Berry Curvature Spectroscopy from Bloch Oscillations.

Artificial crystals such as moiré superlattices can have a real-space periodicity much larger than the underlying atomic scale. This facilitates the presence of Bloch oscillations in the presence of a static electric field. We demonstrate that the optical response of such a system, when dressed with a static field, becomes resonant at the frequencies of Bloch oscillations, which are in the terahertz regime when the lattice constant is of the order of 10 nm. In particular, we show within a semiclassical band-projected theory that resonances in the dressed Hall conductivity are proportional to the lattice Fourier components of the Berry curvature. We illustrate our results with a low-energy model on an effective honeycomb lattice.

Roses in the nonperturbative current response of artificial crystals.

In two-dimensional artificial crystals with large real-space periodicity, the nonlinear current response to a large applied electric field can feature a strong angular dependence, which encodes information about the band dispersion and Berry curvature of isolated electronic Bloch minibands. Within the relaxation-time approximation, we obtain analytic expressions up to infinite order in the driving field for the current in a band-projected theory with time-reversal and trigonal symmetry. For a fixed field strength, the dependence of the current on the direction of the applied field is given by rose curves whose petal structure is symmetry constrained and is obtained from an expansion in real-space translation vectors. We illustrate our theory with calculations on periodically buckled graphene and twisted double bilayer graphene, wherein the discussed physics can be accessed at experimentally relevant field strengths.

Quantum Geometric Oscillations in Two-Dimensional Flat-Band Solids.

Two-dimensional van der Waals heterostructures can be engineered into artificial superlattices that host flat bands with significant Berry curvature and provide a favorable environment for the emergence of novel electron dynamics. In particular, the Berry curvature can induce an oscillating trajectory of an electron wave packet transverse to an applied static electric field. Though analogous to Bloch oscillations, this novel oscillatory behavior is driven entirely by quantum geometry in momentum space instead of band dispersion. While the current from Bloch oscillations can be localized by increasing field strength, the current from the geometric orbits saturates to a nonzero plateau in the strong-field limit. In nonmagnetic materials, the geometric oscillations are even under inversion of the applied field, whereas the Bloch oscillations are odd, a property that can be used to distinguish these two coexisting effects.

Boundary Modes from Periodic Magnetic and Pseudomagnetic Fields in Graphene.

Single-layer graphene subject to periodic lateral strains is an artificial crystal that can support boundary spectra with an intrinsic polarity. This is analyzed by comparing the effects of periodic magnetic fields and strain-induced pseudomagnetic fields that, respectively, break and preserve time-reversal symmetry. In the former case, a Chern classification of the superlattice minibands with zero total magnetic flux enforces single counterpropagating modes traversing each bulk gap on opposite boundaries of a nanoribbon. For the pseudomagnetic field, pairs of counterpropagating modes migrate to the same boundary where they provide well-developed valley-helical transport channels on a single zigzag edge. We discuss possible schemes for implementing this situation and their experimental signatures.

Imaging the Néel vector switching in the monolayer antiferromagnet MnPSe3 with strain-controlled Ising order.

Antiferromagnets are interesting materials for spintronics because of their faster dynamics and robustness against perturbations from magnetic fields. Control of the antiferromagnetic order constitutes an important step towards applications, but has been limited to bulk materials so far. Here, using spatially resolved second-harmonic generation, we show direct evidence of long-range antiferromagnetic order and Ising-type Néel vector switching in monolayer MnPSe3 with large XY anisotropy. In additional to thermally induced switching, uniaxial strain can rotate the Néel vector, aligning it to a general in-plane direction irrespective of the crystal axes. A change of the universality class of the phase transition in the XY model under uniaxial strain causes this emergence of strain-controlled Ising order in the XY magnet MnPSe3. Our discovery is a further ingredient for compact antiferromagnetic spintronic devices in the two-dimensional limit.

Giant topological longitudinal circular photo-galvanic effect in the chiral multifold semimetal CoSi.

The absence of mirror symmetry, or chirality, is behind striking natural phenomena found in systems as diverse as DNA and crystalline solids. A remarkable example occurs when chiral semimetals with topologically protected band degeneracies are illuminated with circularly polarized light. Under the right conditions, the part of the generated photocurrent that switches sign upon reversal of the light's polarization, known as the circular photo-galvanic effect, is predicted to depend only on fundamental constants. The conditions to observe quantization are non-universal, and depend on material parameters and the incident frequency. In this work, we perform terahertz emission spectroscopy with tunable photon energy from 0.2 -1.1 eV in the chiral topological semimetal CoSi. We identify a large longitudinal photocurrent peaked at 0.4 eV reaching  ~550 µ A/V2, which is much larger than the photocurrent in any chiral crystal reported in the literature. Using first-principles calculations we establish that the peak originates only from topological band crossings, reaching 3.3 ± 0.3 in units of the quantization constant. Our calculations indicate that the quantized circular photo-galvanic effect is within reach in CoSi upon doping and increase of the hot-carrier lifetime. The large photo-conductivity suggests that topological semimetals could potentially be used as novel mid-infrared detectors.

Obstruction and Interference in Low-Energy Models for Twisted Bilayer Graphene.

The electronic bands of twisted bilayer graphene (TBLG) with a large-period moiré superlattice fracture to form narrow Bloch minibands that are spectrally isolated by forbidden energy gaps from remote dispersive bands. When these gaps are sufficiently large, one can study a band-projected Hamiltonian that correctly represents the dynamics within the minibands. This inevitably introduces nontrivial geometrical constraints that arise from the assumed form of the projection. Here we show that this choice has a profound consequence in a low-energy experimentally observable signature that therefore can be used to tightly constrain the analytic form of the appropriate low-energy theory. We find that this can be accomplished by a careful analysis of the electron density produced by backscattering of Bloch waves from an impurity potential localized on the moiré superlattice scale. We provide numerical estimates of the effect that can guide experimental work to clearly discriminate between competing models for the low-energy band structure.

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.

Optically Controlled Orbitronics on a Triangular Lattice.

The propagation of electrons in an orbital multiplet dispersing on a lattice can support anomalous transport phenomena deriving from an orbitally induced Berry curvature. In striking contrast to the related situation in graphene, we find that anomalous transport for an L=1 multiplet on the primitive 2D triangular lattice is activated by easily implemented on site and optically tunable potentials. We demonstrate this for dynamics in a Bloch band where point degeneracies carrying opposite winding numbers are generically offset in energy, allowing both an anomalous charge Hall conductance with the sign selected by off-resonance coupling to circularly polarized light and a related anomalous orbital Hall conductance activated by layer buckling.

Photonic crystal for graphene plasmons.

Photonic crystals are commonly implemented in media with periodically varying optical properties. Photonic crystals enable exquisite control of light propagation in integrated optical circuits, and also emulate advanced physical concepts. However, common photonic crystals are unfit for in-operando on/off controls. We overcome this limitation and demonstrate a broadly tunable two-dimensional photonic crystal for surface plasmon polaritons. Our platform consists of a continuous graphene monolayer integrated in a back-gated platform with nano-structured gate insulators. Infrared nano-imaging reveals the formation of a photonic bandgap and strong modulation of the local plasmonic density of states that can be turned on/off or gradually tuned by the applied gate voltage. We also implement an artificial domain wall which supports highly confined one-dimensional plasmonic modes. Our electrostatically-tunable photonic crystals are derived from standard metal oxide semiconductor field effect transistor technology and pave a way for practical on-chip light manipulation.

Dirac-Weyl Semimetal: Coexistence of Dirac and Weyl Fermions in Polar Hexagonal ABC Crystals.

We propose that the noncentrosymmetric LiGaGe-type hexagonal ABC crystal SrHgPb realizes a new type of topological semimetal that hosts both Dirac and Weyl points in momentum space. The symmetry-protected Dirac points arise due to a band inversion and are located on the sixfold rotation z axis, whereas the six pairs of Weyl points related by sixfold symmetry are located on the perpendicular k_{z}=0 plane. By studying the electronic structure as a function of the buckling of the HgPb layer, which is the origin of inversion symmetry breaking, we establish that the coexistence of Dirac and Weyl fermions defines a phase separating two topologically distinct Dirac semimetals. These two Dirac semimetals are distinguished by the Z_{2} index of the k_{z}=0 plane and the corresponding presence or absence of 2D Dirac fermions on side surfaces. We formalize our first-principles calculations by deriving and studying a low-energy model Hamiltonian describing the Dirac-Weyl semimetal phase. We conclude by proposing several other materials in the noncentrosymmetric ABC material class, in particular SrHgSn and CaHgSn, as candidates for realizing the Dirac-Weyl semimetal.

Electrodynamics on Fermi Cyclides in Nodal Line Semimetals.

We study the frequency-dependent conductivity of nodal line semimetals (NLSMs), focusing on the effects of carrier density and energy dispersion on the nodal line. We find that the low-frequency conductivity has a rich spectral structure which can be understood using scaling rules derived from the geometry of their Dupin cyclide Fermi surfaces. We identify different frequency regimes, find scaling rules for the optical conductivity in each, and demonstrate them with numerical calculations of the inter- and intraband contributions to the optical conductivity using a low-energy model for a generic NLSM.

Bending Rules in Graphene Kirigami.

The three-dimensional shapes of graphene sheets produced by nanoscale cut-and-join kirigami are studied by combining large-scale atomistic simulations with continuum elastic modeling. Lattice segments are selectively removed from a graphene sheet, and the structure is allowed to close by relaxing in the third dimension. The surface relaxation is limited by a nonzero bending modulus which produces a smoothly modulated landscape instead of the ridge-and-plateau motif found in macroscopic lattice kirigami. The resulting surface shapes and their interactions are well described by a new set of microscopic kirigami rules that resolve the competition between bending and stretching energies.

Layered Topological Crystalline Insulators.

Topological crystalline insulators (TCIs) are insulating materials whose topological property relies on generic crystalline symmetries. Based on first-principles calculations, we study a three-dimensional (3D) crystal constructed by stacking two-dimensional TCI layers. Depending on the interlayer interaction, the layered crystal can realize diverse 3D topological phases characterized by two mirror Chern numbers (MCNs) (µ1,µ2) defined on inequivalent mirror-invariant planes in the Brillouin zone. As an example, we demonstrate that new TCI phases can be realized in layered materials such as a PbSe (001) monolayer/h-BN heterostructure and can be tuned by mechanical strain. Our results shed light on the role of the MCNs on inequivalent mirror-symmetric planes in reciprocal space and open new possibilities for finding new topological materials.

Bulk Dirac points in distorted spinels.

We report on a Dirac-like Fermi surface in three-dimensional bulk materials in a distorted spinel structure on the basis of density functional theory as well as tight-binding theory. The four examples we provide in this Letter are BiZnSiO4, BiCaSiO4, BiAlInO4, and BiMgSiO4. A necessary characteristic of these structures is that they contain a Bi lattice which forms a hierarchy of chainlike substructures, with consequences for both fundamental understanding and materials design.

Time-reversal-invariant topological superconductivity and Majorana Kramers pairs.

We propose a feasible route to engineer one- and two-dimensional time-reversal-invariant topological superconductors (SCs) via proximity effects between nodeless s(±) wave iron-based SCs and semiconductors with large Rashba spin-orbit interactions. At the boundary of a time-reversal-invariant topological SC, there emerges a Kramers pair of Majorana edge (bound) states. For a Josephson π junction, we predict a Majorana quartet that is protected by mirror symmetry and leads to a mirror fractional Josephson effect. We analyze the evolution of the Majorana pair in Zeeman fields, as the SC undergoes a symmetry class change as well as topological phase transitions, providing an experimental signature in tunneling spectroscopy. We briefly discuss the realization of this mechanism in candidate materials and the possibility of using s and d wave SCs and weak topological insulators.

Topological mirror superconductivity.

We demonstrate the existence of topological superconductors (SCs) protected by mirror and time-reversal symmetries. D-dimensional (D=1, 2, 3) crystalline SCs are characterized by 2(D-1) independent integer topological invariants, which take the form of mirror Berry phases. These invariants determine the distribution of Majorana modes on a mirror symmetric boundary. The parity of total mirror Berry phase is the Z(2) index of a class DIII SC, implying that a DIII topological SC with a mirror line must also be a topological mirror SC but not vice versa and that a DIII SC with a mirror plane is always time-reversal trivial but can be mirror topological. We introduce representative models and suggest experimental signatures in feasible systems. Advances in quantum computing, the case for nodal SCs, the case for class D, and topological SCs protected by rotational symmetries are pointed out.

Surface state magnetization and chiral edge states on topological insulators.

We study the interaction between a ferromagnetically ordered medium and the surface states of a topological insulator with a general surface termination that were identified recently [F. Zhang et al. Phys. Rev. B 86, 081303(R) (2012)]. This interaction is strongly crystal face dependent and can generate chiral states along edges between crystal facets even for a uniform magnetization. While magnetization parallel to quintuple layers shifts the momentum of the Dirac point, perpendicular magnetization lifts the Kramers degeneracy at any Dirac points except on the side face, where the spectrum remains gapless and the Hall conductivity switches sign. Chiral states can be found at any edge that reverses the projection of the surface normal to the stacking direction of quintuple layers. Magnetization also weakly hybridizes noncleavage surfaces.

Dirac semimetal in three dimensions.

We show that the pseudorelativistic physics of graphene near the Fermi level can be extended to three dimensional (3D) materials. Unlike in phase transitions from inversion symmetric topological to normal insulators, we show that particular space groups also allow 3D Dirac points as symmetry protected degeneracies. We provide criteria necessary to identify these groups and, as an example, present ab initio calculations of ß-cristobalite BiO(2) which exhibits three Dirac points at the Fermi level. We find that ß-cristobalite BiO(2) is metastable, so it can be physically realized as a 3D analog to graphene.