle between a pair of interleaved spatially modulated pillar lattices. The resulting quasi-periodic moiré patterns support topological transitions in the isofrequency contours, leading to strong tunability of highly directional waves. The findings illustrate how the rich phenomena enabled by twistronics and moiré physics can be translated over a single-layer metasurface platform, introducing a practical route toward the observation of extreme phenomena in a variety of wave systems, potentially applicable to both quantum and classical settings without multilayered fabrication requirements.

Reproducibility in the fabrication and physics of moiré materials

Overlaying two atomic layers with a slight lattice mismatch or at a small rotation angle creates a moiré superlattice, which has properties that are markedly modified from (and at times entirely absent in) the 'parent' materials. Such moiré materials have progressed the study and engineering of strongly correlated phenomena and topological systems in reduced dimensions. The fundamental understanding of the electronic phases, such as superconductivity, requires a precise control of the challenging fabrication process, involving the rotational alignment of two atomically thin layers with an angular precision below 0.1 degrees. Here we review the essential properties of moiré materials and discuss their fabrication and physics from a reproducibility perspective.

Weyl Semimetal Path to Valley Filtering in Graphene

We propose a device in which a sheet of graphene is coupled to a Weyl semimetal, allowing for the physical access to the study of tunneling from two- to three-dimensional massless Dirac fermions. Because of the reconstructed band structure, we find that this device acts as a robust valley filter for electrons in the graphene sheet. We show that, by appropriate alignment, the Weyl semimetal draws away current in one of the two graphene valleys, while allowing current in the other to pass unimpeded. In contrast to other proposed valley filters, the mechanism of our proposed device occurs in the bulk of the graphene sheet, obviating the need for carefully shaped edges or dimensions.

Moiré-Potential-Induced Band Structure Engineering in Graphene and Silicene

A moiré pattern results from the projection of one periodic pattern to another with relative lattice constant or misalignment and provides great periodic potential to modify the electronic properties of pristine materials. In this Review, recent research on the effect of the moiré superlattice on the electronic structures of graphene and silicene, both of which possess a honeycomb lattice, is focused on. The moiré periodic potential is introduced by the interlayer interaction to realize abundant phenomena, including new generation of Dirac cones, emergence of Van Hove singularities (vHs) at the cross point of two sets of Dirac cones, Mott-like insulating behavior at half-filling state, unconventional superconductivity, and electronic Kagome lattice and flat band with nontrivial edge state. The role of interlayer coupling strength, which is determined by twist angle and buckling degree, in these exotic properties is discussed in terms of both the theoretical prediction and experimental measurement, and finally, the challenges and outlook for this field are discussed.

WKB Estimate of Bilayer Graphene's Magic Twist Angles

Graphene bilayers exhibit zero-energy flatbands at a discrete series of magic twist angles. In the absence of intrasublattice interlayer hopping, zero-energy states satisfy a Dirac equation with a non-Abelian SU(2) gauge potential that cannot be diagonalized globally. We develop a semiclassical WKB approximation scheme for this Dirac equation by introducing a dimensionless Planck's constant proportional to the twist angle, solving the linearized Dirac equation around AB and BA turning points, and connecting Airy function solutions via bulk WKB wave functions. We find zero-energy solutions at a discrete set of values of the dimensionless Planck's constant, which we obtain analytically. Our analytic flatband twist angles correspond closely to those determined numerically in previous work.

Layer Pseudospin Dynamics and Genuine Non-Abelian Berry Phase in Inhomogeneously Strained Moiré Pattern

Periodicity of long wavelength moiré patterns is very often destroyed by the inhomogeneous strain introduced in fabrications of van der Waals layered structures. We present a framework to describe massive Dirac fermions in such distorted moiré pattern of transition metal dichalcogenides homobilayers, accounting for the dynamics of layer pseudospin. In decoupled bilayers, we show two causes of in-plane layer pseudospin precession: By the coupling of layer antisymmetric strain to valley magnetic moment; and by the Aharonov-Bohm effect in the SU(2) gauge potential for the case of R-type bilayer under antisymmetric strain and H-type under symmetric strain. With interlayer coupling in the moiré, its interplay with strain manifests as a non-Abelian gauge field. We show a genuine non-Abelian Aharonov-Bohm effect in such field, where the evolution operators for different loops are noncommutative. This provides an exciting platform to explore non-Abelian gauge field effects on electron, with remarkable tunability of the field by strain and interlayer bias.

Flat Bands in Magic-Angle Vibrating Plates

Twisted bilayer graphene develops quasiflat bands at specific "magic" interlayer rotation angles through an unconventional mechanism connected to carrier chirality. Quasiflat bands are responsible for a wealth of exotic, correlated-electron phases in the system. In this Letter, we propose a mechanical analog of twisted bilayer graphene made of two vibrating plates patterned with a honeycomb mesh of masses and coupled across a continuum elastic medium. We show that flexural waves in the device exhibit vanishing group velocity and quasiflat bands at magic angles in close correspondence with electrons in graphene models. The strong similarities of spectral structure and spatial eigenmodes in the two systems demonstrate the chiral nature of the mechanical flat bands. We derive analytical expressions that quantitatively connect the mechanical and electronic models, which allow us to predict the parameters required for an experimental realization of our proposal.

Moiré Fringe Induced Gauge Field in Photonics

We realize moiré fringe induced gauge field in a double-layer photonic honeycomb metacrystal with mismatched lattice constants. Benefitting from the generated strong effective gauge field, we report direct measurement of the band diagrams of both Landau level flat bands and intermagnetic-domain edge states. Importantly, we observe the correlation between the momentum and orbital position of the Landau modes, serving as an evidence of the noncommuteness between orthogonal components of the momentum. Without complicated time driving mechanics and careful site-by-site engineering, moiré superlattices could emerge as a powerful means to generate effective gauge fields for photonics benefiting from its simplicity and reconfigurability, which can be applied to nonlinearity enhancement and lasing applications at optical frequencies.

Spectroscopic Evidence for a Spin- and Valley-Polarized Metallic State in a Nonmagic-Angle Twisted Bilayer Graphene

In the magic-angle twisted bilayer graphene (MA-TBG), strong electron-electron (e-e) correlations caused by the band-flattening lead to many exotic quantum phases such as superconductivity, correlated insulator, ferromagnetism, and quantum anomalous Hall effects, when its low-energy van Hove singularities (VHSs) are partially filled. Here our high-resolution scanning tunneling microscope and spectroscopy measurements demonstrate that the e-e correlation in a nonmagic-angle TBG with a twist angle θ = 1.49° still plays an important role in determining its electronic properties. Our most interesting observation on that sample is when one of its VHSs is partially filled, the one associated peak in the spectrum splits into four peaks. Simultaneously, the spatial symmetry of electronic states around the split VHSs is broken by the e-e correlation. Our analysis based on the continuum model suggests that such a one-to-four split of the VHS originates from the formation of an interaction-driven spin-valley-polarized metallic state near the VHS, which is a symmetry-breaking phase that has not been identified in the MA-TBG or in other systems.

Dirac-Harper Theory for One-Dimensional Moiré Superlattices

We study a Dirac-Harper model for moiré bilayer superlattices where layer antisymmetric strain periodically modulates the interlayer coupling between two honeycomb lattices in one spatial dimension. Discrete and continuum formulations of this model are analyzed. For a sufficiently long moiré period we find low-energy spectra that host a manifold of weakly dispersive bands arising from a hierarchy of momentum and position-dependent mass inversions. We analyze their charge distributions, mode count, and valley coherence using exact symmetries of the lattice model and approximate symmetries of a four-flavor version of the Jackiw-Rebbi one-dimensional solution.

Emergent magnetic texture in driven twisted bilayer graphene

The transport properties of a twisted bilayer graphene barrier are investigated for various twist angles. Remarkably, for small twist angles around the magic angle θm ∼ 1.05°, the local currents around the AA-stacked regions are strongly enhanced compared to the injected electron rate. Furthermore, the total and counterflow (magnetic) current patterns show high correlations in these regions, giving rise to well-defined magnetic moments that form a magnetic Moiré superlattice. The orientation and magnitude of these magnetic moments change as a function of the gate voltage and possible implications for emergent spin-liquid behaviour are discussed.

Cascade of phase transitions and Dirac revivals in magic-angle graphene

Twisted bilayer graphene near the magic angle^1-4 exhibits rich electron-correlation physics, displaying insulating^3-6, magnetic^7,8 and superconducting phases^4-6. The electronic bands of this system were predicted^1,2 to narrow markedly^9,10 near the magic angle, leading to a variety of possible symmetry-breaking ground states^11-17. Here, using measurements of the local electronic compressibility, we show that these correlated phases originate from a high-energy state with an unusual sequence of band population. As carriers are added to the system, the four electronic 'flavours', which correspond to the spin and valley degrees of freedom, are not filled equally. Rather, they are populated through a sequence of sharp phase transitions, which appear as strong asymmetric jumps of the electronic compressibility near integer fillings of the moiré lattice. At each transition, a single spin/valley flavour takes all the carriers from its partially filled peers, 'resetting' them to the vicinity of the charge neutrality point. As a result, the Dirac-like character observed near charge neutrality reappears after each integer filling. Measurement of the in-plane magnetic field dependence of the chemical potential near filling factor one reveals a large spontaneous magnetization, further substantiating this picture of a cascade of symmetry breaking. The sequence of phase transitions and Dirac revivals is observed at temperatures well above the onset of the superconducting and correlated insulating states. This indicates that the state that we report here, with its strongly broken electronic flavour symmetry and revived Dirac-like electronic character, is important in the physics of magic-angle graphene, forming the parent state out of which the more fragile superconducting and correlated insulating ground states emerge.

Mechanism for Anomalous Hall Ferromagnetism in Twisted Bilayer Graphene

Motivated by the recent observation of an anomalous Hall effect in twisted bilayer graphene, we use a lowest Landau level model to understand the origin of the underlying symmetry-broken correlated state. This effective model is rooted in the occurrence of Chern bands which arise due to the coupling between the graphene device and its encapsulating substrate. Our model exhibits a phase transition from a spin-valley polarized insulator to a partial or fully valley unpolarized metal as the bandwidth is increased relative to the interaction strength, consistent with experimental observations. In sharp contrast to standard quantum Hall ferromagnetism, the Chern number structure of the flat bands precludes an instability to an intervalley coherent phase, but allows for an excitonic vortex lattice at large interaction anisotropy.

Coexistence of Van Hove singularities and pseudomagnetic fields in modulated graphene bilayer

The stacking and bending of graphene are trivial but extremely powerful agents of control over graphene's manifold physics. By changing the twist angle, one can drive the system over a plethora of exotic states via strong electron correlation, thanks to the moiré superlattice potentials, while the periodic or triaxial strains induce discretization of the band structure into Landau levels without the need for an external magnetic field. We fabricated a hybrid system comprising both the stacking and bending tuning knobs. We have grown the graphene monolayers by chemical vapor deposition, using ¹²C and ¹³C precursors, which enabled us to individually address the layers through Raman spectroscopy mapping. We achieved the long-range spatial modulation by sculpturing the top layer (¹³C) over uniform magnetic nanoparticles (NPs) deposited on the bottom layer (¹²C). An atomic force microscopy study revealed that the top layer tends to relax into pyramidal corrugations with C₃ axial symmetry at the position of the NPs, which have been widely reported as a source of large pseudomagnetic fields (PMFs) in graphene monolayers. The modulated graphene bilayer (MGBL) also contains a few micrometer large domains, with the twist angle ∼10°, which were identified via extreme enhancement of the Raman intensity of the G-mode due to formation of van Hove singularities (VHSs). We thereby conclude that the twist-induced VHSs coexist with the PMFs generated in the strained pyramidal objects without mutual disturbance. The graphene bilayer modulated with magnetic NPs is a non-trivial hybrid system that accommodates features of twist-induced VHSs and PMFs in environs of giant classical spins.

Moiré Flat Bands in Twisted Double Bilayer Graphene

We investigate twisted double bilayer graphene (TDBG), a four-layer system composed of two AB-stacked graphene bilayers rotated with respect to each other by a small angle. Our ab initio band structure calculations reveal a considerable energy gap at the charge-neutrality point that we assign to the intrinsic symmetric polarization (ISP). We then introduce the ISP effect into the tight-binding parametrization and perform calculations on TDBG models that include lattice relaxation effects down to very small twist angles. We identify a narrow region around the magic angle characterized by a manifold of remarkably flat bands gapped out from other states even without external electric fields. To understand the fundamental origin of the magic angle in TDBG, we construct a continuum model that points to a hidden mathematical link to the twisted bilayer graphene model, thus indicating that the band flattening is a fundamental feature of TDBG and is not a result of external fields.

Perfect and Controllable Nesting in Minimally Twisted Bilayer Graphene

Parallel ("nested") regions of a Fermi surface (FS) drive instabilities of the electron fluid, for example, the spin density wave in elemental chromium. In one-dimensional materials, the FS is trivially fully nested (a single nesting vector connects two "Fermi dots"), while in higher dimensions only a fraction of the FS consists of parallel sheets. We demonstrate that the tiny angle regime of twist bilayer graphene (TBLG) possesses a phase, accessible by interlayer bias, in which the FS consists entirely of nestable "Fermi lines", the first example of a completely nested FS in a two-dimensional (2D) material. This nested phase is found both in the ideal as well as relaxed structure of the twist bilayer. We demonstrate excellent agreement with recent STM images of topological states in this material and elucidate the connection between these and the underlying Fermiology. We show that the geometry of the Fermi lines network is controllable by the strength of the applied interlayer bias, and thus TBLG offers unprecedented access to the physics of FS nesting in 2D materials.

Large-area, periodic, and tunable intrinsic pseudo-magnetic fields in low-angle twisted bilayer graphene

A properly strained graphene monolayer or bilayer is expected to harbour periodic pseudo-magnetic fields with high symmetry, yet to date, a convincing demonstration of such pseudo-magnetic fields has been lacking, especially for bilayer graphene. Here, we report a definitive experimental proof for the existence of large-area, periodic pseudo-magnetic fields, as manifested by vortex lattices in commensurability with the moiré patterns of low-angle twisted bilayer graphene. The pseudo-magnetic fields are strong enough to confine the massive Dirac electrons into circularly localized pseudo-Landau levels, as observed by scanning tunneling microscopy/spectroscopy, and also corroborated by tight-binding calculations. We further demonstrate that the geometry, amplitude, and periodicity of the pseudo-magnetic fields can be fine-tuned by both the rotation angle and heterostrain. Collectively, the present study substantially enriches twisted bilayer graphene as a powerful enabling platform for exploration of new and exotic physical phenomena, including quantum valley Hall effects and quantum anomalous Hall effects.

Giant magnetic field from moiré induced Berry phase in homobilayer semiconductors

When quasiparticles move in condensed matters, the texture of their internal quantum structure as a function of position and momentum can give rise to Berry phases that have profound effects on the material's properties. Seminal examples include the anomalous Hall and spin Hall effects from the momentum-space Berry phases in homogeneous crystals. Here, we explore a conjugate form of the electron Berry phase arising from the moiré pattern: the texture of atomic configurations in real space. In homobilayer transition metal dichalcogenides, we show that the real-space Berry phase from moiré patterns manifests as a periodic magnetic field with magnitudes of up to hundreds of Tesla. This quantity distinguishes moiré patterns from different origins, which can have an identical potential landscape, but opposite quantized magnetic flux per supercell. For low-energy carriers, the homobilayer moirés realize topological flux lattices for the quantum-spin Hall effect. An interlayer bias can continuously tune the spatial profile of the moiré magnetic field, whereas the flux per supercell is a topological quantity that can only have a quantized jump observable at a moderate bias. We also reveal the important role of the non-Abelian Berry phase in shaping the energy landscape in small moiré patterns. Our work points to new possibilities to access ultra-high magnetic fields that can be tailored to the nanoscale by electrical and mechanical controls.

Localization and delocalization of light in photonic moiré lattices

Moiré lattices consist of two superimposed identical periodic structures with a relative rotation angle. Moiré lattices have several applications in everyday life, including artistic design, the textile industry, architecture, image processing, metrology and interferometry. For scientific studies, they have been produced using coupled graphene-hexagonal boron nitride monolayers^1,2, graphene-graphene layers^3,4 and graphene quasicrystals on a silicon carbide surface⁵. The recent surge of interest in moiré lattices arises from the possibility of exploring many salient physical phenomena in such systems; examples include commensurable-incommensurable transitions and topological defects², the emergence of insulating states owing to band flattening^3,6, unconventional superconductivity⁴ controlled by the rotation angle^7,8, the quantum Hall effect⁹, the realization of non-Abelian gauge potentials¹⁰ and the appearance of quasicrystals at special rotation angles¹¹. A fundamental question that remains unexplored concerns the evolution of waves in the potentials defined by moiré lattices. Here we experimentally create two-dimensional photonic moiré lattices, which-unlike their material counterparts-have readily controllable parameters and symmetry, allowing us to explore transitions between structures with fundamentally different geometries (periodic, general aperiodic and quasicrystal). We observe localization of light in deterministic linear lattices that is based on flat-band physics⁶, in contrast to previous schemes based on light diffusion in optical quasicrystals¹², where disorder is required¹³ for the onset of Anderson localization¹⁴ (that is, wave localization in random media). Using commensurable and incommensurable moiré patterns, we experimentally demonstrate the two-dimensional localization-delocalization transition of light. Moiré lattices may feature an almost arbitrary geometry that is consistent with the crystallographic symmetry groups of the sublattices, and therefore afford a powerful tool for controlling the properties of light patterns and exploring the physics of periodic-aperiodic phase transitions and two-dimensional wavepacket phenomena relevant to several areas of science, including optics, acoustics, condensed matter and atomic physics.

Intrinsically undamped plasmon modes in narrow electron bands

Surface plasmons in 2-dimensional electron systems with narrow Bloch bands feature an interesting regime in which Landau damping (dissipation via electron-hole pair excitation) is completely quenched. This surprising behavior is made possible by strong coupling in narrow-band systems characterized by large values of the "fine structure" constant [Formula: see text] Dissipation quenching occurs when dispersing plasmon modes rise above the particle-hole continuum, extending into the forbidden energy gap that is free from particle-hole excitations. The effect is predicted to be prominent in moiré graphene, where at magic twist-angle values, flat bands feature [Formula: see text] The extinction of Landau damping enhances spatial optical coherence. Speckle-like interference, arising in the presence of disorder scattering, can serve as a telltale signature of undamped plasmons directly accessible in near-field imaging experiments.

Electrically Tunable Flat Bands and Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers provide a versatile platform to engineer metamaterials with novel emergent properties by exploiting the resulting geometric moiré superlattice. Such superlattices are known to host bulk valley currents at tiny angles (α≈0.3°) and flat bands at magic angles (α≈1°). We show that tuning the twist angle to α^{*}≈0.8° generates flat bands away from charge neutrality with a triangular superlattice periodicity. When doped with ±6 electrons per moiré cell, these bands are half-filled and electronic interactions produce a symmetry-broken ground state (Stoner instability) with spin-polarized regions that order ferromagnetically. Application of an interlayer electric field breaks inversion symmetry and introduces valley-dependent dispersion that quenches the magnetic order. With these results, we propose a solid-state platform that realizes electrically tunable strong correlations.

Flatbands and Perfect Metal in Trilayer Moiré Graphene

We investigate the electronic structure of a twisted multilayer graphene system forming a moiré pattern. We consider small twist angles separating the graphene sheets and develop a low-energy theory to describe the coupling of Dirac Bloch states close to the K point in each individual plane. Extending beyond the bilayer case, we show that, when the ratio of the consecutive twist angles is rational, a periodicity emerges in quasimomentum space with moiré Bloch bands even when the system does not exhibit a crystalline lattice structure in real space. For a trilayer geometry, we find flatbands in the spectrum at certain rotation angles. Performing a symmetry analysis of the band model for the trilayer, we prove that the system is a perfect metal in the sense that it is gapless at all energies. This striking result originates from the three Dirac cones which can only gap in pairs and produce bands with an infinite connectivity. It also holds quite generally for multilayer graphene with an odd number of planes under the condition of C_{2z}T symmetry.

Origin of Magic Angles in Twisted Bilayer Graphene

Twisted bilayer graphene (TBG) was recently shown to host superconductivity when tuned to special "magic angles" at which isolated and relatively flat bands appear. However, until now the origin of the magic angles and their irregular pattern have remained a mystery. Here we report on a fundamental continuum model for TBG which features not just the vanishing of the Fermi velocity, but also the perfect flattening of the entire lowest band. When parametrized in terms of α∼1/θ, the magic angles recur with a remarkable periodicity of Δα≃3/2. We show analytically that the exactly flat band wave functions can be constructed from the doubly periodic functions composed of ratios of theta functions-reminiscent of quantum Hall wave functions on the torus. We further report on the unusual robustness of the experimentally relevant first magic angle, address its properties analytically, and discuss how lattice relaxation effects help justify our model parameters.

Kohn-Luttinger Superconductivity in Twisted Bilayer Graphene

We show that the recently observed superconductivity in twisted bilayer graphene (TBG) can be explained as a consequence of the Kohn-Luttinger (KL) instability which leads to an effective attraction between electrons with originally repulsive interaction. Usually, the KL instability takes place at extremely low energy scales, but in TBG, a doubling and subsequent strong coupling of the van Hove singularities (vHS) in the electronic spectrum occurs as the magic angle is approached, leading to extended saddle points in the highest valence band with almost perfect nesting between states belonging to different valleys. The highly anisotropic screening induces an effective attraction in a p-wave channel with odd parity under the exchange of the two disjoined patches of the Fermi line. We also predict the appearance of a spin-density wave instability, adjacent to the superconducting phase, and the opening of a gap in the electronic spectrum from the condensation of spins with wave vector corresponding to the nesting vector close to the vHS.

Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers

Bilayer graphene twisted by a small angle shows a significant charge modulation away from neutrality, as the charge in the narrow bands near the Dirac point is mostly localized in a fraction of the Moiré unit cell. The resulting electrostatic potential leads to a filling-dependent change in the low-energy bands, of a magnitude comparable to or larger than the bandwidth. These modifications can be expressed in terms of new electron-electron interactions, which, when expressed in a local basis, describe electron-assisted hopping terms. These interactions favor superconductivity at certain fillings.

Electrically Tunable Gauge Fields in Tiny-Angle Twisted Bilayer Graphene

Twisted bilayer graphene has recently attracted a lot of attention for its rich electronic properties and tunability. Here we show that for very small twist angles, α≪1°, the application of a perpendicular electric field is mathematically equivalent to a new kind of artificial gauge field. This identification opens the door for the generation and detection of pseudo-Landau levels in graphene platforms within robust setups, which do not depend on strain engineering and therefore can be realistically harvested for technological applications. Furthermore, this new artificial gauge field leads to the development of highly localized modes associated with flat bands close to charge neutrality, which form an emergent kagome lattice in real space. Our findings indicate that for tiny angles biased twisted bilayer graphene is a promising platform that can realize frustrated lattices of highly localized states, opening a new direction for the investigation of strongly correlated phases of matter.

Electrically Controllable Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers develop highly localized states around AA-stacked regions for small twist angles. We show that interaction effects may induce either an antiferromagnetic or a ferromagnetic (FM) polarization of said regions, depending on the electrical bias between layers. Remarkably, FM-polarized AA regions under bias develop spiral magnetic ordering, with a relative 120° misalignment between neighboring regions due to a frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges naturally without the need of spin-orbit coupling, and competes with the more conventional lattice-antiferromagnetic instability, which interestingly develops at smaller bias under weaker interactions than in monolayer graphene, due to Fermi velocity suppression. This rich and electrically controllable magnetism could turn twisted bilayer graphene into an ideal system to study frustrated magnetism in two dimensions.

NanoARPES of twisted bilayer graphene on SiC: absence of velocity renormalization for small angles

The structural and electronic properties of twisted bilayer graphene (TBG) on SiC(000) grown by Si flux-assisted molecular beam epitaxy were investigated using scanning tunneling microscopy (STM) and angle-resolved photoelectron spectroscopy with nanometric spatial resolution. STM images revealed a wide distribution of twist angles between the two graphene layers. The electronic structure recorded in single TBG grains showed two closely-spaced Dirac π bands associated to the two stacked layers with respective twist angles in the range 1–3°. The renormalization of velocity predicted in previous theoretical calculations for small twist angles was not observed.

Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis

Extensive scanning tunneling microscopy and spectroscopy experiments complemented by first-principles and parametrized tight binding calculations provide a clear answer to the existence, origin, and robustness of van Hove singularities (vHs) in twisted graphene layers. Our results are conclusive: vHs due to interlayer coupling are ubiquitously present in a broad range (from 1° to 10°) of rotation angles in our graphene on 6H-SiC(000-1) samples. From the variation of the energy separation of the vHs with the rotation angle we are able to recover the Fermi velocity of a graphene monolayer as well as the strength of the interlayer interaction. The robustness of the vHs is assessed both by experiments, which show that they survive in the presence of a third graphene layer, and by calculations, which test the role of the periodic modulation and absolute value of the interlayer distance. Finally, we clarify the role of the layer topographic corrugation and of electronic effects in the apparent moiré contrast measured on the STM images.