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.

Topologically Protected Helical States in Minimally Twisted Bilayer Graphene.

In minimally twisted bilayer graphene, a moiré pattern consisting of AB and BA stacking regions separated by domain walls forms. These domain walls are predicted to support counterpropogating topologically protected helical (TPH) edge states when the AB and BA regions are gapped. We fabricate designer moiré crystals with wavelengths longer than 50 nm and demonstrate the emergence of TPH states on the domain wall network by scanning tunneling spectroscopy measurements. We observe a double-line profile of the TPH states on the domain walls, only occurring when the AB and BA regions are gapped. Our results demonstrate a practical and flexible method for TPH state network construction.

Tunable Γ-K Valley Populations in Hole-Doped Trilayer WSe_{2}.

We present a combined experimental and theoretical study of valley populations in the valence bands of trilayer WSe_{2}. Shubnikov-de Haas oscillations show that trilayer holes populate two distinct subbands associated with the K and Γ valleys, with effective masses 0.5m_{e} and 1.2m_{e}, respectively; m_{e} is the bare electron mass. At a fixed total hole density, an applied transverse electric field transfers holes from Γ orbitals to K orbitals. We are able to explain this behavior in terms of the larger layer polarizability of the K orbital subband.

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.

Complex Quasi-Two-Dimensional Crystalline Order Embedded in VO_{2} and Other Crystals.

Metal oxides such as VO_{2} undergo structural transitions to low-symmetry phases characterized by intricate crystalline order, accompanied by rich electronic behavior. We derive a minimal ionic Hamiltonian based on symmetry and local energetics which describes structural transitions involving all four observed phases, in the correct order. An exact analysis shows that complexity results from the symmetry-induced constraints of the parent phase, which forces ionic displacements to form multiple interpenetrating groups using low-dimensional pathways and distant neighbors. Displacements within each group exhibit independent, quasi-two-dimensional order, which is frustrated and fragile. This selective ordering mechanism is not restricted to VO_{2}: it applies to other oxides that show similar complex order.