We analyze competition between different ordered states in bilayer graphene (BLG). Combining arguments based on SU(4) spin-valley flavor symmetry with a mean field analysis, we identify the lowest energy state with the anomalous Hall insulator (AHI). This state is an SU(4) singlet excitonic insulator with broken time reversal symmetry, exhibiting quantized Hall effect in the absence of external magnetic field. Applied electric field drives an Ising-type phase transition, restoring time reversal symmetry. Applied magnetic field drives a transition from the AHI state to a quantum Hall ferromagnet state. We estimate energies of these states, taking full account of screening, and predict the phase diagram.
We report the nanoscale imaging of Ti-doped bilayer calcium ruthenates during the Mott metal-insulator transition by microwave impedance microscopy. Different from a typical first-order phase transition where coexistence of the two terminal phases takes place, a new metallic stripe phase oriented along the in-plane crystalline axes emerges inside both the G-type antiferromagnetic insulating state and paramagnetic metallic state. The effect of this electronic state can be observed in macroscopic measurements, allowing us to construct a phase diagram that takes into account the energetically com…
We study the fractional quantum Hall effect in the central Landau level of bilayer graphene. By tuning the external applied magnetic field and the electric bias between the two layers one can access a regime where there is a degeneracy between Landau levels with orbital characters corresponding to N=0 and N=1 Galilean Landau levels. While the Laughlin state is generically the ground state for filling $ν=1/3$ we find that it can be destroyed and replaced by a Wigner crystal at the same filling factor by tuning the bias and applied field. This competition does not take place at $ν=2/3$ where the…
Near a magic twist angle, the lowest energy conduction and valence bands of bilayer graphene moiré superlattices become extremely narrow. The band dispersion that remains is sensitive to the moiré's strain pattern, nonlocal tunneling between layers, and filling-factor dependent Hartree and exchange band renormalizations. In this article we analyze the influence of these band-structure details on the pattern of flavor-symmetry-breaking observed in this narrow band system, and on the associated pattern of Fermi surface reconstructions revealed by weak-field-Hall and Schubnikov-de Haas magneto-tr…
We use a perturbative renormalization group approach with short-range continuum model interactions to analyze the competition between isotropic gapped and anisotropic gapless ordered states in bilayer graphene, commenting specifically on the role of exchange and on the importance of spin and valley flavor degeneracy. By comparing the divergences of the corresponding susceptibilities, we conclude that this approach predicts gapped states for flavor numbers N=1,2,4. We also comment briefly on the related gapped states expected in chiral (ABC) trilayer graphene.
We implement a self-consistent Hartree-Fock approximation based on a microscopic model in real space, which allows us to consider the interplay between the Hubbard and the extended Coulomb interaction in twisted bilayer graphene at the magic angle. These two interactions tend to favor different symmetry breaking patterns, having therefore complementary roles in the regimes where one or the other dominates. We show that, for sufficiently large values of the on-site Hubbard repulsion, magic angle graphene has an antiferromagnetic ground state at the charge neutrality point, while at half-filling…
We study the phase diagram of an effective model of competing nematic and antiferromagnetic orders of interacting electrons on the Bernal-stacked honeycomb bilayer, as relevant for bilayer graphene. In the noninteracting limit, the model features a semimetallic ground state with quadratic band touching points at the Fermi level. Taking the effects of short-range interactions into account, we demonstrate the existence of an extended region in the mean-field phase diagram characterized by coexisting nematic and antiferromagnetic orders. By means of a renormalization group approach, we reveal tha…
We analyze the phase diagram of the Bilayer graphene (BLG) at zero temperature and doping. Assuming that at the high energies the electronic system of BLG can be described within a weak coupling theory (consistent with the experimental evidence), we systematically study the evolution of the couplings with going from high to low energies. The divergences of the couplings at some energies indicates the tendency towards certain symmetry breakings. Carrying out this program, we found that the phase diagram is determined by microscopic couplings defined on the short distances (initial conditions). …
Because of its large density-of-states and the 2π Berry phase near its low-energy band-contact points, neutral bilayer graphene (BLG) at zero magnetic field (B) is susceptible to chiral-symmetry breaking, leading to a variety of gapped spontaneous quantum Hall states distinguished by valley and spin-dependent quantized Hall conductivities. Among these, the layer antiferromagnetic state, which has quantum valley Hall (QVH) effects of opposite sign for opposite spins, appears to be the thermodynamic ground state. Though other gapped states have not been observed experimentally at B=0, they can b…
Flat band moiré superlattices have recently emerged as unique platforms for investigating the interplay between strong electronic correlations, nontrivial band topology, and multiple isospin 'flavor' symmetries. Twisted monolayer-bilayer graphene (tMBG) is an especially rich system owing to its low crystal symmetry and the tunability of its bandwidth and topology with an external electric field. Here, we find that orbital magnetism is abundant within the correlated phase diagram of tMBG, giving rise to the anomalous Hall effect (AHE) in correlated metallic states nearby most odd integer fillin…
Graphene's electronic structure can be fundamentally altered when a substrate- or adatom-induced Kekulé superlattice couples the valley and isospin degrees of freedom. Here, we show that the band structure of Kekulé-textured graphene can be re-engineered through layer stacking. We predict a family of Kekulé graphene bilayers that exhibit band structures with up to six valleys, and room-temperature Dirac quasiparticles whose masses can be tuned electrostatically. Fermi velocities half as large as in pristine graphene put this system in the strongly coupled regime, where correlated ground states…
The competing ground states of bilayer graphene are studied by applying renormalization group techniques to a bilayer honeycomb lattice with nearest neighbor hopping. In the absence of interactions, the Fermi surface of this model at half-filling consists of two nodal points with momenta $\mathbf{K}$, $\mathbf{K}'$, where the conduction band and valence band touch each other, yielding a semi-metal. Since near these two points the energy dispersion is quadratic with perfect particle-hole symmetry, excitonic instabilities are inevitable if inter-band interactions are present. Using a perturbativ…
Motivated by recent experiments [H. Zhou, \emph{et al.}, Science {\bf 375}, 774 (2022) and S. C. de la Barrera, \emph{et al.}, arXiv:2110.13907], here we propose a general mechanism for valley and/or spin degeneracy lifting of the electronic bands in doped Bernal bilayer graphene, subject to electric displacement ($D$) fields. A $D$-field induced layer polarization (LP), when accompanied by Hubbard repulsion driven layer antiferromagnet (LAF) and next-nearest-neighbor repulsion driven quantum anomalous Hall (QAH) orders, lifts the four-fold degeneracy of electronic bands, yielding a quarter me…
Performing a leading-order renormalization group analysis, here we compute the effects of generic local or short-range electronic interactions in monolayer and Bernal bilayer graphene. Respectively in these two systems chiral quasiparticles display linear and biquadratic band touching, leading to linearly vanishing and constant DOS. Consequently, the former system remains stable for weak enough local interactions, and supports a variety of ordered phases only beyond a critical strength of interactions. By contrast, ordered phases can nucleate for sufficiently weak interactions in bilayer graph…
In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and flavor dependent spontaneous inter layer charge transfers. We use a lattice Hartree-Fock model to explore some of the physics which controls whether or not this type of broken symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature. We find that inversion symmetry is still broken in the lattice model and estimate that transferred areal densities are $\sim 10^{-5}$ electrons per carbon atom, that the associated energy gaps are…
The discovery of magic angle twisted bilayer graphene (MATBG) has unveiled a rich variety of superconducting, magnetic and topologically nontrivial phases. The existence of all these phases in one material, and their tunability, has opened new pathways for the creation of unusual gate tunable junctions. However, the required conditions for their creation - gate induced transitions between phases in zero magnetic field - have so far not been achieved. Here, we report on the first experimental demonstration of a device that is both a zero-field Chern insulator and a superconductor. The Chern ins…
We introduce and analyze a model that sheds light on the interplay between correlated insulating states, superconductivity, and flavor-symmetry breaking in magic angle twisted bilayer graphene. Using a variational mean-field theory, we determine the normal-state phase diagram of our model as a function of the band filling. The model features robust insulators at even integer fillings, occasional weaker insulators at odd integer fillings, and a pattern of flavor-symmetry breaking at non-integer fillings. Adding a phonon-mediated inter-valley retarded attractive interaction, we obtain strong-cou…
We perform an extensive exact diagonalization study of interaction driven insulators in spin- and valley-polarized moiré flat bands of twisted bilayer graphene aligned with its hexagonal boron nitride substrate. In addition to previously reported fractional Chern insulator phases, we provide compelling evidence for competing charge-density-wave phases at multiple fractional fillings of a realistic single-band model. A thorough analysis at different interlayer hopping parameters, motivated by experimental variability, and the role of kinetic energy at various Coulomb interaction strengths highl…
The $E = 0$ octet of bilayer graphene in the filling factor range of -4 < $ν$ < 4 is a fertile playground for many-body phenomena, yet a Landau level diagram is missing due to strong interactions and competing quantum degrees of freedom. We combine measurements and modeling to construct an empirical and quantitative spectrum. The single-particle-like diagram incorporates interaction effects effectively and provides a unified framework to understand the occupation sequence, gap energies and phase transitions observed in the octet. It serves as a new starting point for more sophisticated calcula…
We study the interaction energy between two graphene nanoribbons by first principles calculations, including van der Waals interactions and spin polarization. For ultranarrow zigzag nanoribbons, the direct stacking is even more stable than Bernal, competing in energy for wider ribbons. This behavior is due to the magnetic interaction between edge states. We relate the reduction of the magnetization in zigzag nanoribbons with increasing ribbon width to the structural changes produced by the magnetic interaction, and show that when deposited on a substrate, zigzag bilayer ribbons remain magnetic…
In two-dimensional (2D) ferromagnets, anisotropy is essential for the magnetic ordering as dictated by the Mermin-Wagner theorem. But when competing anisotropies are present, the phase transition becomes nontrivial. Here, utilizing highly sensitive susceptometry of scanning superconducting quantum interference device microscopy, we probe the spin correlations of ABC-stacked CrBr3 under zero magnetic field. We identify a plateau feature in susceptibility above the critical temperature (Tc) in thick samples. It signifies a crossover regime induced by the competition between easy-plane intralayer…
We provide numerical evidence for composite fermion pairing in quantum Hall bilayer systems at filling $ν=1/2 + 1/2$ for intermediate spacing between the layers. We identify the phase as $p_x + i p_y$ pairing, and construct high accuracy trial wavefunctions to describe the groundstate on the sphere. For large distances between the layers, and for finite systems, a competing "Hund's rule" state, or composite fermion liquid, prevails for certain system sizes. We argue that for larger systems, the pairing phase will persist to larger layer spacing.
We investigate the effect of spin-orbit coupling on the band structure of graphene-based two-dimensional Dirac fermion gases in the quantum Hall regime. Taking monolayer graphene as our first candidate, we show that a quantum phase transition between two distinct topological states -- the quantum Hall and the quantum spin Hall phases -- can be driven by simply tuning the Fermi level with a gate voltage. This transition is characterized by the existence of a chiral spin-polarized edge state propagating along the interface separating the two topological phases. We then apply our analysis to the …
We present a symmetry-based analysis of competition between different gapped states that have been proposed in bilayer graphene (BLG), which are all degenerate on a mean field level. We classify the states in terms of a hidden SU(4) symmetry, and distinguish symmetry protected degeneracies from accidental degeneracies. One of the states, which spontaneously breaks discrete time reversal symmetry but no continuous symmetry, is identified as a Quantum Anomalous Hall (QAH) state, which exhibits quantum Hall effect at zero magnetic field. We investigate the lifting of the accidental degeneracies b…
We show that the insulating states of magic-angle twisted bilayer graphene support a series of collective modes corresponding to local particle-hole excitations on triangular lattice sites. Our theory is based on a continuum model of the magic angle flat bands. When the system is insulating at moiré band filling $ν=-3$, our calculations show that the ground state supports seven low-energy modes that lie well below the charge gap throughout the moiré Brillouin zone, one of which couples strongly to THz photons. The low-energy collective modes are faithfully described by a model with a local $SU…
We investigate the full doping and strain-dependent phase diagram of the normal state of magic-angle twisted bilayer graphene (TBG). Using comprehensive Hartree-Fock calculations, we show that at temperatures where superconductivity is absent the global phase structure can be understood based on the competition and coexistence between three types of intertwined orders: a fully symmetric phase, spatially uniform flavor-symmetry-breaking states, and an incommensurate Kekulé spiral (IKS) order. For small strain, the IKS phase, recently proposed as a candidate order at all non-zero integer filling…
Inter-layer excitonic coherence in a quantum Hall bilayer with negligible tunneling is monitored by measurements of low-lying spin excitations. At $ν_T =1$ new quasiparticle excitations are observed above a transition temperature revealing a competing metallic phase. For magnetic fields above an onset Zeeman energy this metallic phase has full spin polarization. A phase diagram in the parameter space of temperature and Zeeman energy reveals that the transition temperature increases at higher fields. This unexpected result suggests intriguing impacts of spin polarization in the highly correlate…
Recent studies evidence the emergence of asymmetric electron transport in layered conductors owing to the interplay between electrical conductivity, magnetization, and the spin Hall or Rashba- Edelstein effects. Here, we investigate the unidirectional magnetoresistance (UMR) caused by the current-induced spin accumulation in Co/Pt and CoCr/Pt bilayers. We identify three competing mechanisms underpinning the resistance asymmetry, namely interface and bulk spin-dependent electron scattering and electron-magnon scattering. Our measurements provide a consistent description of the current, magnetic…
Gapless bilayer graphene is susceptible to a variety of spontaneously gapped states. As predicted by theory and observed by experiment, the ground state is however topologically trivial, because a valley-independent gap is energetically favorable. Here, we show that under the application of interlayer electric field and circularly polarized light, one valley can be selected to exhibit the original interaction instability while the other is frozen out. Tuning this Floquet system stabilizes multiple competing topological ordered states, distinguishable by edge transport and circular dichroism. N…
Using a weak-coupling renormalization group formalism, we study competing ordered phases for repulsively interacting fermions on the bilayer honeycomb lattice away from half-filling, which is realized experimentally as doped bilayer graphene. As electrons are added to the system, excitonic order is suppressed, and unconventional superconductivity appears generically in its place. In general it is found that the maximum critical temperature for superconductivity appears directly adjacent to the dome of particle-hole order, illustrating the importance of fluctuations in these channels for the fo…
Bilayer graphene and its thicker cousins with Rhombohedral stacking have attracted considerable attention because of their susceptibility to a variety of broken chiral symmetry states. Due to large density-of-states and quantized Berry phases near their gapless band touching points, each spin-valley flavor spontaneously transfers charge between layers to yield opening of energy gaps in quasiparticle spectra and spreading of momentum-space Berry curvatures. In this article we review the development of theories that predicted such chiral symmetry breaking and classified the possible topological …
When bilayer graphene is rotationally faulted to an angle $θ\approx 1.1^\circ$, theory predicts the formation of a flat electronic band and correlated insulating, superconducting, and ferromagnetic states have all been observed at partial band filling. The proximity of superconductivity to correlated insulators has suggested a close relationship between these states, reminiscent of the cuprates where superconductivity arises by doping a Mott insulator. Here, we show that superconductivity can appear without correlated insulating states. While both superconductivity and correlated insulating be…
Ferromagnetic resonance (FMR) in exchange-coupled ferromagnet-antiferromagnet (FM/AFM) bilayers commonly shows a moderate increase in the resonance frequency owing to the induced unidirectional anisotropy. Here we report a large FMR frequency enhancement toward the sub-THz range observed in Py/FeMn with ultrathin AFM FeMn. The effect is connected with a sizable induced magnetic moment in FeMn caused by the magnetic proximity effect from the Py layer. The observed FMR properties are explained as due to the competing intrinsic antiferromagnetic order and the ferromagnetic proximity effect in nan…
In crystalline solids the interactions of charge and spin can result in a variety of emergent quantum ground states, especially in partially filled, topological flat bands such as Landau levels or in 'magic-angle' bilayer graphene. Much less explored is rhombohedral graphite (RG), perhaps the simplest and structurally most perfect condensed matter system to host a flat band protected by symmetry. By scanning tunneling microscopy we map the flat band charge density of 8, 10 and 17 layers and identify a domain structure emerging from a competition between a sublattice antiferromagnetic insulator…
SU($N$) Mott insulators have been proposed and/or realized in solid-state materials and with ultracold atoms on optical lattices. We study the two-dimensional SU($N$) antiferromagnets on the triangular lattice. Starting from an SU($N$) Heisenberg model with the fundamental representation on each site in the large-$N$ limit, we perform a self-consistent calculation and find a variety of ground states including the valence cluster states, stripe ordered states with a doubled unit-cell and topological chiral spin liquids. The system favors a cluster or ordered ground state when the number of flav…
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two- component FQH systems at total filling fraction $ν= n+ 2/3$, for integer $n$. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here, we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $ν…
We analyze the many-body properties of bilayer graphene (BLG) at charge neutrality, governed by long range interactions between electrons. Perturbation theory in a large number of flavors is used in which the interactions are described within a random phase approximation, taking account of dynamical screening effect. Crucially, the dynamically screened interaction retains some long range character, resulting in $\log^2$ renormalization of key quantities. We carry out the perturbative renormalization group calculations to one loop order, and find that BLG behaves to leading order as a marginal …
The quantum anomalous Hall (QAH) effect - a macroscopic manifestation of chiral band topology at zero magnetic field - has only been experimentally realized by magnetic doping of topological insulators (1 - 3) and delicate design of Moire heterostructures (4 - 8). However, the seemingly simple bilayer graphene without magnetic doping or Moire engineering has long been predicted to host competing ordered states with QAH effects (9 - 11). Here, we explore states in bilayer graphene with conductance of 2 e2/h that not only survive down to anomalously small magnetic fields and up to temperatures o…
Recent experiments have observed possible spin- and valley-polarized insulators and spin-triplet superconductivity in twisted double bilayer graphene, a moire structure consisting of a pair of Bernal-stacked bilayer graphene. Besides the continuously tunable band widths controlled by an applied displacement field and twist angle, these moire bands also possess van Hove singularities near the Fermi surface and a field-dependent nesting which is far from perfect. Here we carry out a perturbative renormalization group analysis to unbiasedly study the competition among all possible instabilities i…
We study magic angle graphene in the presence of both strain and particle-hole symmetry breaking due to non-local inter-layer tunneling. We perform a self-consistent Hartree-Fock study that incorporates these effects alongside realistic interaction and substrate potentials, and explore a comprehensive set of competing orders including those that break translational symmetry at arbitrary wavevectors. We find that at all non-zero integer fillings very small strains, comparable to those measured in scanning tunneling experiments, stabilize a fundamentally new type of time-reversal symmetric and s…
Magic-angle twisted bilayer graphene (tBLG) has been studied extensively owing to its wealth of symmetry-broken phases, correlated Chern insulators, orbital magnetism, and superconductivity. In particular, the anomalous Hall effect (AHE) has been observed at odd integer filling factors ($ν=1$ and $3$) in a small number of tBLG devices, indicating the emergence of a zero-field orbital magnetic state with spontaneously broken time-reversal symmetry. However, the AHE is typically not anticipated at half filling ($ν=2$) owing to competing intervalley coherent states, as well as spin-polarized and …
Owing to the spin, valley, and orbital symmetries, the lowest Landau level (LL) in bilayer graphene exhibits multicomponent quantum Hall ferromagnetism. Using transport spectroscopy, we investigate the energy gaps of integer and fractional quantum Hall states in bilayer graphene with controlled layer polarization. The state at filling factor ν=1 has two distinct phases: a layer polarized state that has a larger energy gap and is stabilized by high electric field, and a hitherto unobserved interlayer coherent state with a smaller gap that is stabilized by large magnetic field. In contrast, the …
The advent of Moiré materials has galvanized interest in the nature of charge carriers in topological bands. In contrast to conventional materials where charge carriers are electron-like quasiparticles, topological bands allow for more exotic possibilities where charge is carried by nontrivial topological textures, such as skyrmions. However, the real space description of skyrmions is ill-suited to address the limit of small or `baby' skyrmions which consist of an electron and a few spin flips. Here, we study the formation of the smallest skyrmions -- spin polarons, formed as bound states of a…
