Phonon Hall Viscosity of Ionic Crystals.

When time-reversal symmetry is broken, the low-energy description of acoustic lattice dynamics allows for a dissipationless component of the viscosity tensor, the phonon Hall viscosity, which captures how phonon chirality grows with the wave vector. In this work, we show that, in ionic crystals, a phonon Hall viscosity contribution is produced by the Lorentz forces on moving ions. We calculate typical values of the Lorentz force contribution to the Hall viscosity using a simple square lattice toy model, and we compare it with literature estimates of the strengths of other Hall-viscosity mechanisms.

Weak-Field Hall Resistivity and Spin-Valley Flavor Symmetry Breaking in Magic-Angle Twisted Bilayer Graphene.

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 Letter, 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 Shubnikov-de Haas magnetotransport measurements.

Exact Diagonalization for Magic-Angle Twisted Bilayer Graphene.

We report on finite-size exact-diagonalization calculations in a Hilbert space defined by the continuum-model flat moiré bands of magic angle twisted bilayer graphene. For moiré band filling 3>|ν|>2, where superconductivity is strongest, we obtain evidence that the ground state is a spin ferromagnet. Near |ν|=3, we find Chern insulator ground states that have spontaneous spin, valley, and sublattice polarization, and demonstrate that the anisotropy energy in this order-parameter space is strongly band-filling-factor dependent. We emphasize that inclusion of the remote band self-energy is necessary for a reliable description of magic angle twisted bilayer graphene flat band correlations.

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.

Voltage-Controlled Magnetic Reversal in Orbital Chern Insulators.

Chern insulator ferromagnets are characterized by a quantized anomalous Hall effect and have so far been identified experimentally in magnetically doped topological insulator thin films and in bilayer graphene moiré superlattices. We classify Chern insulator ferromagnets as either spin or orbital, depending on whether the orbital magnetization results from spontaneous spin polarization combined with spin-orbit interactions, as in the magnetically doped topological insulator case, or directly from spontaneous orbital currents, as in the moiré superlattice case. We argue that, in a given magnetic state, characterized, for example, by the sign of the anomalous Hall effect, the magnetization of an orbital Chern insulator will often have opposite signs for weak n and weak p electrostatic or chemical doping. This property enables pure electrical switching of a magnetic state in the presence of a fixed magnetic field.

Electrical switching of magnetic order in an orbital Chern insulator.

Magnetism typically arises from the joint effect of Fermi statistics and repulsive Coulomb interactions, which favours ground states with non-zero electron spin. As a result, controlling spin magnetism with electric fields-a longstanding technological goal in spintronics and multiferroics1,2-can be achieved only indirectly. Here we experimentally demonstrate direct electric-field control of magnetic states in an orbital Chern insulator3-6, a magnetic system in which non-trivial band topology favours long-range order of orbital angular momentum but the spins are thought to remain disordered7-14. We use van der Waals heterostructures consisting of a graphene monolayer rotationally faulted with respect to a Bernal-stacked bilayer to realize narrow and topologically non-trivial valley-projected moiré minibands15-17. At fillings of one and three electrons per moiré unit cell within these bands, we observe quantized anomalous Hall effects18 with transverse resistance approximately equal to h/2e2 (where h is Planck's constant and e is the charge on the electron), which is indicative of spontaneous polarization of the system into a single-valley-projected band with a Chern number equal to two. At a filling of three electrons per moiré unit cell, we find that the sign of the quantum anomalous Hall effect can be reversed via field-effect control of the chemical potential; moreover, this transition is hysteretic, which we use to demonstrate non-volatile electric-field-induced reversal of the magnetic state. A theoretical analysis19 indicates that the effect arises from the topological edge states, which drive a change in sign of the magnetization and thus a reversal in the favoured magnetic state. Voltage control of magnetic states can be used to electrically pattern non-volatile magnetic-domain structures hosting chiral edge states, with applications ranging from reconfigurable microwave circuit elements to ultralow-power magnetic memories.

Nature of the Correlated Insulator States in Twisted Bilayer Graphene.

We use self-consistent Hartree-Fock calculations performed in the full π-band Hilbert space to assess the nature of the recently discovered correlated insulator states in magic-angle twisted bilayer graphene (TBG). We find that gaps between the flat conduction and valence bands open at neutrality over a wide range of twist angles, sometimes without breaking the system's valley projected C_{2}T symmetry. Broken spin-valley flavor symmetries then enable gapped states to form not only at neutrality, but also at total moiré band filling n=±p/4 with integer p=1, 2, 3, when the twist angle is close to the magic value at which the flat bands are most narrow. Because the magic-angle flat band quasiparticles are isolated from remote band quasiparticles only for effective dielectric constants larger than ∼20, the gapped states do not necessarily break C_{2}T symmetry and as a consequence the insulating states at n=±1/4 and n=±3/4 need not exhibit a quantized anomalous Hall effect.

Magneto-Coulomb Drag and Hall Drag in Double-Layer Dirac Systems.

We develop a theory of Coulomb drag due to momentum transfer between graphene layers in a strong magnetic field. The theory is intended to apply in systems with disorder that is weak compared to Landau level separation, so that Landau level mixing is weak but strong compared to correlation energies within a single Landau level, so that fractional quantum Hall physics is not relevant. We find that, in contrast to the zero-field limit, the longitudinal magneto-Coulomb drag is finite and, in fact, attains a maximum at the simultaneous charge neutrality point (CNP) of both layers. Our theory also predicts a sizable Hall drag resistivity at densities away from the CNP.

Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers.

We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in ±K valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe_{2}. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe_{2} at the +K (-K) valley can be described using a two-band model with a layer-pseudospin magnetic field Δ(r) that has the moiré period. We show that Δ(r) has a topologically nontrivial skyrmion lattice texture in real space, and that the topmost moiré valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moiré cell.

Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands.

Flexible long period moiré superlattices form in two-dimensional van der Waals crystals containing layers that differ slightly in lattice constant or orientation. In this Letter we show theoretically that isolated flat moiré bands described by generalized triangular lattice Hubbard models are present in twisted transition metal dichalcogenide heterobilayers. The hopping and interaction strength parameters of the Hubbard model can be tuned by varying the twist angle and the three-dimensional dielectric environment. When the flat moiré bands are partially filled, candidate many-body ground states at some special filling factors include spin-liquid states, quantum anomalous Hall insulators, and chiral d-wave superconductors.

Electrical Reservoirs for Bilayer Excitons.

The ground state of two-dimensional (2D) electron systems with equal low densities of electrons and holes in nearby layers is an exciton fluid. We show that a reservoir for excitons can be established by contacting the two layers separately and maintaining the chemical potential difference at a value less than the spatially indirect band gap, thereby avoiding the presence of free carriers in either layer. Equilibration between the exciton fluid and the contacts proceeds via a process involving virtual intermediate states in which an unpaired electron or hole virtually occupies a free carrier state in one of the 2D layers. We derive an approximate relationship between the exciton-contact equilibration rate and the electrical conductances between the contacts and individual 2D layers when the contact chemical potentials align with the free-carrier bands, and explain how electrical measurements can be used to measure thermodynamic properties of the exciton fluids.

Time-Reversal Symmetry-Breaking Nematic Insulators near Quantum Spin Hall Phase Transitions.

We study the phase diagram of a model quantum spin Hall system as a function of band inversion and band-coupling strength, demonstrating that when band hybridization is weak, an interaction-induced nematic insulator state emerges over a wide range of band inversion. This property is a consequence of the long-range Coulomb interaction, which favors interband phase coherence that is weakly dependent on momentum and therefore frustrated by the single-particle Hamiltonian at the band inversion point. For weak band hybridization, interactions convert the continuous gap closing topological phase transition at inversion into a pair of continuous phase transitions bounding a state with broken time-reversal and rotational symmetries. At intermediate band hybridization, the topological phase transition proceeds instead via a quantum anomalous Hall insulator state, whereas at strong hybridization interactions play no role. We comment on the implications of our findings for InAs/GaSb and HgTe/CdTe quantum spin Hall systems.

Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene.

We present a theory of phonon-mediated superconductivity in near magic angle twisted bilayer graphene. Using a microscopic model for phonon coupling to moiré band electrons, we find that phonons generate attractive interactions in both s- and d-wave pairing channels and that the attraction is strong enough to explain the experimental superconducting transition temperatures. Before including Coulomb repulsion, the s-wave channel is more favorable; however, on-site Coulomb repulsion can suppress s-wave pairing relative to d wave. The pair amplitude varies spatially with the moiré period, and is identical in the two layers in the s-wave channel but phase shifted by π in the d-wave channel. We discuss experiments that can distinguish the two pairing states.

Topological Exciton Bands in Moiré Heterojunctions.

Moiré patterns are common in van der Waals heterostructures and can be used to apply periodic potentials to elementary excitations. We show that the optical absorption spectrum of transition metal dichalcogenide bilayers is profoundly altered by long period moiré patterns that introduce twist-angle dependent satellite excitonic peaks. Topological exciton bands with nonzero Chern numbers that support chiral excitonic edge states can be engineered by combining three ingredients: (i) the valley Berry phase induced by electron-hole exchange interactions, (ii) the moiré potential, and (iii) the valley Zeeman field.

Two-dimensional chiral topological superconductivity in Shiba lattices.

The chiral p-wave superconductor is the archetypal example of a state of matter that supports non-Abelian anyons, a highly desired type of exotic quasiparticle. With this, it is foundational for the distant goal of building a topological quantum computer. While some candidate materials for bulk chiral superconductors exist, they are subject of an ongoing debate about their actual paring state. Here we propose an alternative route to chiral superconductivity, consisting of the surface of an ordinary superconductor decorated with a two-dimensional lattice of magnetic impurities. We furthermore identify a promising experimental platform to realize this proposal.

SU(3) Quantum Hall Ferromagnetism in SnTe.

The (111) surface of SnTe hosts one isotropic Γ[over ¯]-centered and three degenerate anisotropic M[over ¯]-centered Dirac surface states. We predict that a nematic phase with spontaneously broken C_{3} symmetry will occur in the presence of a perpendicular magnetic field when the N=0 M[over ¯] Landau levels are 1/3 or 2/3 filled. The nematic state phase boundary is controlled by a competition between intravalley Coulomb interactions that favor a valley-polarized state and weaker intervalley scattering processes that increase in relative strength with magnetic field. An in-plane Zeeman field alters the phase diagram by lifting the threefold M[over ¯] Landau-level degeneracy, yielding a ground state energy with 2π/3 periodicity as a function of Zeeman-field orientation angle.

Spontaneous layer-pseudospin domain walls in bilayer graphene.

Bilayer graphene is susceptible to a family of unusual broken symmetry states with spin and valley dependent layer polarization. We report on a microscopic study of the domain walls in these systems, demonstrating that they have interesting microscopic structure related to the topological character of the ordered states. We use our results to show that the metal-insulator transition temperature in bilayer graphene is reduced from mean-field estimates by thermal excitation of domain walls.

Competing ordered states with filling factor two in bilayer graphene.

The quantum Hall effect, in which a two-dimensional sample's Hall conductivities become quantized, is a remarkable transport anomaly commonly observed at strong magnetic fields. However, it may also appear at zero magnetic field if time-reversal symmetry is broken. Charge-neutral bilayer graphene is unstable to a variety of competing and closely related broken symmetry states, some of which have non-zero quantized Hall conductivities. Here we explore those states by stabilizing them with external fields. Transport spectroscopy measurements reveal two distinct states that have two quantum units of Hall conductivity, stabilized by large magnetic and electric fields, respectively. The majority spins of both phases form a quantum anomalous Hall state, and the minority spins constitute a Kekulé state with spontaneous valley coherence for phase I and a quantum valley Hall state for phase II. Our results shed light on the rich set of competing ordered states in bilayer graphene.

Impurity cyclotron resonance of anomalous Dirac electrons in graphene.

We have investigated a new feature of impurity cyclotron resonances common to various localized potentials of graphene. A localized potential can interact with a magnetic field in an unexpected way in graphene. It can lead to formation of anomalous boundstates that have a sharp peak with a width R in the probability density inside the potential and a broad peak of size magnetic length ℓ outside the potential. We investigate optical matrix elements of anomalous states and find that they are unusually small and depend sensitively on the magnetic field. The effect of many-body interactions on their optical conductivity is investigated using a self-consistent time-dependent Hartree-Fock approach. For a completely filled Landau level we find that an excited electron-hole pair, originating from the optical transition between two anomalous impurity states, is nearly uncorrelated with other electron-hole pairs, although it displays substantial exchange self-energy effects. This absence of correlation is a consequence of a small vertex correction in comparison to the difference between renormalized transition energies computed within the one electron-hole pair approximation. However, an excited electron-hole pair originating from the optical transition between a normal and an anomalous impurity state can be substantially correlated with other electron-hole states with a significant optical strength.

Broken SU(4) symmetry and the fractional quantum Hall effect in graphene.

We describe a variational theory for incompressible ground states and charge gaps in the N=0 Landau level of graphene that accounts for the fourfold Landau level degeneracy and the short-range interactions that break SU(4) spin-valley invariance. Our approach explains the experimental finding that gaps at odd numerators are weak for 1<|ν|<2 and strong for 0<|ν|<1. We find that in the SU(4) invariant case the incompressible ground state at |ν|=1/3 is a three-component incompressible state, not the Laughlin state, and discuss the competition between these two states in the presence of SU(4) spin-valley symmetry-breaking terms.