Twisted bilayer graphene (TBG) hosts a rich landscape of electronic phases arising from the interplay between strong electron-electron interactions and nontrivial band topology. While the flat bands near zero energy are central to many correlated phenomena, their interaction with higher-energy remote bands remains less understood. Here, we investigate these hybridization processes as a function of the twist angle and analyze their impact on the charge distribution, topological properties such as Chern number, quantum metric, and orbital magnetic energy. We identify multiple topological phase transitions between magic angles, driven by band inversions at high-symmetry points in momentum space. Notably, the central bands can exhibit phases with Chern numbers C = 2, revealing previously unreported topological states in TBG.

The experimental observations of many interaction-driven electronic phases in moiré superlattices have stimulated intense theoretical and experimental efforts to understand and engineer these correlated physics. Strain is a powerful tool for manipulating and controlling the geometrical and electronic structures of moiré superlattices. This review provides a comprehensive introduction to the geometry of strained moiré superlattices. First, starting from the linear elasticity theory, we briefly introduce the general formalism of small deformations in two-dimensional materials, and discuss the particular cases of uniaxial, shear and biaxial strain. Then, we apply the theory to twisted and strained moiré materials, mainly focusing on the hexagonal homobilayers, hexagonal heterobilayers and monoclinic lattices. Special moiré geometries, like the quasi-unidimensional patterns, square patterns and hexagonal, are theoretically predicted by manipulating the strain and twist. Finally, we review recently developed strain techniques and the special moiré geometries realized via these approaches. This review aims at equipping the reader with a robust understanding on the description and implementation of strain in moiré materials, as well as highlight some major breakthroughs in this active field.

In twisted bilayer graphene (TBG) devices, local strain frequently coexists with the twist-angle-dependent moiré superlattice and strongly influences the electronic properties, yet their combined effects remain incompletely understood. Here, using low-temperature scanning tunneling microscopy, we study a TBG device exhibiting both a continuous twist-angle gradient from 0.35° to 1.30° and spatially varying strain fields, spanning the first (1.1°), second (0.5°), and third (0.3°) magic angles. We directly visualize the evolution of flat and remote bands in both energy and real space with atomic resolution. By comparing regions dominated by shear, uniaxial, and mixed strain, we find that shear strain plays a decisive role in controlling flat-band separation, linewidth, and spectral-weight redistribution. Near the first magic angle, this manifests as an anomalous transfer of spectral weight between the two flat-band peaks, accompanied by an unusual spatial dispersion of flat-band states within a moiré unit cell. In contrast, the energy of the remote bands provides a robust, strain-insensitive indicator of the local twist angle. Structural analysis reveals that shear strain dominates over large regions of the sample, consistent with its lower elastic energy cost. All observations are quantitatively reproduced by a continuum model incorporating heterostrain and electron-electron interactions, establishing shear strain as a central ingredient in shaping the low-energy electronic landscape of TBG.

For the unconventional superconducting phases in moire materials, a critical question is the role played by electronic interactions in the formation of Cooper pairs. In twisted bilayer graphene (tBLG), the strength of electronic interactions can be reduced by increasing the twist angle or screening provided by the dielectric medium. In this work, we place tBLG at 3-4 nm above bulk SrTiO3 substrates, which have a large yet tunable dielectric constant. By raising the dielectric constant in situ in a magic angle device, we observe suppression of both the height and the width of the entire superconducting dome, thus demonstrating that, unlike conventional superconductors, the pairing mechanism in tBLG is strongly dependent on electronic interactions. Interestingly, in contrast to the absence of superconductivity in devices on SiO2 with angle>1.3 deg, we observe a superconducting pocket in a large-angle (angle=1.4 deg) tBLG/STO device while the correlated insulating states are absent. These experimental results are in qualitative agreement with a theoretical model in which the pairing mechanism arises from Coulomb interactions that are screened by plasmons, electron-hole pairs, and longitudinal acoustic phonons. Our results highlight the unconventional nature of the superconductivity in tBLG, the double-edged role played by electronic interactions in its formation, as well as their complex interplay with the correlated insulating states.

Perturbations in moir & eacute; materials, such as due to substrates or strain, are common in many experiments and can significantly modify the electronic properties of the system. Here, we show that perturbations in twisted bilayer graphene tend to be transferred between the coupled Dirac cones, eventually reaching an equilibrium near the magic angle. We connect our results to experiments and show that this equilibrium behavior remains robust even when the moir & eacute; potential itself is perturbed. Our findings extend the notion of the magic angle to a more general regime governed by moir & eacute;-driven equilibrium.

We introduce twisted anisotropic homobilayers as a distinct class of moiré systems, characterized by a distinctive "magic angle," [Formula: see text], where the moiré unit cell collapses. Unlike conventional studies of moiré materials, which primarily focus on small lattice misalignments, we demonstrate that this moiré collapse occurs at large twist angles in generic twisted anisotropic homobilayers. The collapse angle, [Formula: see text], is likely to give rise quasi-crystal behavior as well as to the formation of strongly correlated states, that arise not from flat bands, but from the presence of ultra-anisotropic electronic states, where non-Fermi liquid phases can be stabilized. In this work, we develop a continuum model for electrons based on extensive ab initio calculations for twisted bilayer black phosphorus, enabling a detailed study of the emerging moiré collapse features in this prototypical system. We show that the (temperature) stability criterion for the emergence of (sliding) Luttinger liquids is generally met as the twist angle approaches [Formula: see text]. Furthermore, we explicitly formulate the collapsed single-particle one-dimensional (1D) continuum Hamiltonian and provide the fully interacting, Hamiltonian applicable at low doping levels. Our analysis reveals a rich landscape of multichannel Luttinger liquids, potentially enhanced by valley degrees of freedom at large twist angles.

The interplay of twist and strain in bilayer graphene enables the formation of moiré patterns and narrow bands that host correlated and topological phases. While magic-angle twisted bilayer graphene has been widely studied, strain provides an additional and realistic control knob for band engineering. In this work, we first generate a global method to construct commensurate supercells for arbitrary twist and heterostrain. Then, using atomistic tight-binding and strain-extended continuum models to study the commensurate structures, we identify configurations that minimize the bandwidth beyond the magic angle. The results reveal a strong dependence of band narrowing and topology on strain type, magnitude, direction, and lattice relaxation. Particularly, shear strain produces a stronger distortion than uniaxial strain. Including electron-electron interactions through a self-consistent Hartree potential shows that strain broadens the bare bands while reducing electrostatic renormalization. Strain also drives topological transitions as the narrow and remote bands hybridize, establishing twisted and strained bilayer graphene as a tunable platform for flat-band and topological phenomena.

We consider plasmon-assisted electron tunneling in a quantum twisting microscope (QTM). The dependence of the differential conductance on the two control parameters of the QTM -- the twist angle and bias -- reveals the plasmon spectrum as well as the strength of plasmon-electron interactions in the sample. We perform microscopic calculations for twisted bilayer graphene (TBG), to predict the plasmon features in the tunneling spectra of TBG close to the magic angle for different screening environments. Our work establishes a general framework for inelastic tunneling spectroscopy of collective electronic excitations using the quantum twisting microscope.

The overlap matrix of electronic energy eigenstates, sometimes referred to as the form factor, determines the quantum geometric tensor of electrons in solids. Here, we show that the variation in the overlap of two eigenstates with opposite momenta can be directly observed via quasiparticle interference (QPI) imaging. We study the QPI in twisted bilayer graphene using a real-space tight-binding model combined with the kernel polynomial method. The resulting QPI patterns, which are largely independent of whether the two graphene layers are commensurate or incommensurate, reveal all intralayer and interlayer interference processes. While the intralayer interference signals resemble those of monolayer graphene, the interlayer interference - which vanishes at large twist angles - displays a chiral structure that reverses between the two layers and between the valence and conduction bands. Furthermore, the QPI patterns explicitly demonstrate the approximate translational symmetries and valley charge conservation in twisted bilayer graphene, validating the topological obstruction to constructing the Wannier orbitals of states at the Dirac cones. Using a continuum model of twisted bilayer graphene, we show that all characteristics of the observed QPI patterns can be explained by the form factor of eigenstates projected onto a single layer. Our results provide fundamental insights into the electronic spectrum and wave functions of twisted bilayer graphene, and establish QPI as an experimental probe for the form factor of back-scattering states.

Moir\'e materials represent strongly interacting electron systems bridging topological and correlated physics. Despite significant advances, decoding wavefunction properties underlying the quantum geometry remains challenging. Here, we utilize polarization-resolved photocurrent measurements to probe magic-angle twisted bilayer graphene, leveraging its sensitivity to the Berry connection that encompasses quantum "textures" of electron wavefunctions. Using terahertz light resonant with optical transitions of its flat bands, we observe bulk photocurrents driven by broken symmetries and reveal the interplay between electron interactions and quantum geometry. We observe inversion-breaking gapped states undetectable through quantum transport, sharp changes in the polarization axes caused by interaction-induced band renormalization, and recurring photocurrent patterns at integer fillings of the moir\'e unit cell that track the evolution of quantum geometry through the cascade of phase transitions. The large and tunable terahertz response intrinsic to flat-band systems offers direct insights into the quantum geometry of interacting electrons and paves the way for innovative terahertz quantum technologies.

Since its discovery, graphene has been one of the most prominent 2D materials due to its unique properties and broad range of possible applications. In particular, the half-integer Quantum Hall Effect (HI-QHE) characterized by the quantization of Hall resistivity as a function of applied magnetic field, offers opportunities for advancements in quantum metrology and the understanding of topological quantum states in this 2D material. While the role of disorder in stabilizing quantum Hall plateaus (QHPs) is widely recognized, the precise interplay between the plateaus width, disorder, mobility and carrier density remains less explored. In this work, we investigate the width of the ν=6 QHP in epitaxial graphene Hall bars, focusing on two distinct regions of the device with markedly different electronic mobilities. Depending on the storage conditions, it is possible to modify the carrier density of graphene QHE devices and consequently increase or reduce the mobility. Our experiments reveal mobility variations of up to 200% from their initial value. In particular, the sample storage time and ambient conditions cause also noticeable changes in the positions and extension of the QHPs. Our results show that the QHP extension for ν=6 differs significantly between the two regions, influenced by both mobility and disorder, rather than solely by carrier density. Transport simulations based on the Landauer-Büttiker formalism with Anderson disorder in a scaled model reveal the critical role of impurities in shaping graphene transport properties defining the extension of the QHPs. This study provides valuable insights into the interplay between mobility, disorder, and quantum transport in graphene systems.

Using a formalism based on the non-Abelian Berry connection, we explore quantum geometric signatures of Wannier-Stark spectra in two-dimensional superlattices. The Stark energy can be written as intraband Berry phases, while Zener tunneling is given by interband Berry connections. We suggest that the gaps induced by interband hybridization can be probed by THz optical absorption and emission spectroscopy. This is especially relevant to modern moir & eacute; materials wherein mini-bands are often spectrally entangled, leading to strong interband hybridization in the Wannier-Stark regime. Furthermore, owing to their large superlattice constants, both the low-field and high-field regimes can be accessed in these materials using presently available technology. Importantly, even at moderate electric fields, we find that stimulated emission can dominate absorption, raising the possibility of lasing at practically relevant parameter regimes.

Magnetic interactions at the surface of topological insulators provide a versatile route to engineer exotic quantum states. Breaking time-reversal symmetry (TRS) at the topological surface state (TSS) enables the opening of a Dirac gap, which is essential for realizing quantum anomalous Hall physics. This work investigates the impact of submonolayer deposition of magnetic rare-earth adatoms on the prototypical topological insulator Bi2Te3. Scanning tunneling microscopy (STM) supported by first-principle calculations, core-level photoemission spectroscopy (XPS), angle-resolved photoemission spectroscopy (ARPES), X-ray magnetic circular dichroism (XMCD) and quasiparticle interference (QPI) mapping are combined to reveal direct evidence of local interactions between erbium (Er) atoms and the substrate, leading to significant modifications of the TSS. XMCD measurements confirm the out-of-plane magnetic anisotropy for Er adatoms on Bi2Te3 , which induces a warping transition of the Fermi surface from a snowflake to a star-of-David-like geometry, along with a Dirac point gap opening and spectral splitting near the Γ point. QPI maps confirm the reconstructed surface band topology through modified scattering patterns consistent with TRS breaking. Our results identify a microscopic mechanism for magnetic interaction at the surface of a topological insulator and establish magnetic rare-earth doping as an effective strategy to tailor topological electronic states with atomic-scale control.

Recently, exotic superconductivity emerging from a spin-and-valley-polarized metallic phase has been discovered in rhombohedral tetralayer graphene. To explain this observation, we study the role of electron-electron interactions in driving flavor symmetry breaking, using the Hartree-Fock (HF) approximation, and in stabilizing superconductivity mediated by repulsive interactions. Though mean-field HF correctly predicts the isospin flavors and reproduces the experimental phase diagram, it overestimates the band renormalization near the Fermi energy and suppresses superconducting instabilities. To address this, we introduce a physically motivated scheme that includes internal screening in the HF calculation. Using this formalism, we find superconductivity arising from the spin-valley polarized phase, on the electron-doped side, for a range of electric fields and electron dopings. Our findings reproduce the experimental observations and reveal a p-wave, finite-momentum, time-reversal-symmetry-broken superconducting state, encouraging further investigation into exotic phases in graphene multilayers.

Moiré superlattices have emerged as a versatile platform for exploring a wide range of ex- otic quantum phenomena. Unlike angstrom-scale materials, the moiré length-scale system contains a large number of atoms, and its electronic structure is significantly modulated by the lattice relaxation. These features pose a huge theoretical challenge. Among the available theoretical approaches, tight-binding (TB) methods are widely employed to predict the electronic, transport, and optical properties of systems such as twisted graphene, twisted transition-metal dichalcogenides (TMDs), and related moiré materials. In this review, we pro- vide a comprehensive overview of atomistic TB Hamiltonians and the numerical techniques commonly used to model graphene-based, TMD-based and hBN-based moiré superlattices. We also discuss the connection between atomistic TB descriptions and effective low-energy continuum models. Two examples of different moiré materials and geometries are provided to emphasize the advantages of the TB methods. This review is intended to serve as a theoretical and practical guide for those seeking to apply TB methods to the study of various properties of moiré superlattices.

We report the first observation of controlled, strain-induced square moire patterns in stacked graphene. By selectively displacing native wrinkles, we drive a reversible transition from the usual trigonal to square moire order. Scanning tunneling microscopy reveals elliptically shaped AA domains, while spectroscopy shows strong electronic correlation in the form of narrow bands with split Van Hove singularities near the Fermi level. A continuum model with electrostatic interactions reproduces these features under the specific twist-strain combination that minimizes elastic energy. This work demonstrates that the combination of twist and strain, or twistraintronics, enables the realization of highly correlated electronic states in moire heterostructures with geometries that were previously inaccessible.

Heavily doping graphene by intercalation can raise its Fermi level near an extended van Hove singularity, potentially inducing correlated electronic phases. Intercalation also modifies the band structure: dopants may hybridize with carbon orbitals and order into √(3)×√(3) or 2×2 superstructures, introducing periodic potentials that fold the graphene π bands. Angle-resolved photoemission spectroscopy further shows a pronounced flattening of the conduction band near the M points, producing higher-order van Hove singularities. These effects depend strongly on the dopant species and substrate, with implications for both many-body physics and transport. We construct effective tight-binding models that incorporate dopant ordering, carbon-dopant hybridization, and π-band renormalization. Model parameters are obtained from density functional theory and reproduce dispersions observed in photoemission experiments. Using these models, we compute the optical conductivity and identify characteristic signatures associated with dopant ordering and hybridization. Our results provide a framework to interpret experimental spectra and to probe the superlattice symmetry of highly doped monolayer graphene.

Rhombohedral (ABC‐stacked) graphene systems with different number of layers feature an abundance of correlated phases and superconducting states in experimental measurements with different doping and displacement fields. Some of the superconducting pockets can emerge from ‐ or close to ‐ one of the correlated states. Therefore, studying the phase diagram of these phases for varying number of layers can be useful to interpret the experimental observations. To achieve this, systematic Hartree–Fock calculations have been performed in the presence of long‐range Coulomb interactions. By varying the external displacement field and carrier density, a cascade of metallic partially‐isospin‐polarized phases that spontaneously break spin and/or valley (flavor) symmetries is found. In addition, these states can present nematicity, stabilized by electron–electron interactions, exhibiting rich internal complexity. Polarized states are more stable for electron doping, and they are found for systems with up to 20 layers. Moreover, the tunability of the phase diagram via substrate screening and spin–orbit coupling proximity effects is studied. The results offer new insights into the role of correlations and symmetry breaking in graphitic systems, which will motivate future experimental and theoretical works.

The half-integer quantum Hall effect in graphene, characterized by the quantization of Hall resistivity as a function of applied magnetic field, offers opportunities for advancements in quantum metrology and the understanding of topological quantum states. While the role of disorder in stabilizing quantum Hall plateaus (QHPs) is widely recognized, the precise interplay between the plateaus’ width, disorder, mobility, and carrier density remains less explored. In this work, we investigate the width of the ν = 6 QHP in epitaxial graphene, focusing on two distinct regions of the device with markedly different electronic mobilities. Depending on the storage conditions, it is possible to modify the carrier density of graphene QHE devices and consequently increase or reduce the mobility. Our experiments reveal mobility variations of up to 200% from their initial value. In particular, the sample storage time and ambient conditions also cause noticeable changes in the positions and extension of the QHPs. Our results show that the QHP extension for ν = 6 differs significantly between the two regions, influenced by both mobility and disorder, rather than solely by carrier density. Transport simulations based on the Landauer–Büttiker formalism with Anderson disorder in a scaled model reveal the critical role of impurities in shaping graphene transport properties, defining the extension of the QHPs. This study provides valuable insights into the interplay between mobility, disorder, and quantum transport in graphene systems.

Marginally twisted bilayer graphene having small twist angles is predicted to exhibit unique structural and electronic properties, though experimental characterization remains limited. Using scanning tunneling microscopy, we investigate such systems with twist angles of 0.06°–0.35°. AA-stacked regions reveal a pronounced tunneling spectral peak signifying highly localized electronic states. Conversely, AB domains display uniform multiple spectral peaks, indicative of strong lattice reconstruction and enhanced electronic homogeneity. We identify two distinct strain-induced domain walls: one exhibits a sharp −120 meV spectral peak (shear type), while the other shows distinct spectral characteristics (mixed shear-tensile type). Tight-binding calculations verify strain-driven transformations of both domain wall types and confirm direct observation of strain-mediated domain wall transitions. These results elucidate the electronic structure of marginally twisted bilayer graphene and establish strain as a control parameter for domain wall states.