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Tutorial 4 — Twisted Bilayer Graphene

Primary notebook: extra/tutorial/Tutorial4_Twisted.ipynb Experimental variants: Tutorial4_Twisted2.ipynb, Tutorial4_Twisted3.ipynb

This tutorial family covers moiré superlattices and the resource-saving tricks needed to handle large twisted bilayer graphene (TBG) systems. The main notebook focuses on a validated workflow, while the follow-up variants capture ongoing experiments—treat them as provisional until you reproduce the outputs.

Learning goals

  • Construct twisted honeycomb geometries with Geometries.honeycomb_twisted(θ_steps).
  • Assemble continuum-inspired tight-binding models tuned for small twist angles.
  • Balance accuracy and runtime by adjusting reciprocal cutoffs, symmetry reductions, and sparse/dense representations.
  • Post-process bandstructures and real-space density profiles for moiré cells.

Prerequisites

  • Comfort with Tutorials 1–3.
  • Access to sufficient RAM/CPU; large twist indices can require several GB.

Workflow outline

  1. Geometry generation — Pick a twist index (e.g. θ = 21) and generate bilayer lattices with precise stacking.
  2. Hamiltonian strategy — Decide between tight-binding truncation and continuum approximations; the notebook demonstrates both.
  3. Symmetry reduction — Use point-group symmetries and Brillouin-zone sampling strategies to limit the matrix dimension.
  4. Band computation — Leverage Spectrum.getbands with sparse matrices or switch to iterative solvers when dense diagonalisation is infeasible.
  5. Observables — Inspect layer and valley content with custom operators; export data for SCF refinement or visualisation.

Live example

figdir = joinpath(pwd(), "figures")
mkpath(figdir)
println("Output figures stored in ", figdir)
Output figures stored in /home/runner/work/LatticeQM.jl/LatticeQM.jl/extra/docs/build/tutorials/figures
println("Number of sites in mini Brillouin zone = ", size(positions, 2))
println("Twist lattice vectors:")
Structure.Lattices.getA(lat_twisted)
3×2 Matrix{Float64}:
 19.5        9.0
  0.866025  17.3205
  0.0        0.0
mask_pos = positions[3, :] .>= 0
mask_neg = .!mask_pos
p = plot(size=(440, 380), aspect_ratio=1, xlabel="x (a)", ylabel="y (a)", legend=false, title="Twisted bilayer moiré cell (N=6)")
scatter!(positions[1, mask_pos], positions[2, mask_pos]; ms=3.0, markerstrokewidth=0, color=:royalblue)
scatter!(positions[1, mask_neg], positions[2, mask_neg]; ms=3.0, markerstrokewidth=0, color=:tomato)
savefig(p, joinpath(figdir, "tbg_positions.svg"))
nothing

Band structure (N=5)

lat5 = Geometries.honeycomb_twisted(5, 1.0, 3.0)
H5 = Operators.graphene(lat5; format=:sparse, cellrange=2)
ks5 = Structure.kpath(lat5; num_points=140)
b5 = Spectrum.getbands(H5, ks5; format=:sparse, num_bands=40)
p_bands = plot(b5; size=(520, 360), title="TBG bands (N=5)")
savefig(p_bands, joinpath(figdir, "tbg_bands_N5.svg"))
nothing

Diagonalization   1%|▌                                   |  ETA: 0:04:05
Diagonalization  26%|█████████▎                          |  ETA: 0:00:13
Diagonalization  51%|██████████████████▎                 |  ETA: 0:00:05
Diagonalization  76%|███████████████████████████▎        |  ETA: 0:00:02
Diagonalization 100%|████████████████████████████████████| Time: 0:00:07

Validation checklist

  • Reproduce low-energy flat bands around the magic angle used in the notebook.
  • Monitor memory usage; if it spikes unexpectedly, confirm sparse conversions are applied (SparseHops or DenseHops as appropriate).
  • Archive resulting figures and data files under extra/tutorial/output/tbg/ (or similar) for comparison across reruns.

Common pitfalls

  • Large twist indices explode the matrix size. Start with smaller N and increase gradually, using format=:sparse and limiting num_bands.
  • Ensure the k-path is defined for the reduced Brillouin zone of the moiré lattice; mismatches cause misleading band plots.

Handling experimental notebooks (Tutorial4_Twisted2/3)

  • Expect partially completed code paths; read cell notes carefully.
  • When results differ from baseline expectations, note findings in your project tracker and decide whether to promote changes into the main tutorial.
  • Promote stable improvements back into Tutorial4_Twisted.ipynb once validated, then retire the experimental copy.

Suggested extensions

  • Couple the workflow to the twistedgraphene_scf example for mean-field studies.
  • Prepare SLURM job scripts using the templates in extra/examples/twistedgraphene_slurm.
  • Compare to Meanfield observables to investigate superconducting phases.