Twisted Bilayer Graphene

Two sheets of graphene that are rotated relative to each other. Graphene has two sublattices $A$ and $B$ , and forms a honeycomb lattice. High symmetry points in the BZ are denoted by $\Gamma$ at the center and $K$/$K^{\prime }$ at the edge of the BZ that are related by TRS.

Low energy effective (Dirac) Hamiltonian. $H_{\tau }(\vec{p})=v_F(\tau p_x{\sigma }_x+p_y{\sigma }_y)$ Breaking $C_2$ symmetry will gap the Dirac points, generating Berry curvature. Berry flux of $\pi$ and $-\pi$ in conduction and valence bands respectively. To generate Chern bands, need TRS breaking Haldane mass.

Moire periodicity

Period is much larger than single layer.

• AA stacking
• BA stacking
• AB Stacking Momenta related by a Moire reciprocal lattice vector are equivalent. Mixing between different layers.

Moire bands will be folded. Bands come in spin and valley ‘flavors’. Filling factor $\nu$ measures number of electrons per moire unit cell.

Flat bands and magic angle

Layers decoupled will have Dirac points at $K_M^{\prime }$ and $K_M$. They will intercept at $\approx 100$ meV between the Dirac points. Results in avoided crossings with magnitude of roughly the interlayer hopping $t_{\perp }\approx 100meV$. This will flatten the bands since the energy scales match. Correlation effects occur because of band flatness.

Dirac points remain, and they have the same winding number. Apply sublattice mass ${\sigma }_z$ and the Dirac points will gap, further flattening the bands, and generating opposite Chern numbers for lower and upper bands.

Ref: Intrinsic Quantized AHE in …

Bistritzer-MacDonald model

$H_K^{BM}(\vec{k},{\vec{k}}^{\prime })=\left(\begin{array}{cc}{v}_F(\vec{k}-{\vec{K}}_D^{(1)})\cdot \sigma & T(\vec{k},{\vec{k}}^{\prime })\\ T^{†}& v_F(k-K_D^{(2)})\cdot \sigma \end{array}\right)$ with $T$ being interlayer hopping.

Superconductivity

At low temperatures, certain angles, and for certain densities (filling), can have superconducting phases.

Unknowns

• How do phonons affect the Moire material?
• Twist angle disorder
• Chiral ratio/modelling (gap from other bands)

Incommensurate Kekule spiral

• IKS dominates all non-zero integers with strain
• explains (semi) metallic behavior at $\nu =0,1$

Other materials

• Transition metal dichalcogenides
• Twisted magnets ($Cr{I}_3$ )
• Twisted cuprates
• Variations on graphene
  • alternating-twist graphene multilayers
  • $M+N$ twisted graphene multilayers
  • helical multilayer graphene
  • multilayer graphene + $hBN/TMD$

Fractional Chern Insulator

TBG is predicted to exhibit the Fractional Quantum Hall Effect.

Fractal energy spectrum

TBG is found to exhibit Hofstader butterfly spectrum.