How to run the code

Open Julia with

julia -t 16

to obtain 16 threads (choose the number of threads you want to run). Call

include("apply_graphene.jl")

to produce the monolayer functions u1, u2, V and Vint. Call

include("apply_effpot.jl")

to produce the effective potentials and compute the bands of the effective Hamiltonian. Computation of Vint and band diagrams production are the two steps requiring more ressources, they are paralellized, naturally on DFTK for the first case, and with Threads.@threads on the CPU for band diagrams.

To produce the band diagrams of the effective model and of the BM model, run

include("apply_bands.jl")

Installation

In the pkg mode that one can enter with writing ], one can add the necessary packages by

add FFTW, DFTK, JLD, LaTeXStrings, Plots, CairoMakie, DelimitedFiles, DataFrames, Optim, BlockArrays

The code works with DFTK v0.5.10, with more recent versions it could be needed to update some DFTK functions calls

Organization of the code

Scripts to compute the monolayer functions

graphene.jl computes the potentials and Dirac eigenfunctions of the monolayer graphene, and the KS potential of the shifted bilayer Vint

apply_graphene.jl applies the functions of the previous script, with some definite parameters, and exports the functions. It computes SCF of the monolayer to get the Dirac eigenvalues and the effective potential, verifies their symmetries (2pi/3 rotation, mirror, z -> -z), exports them, and computes Vint which is the long step

Scripts to compute effective potentials

effective_potentials.jl computes the effective potentials

apply_effpot.jl applies the previous script with definite parameters. It builds the effective potentials, verifies symmetries on them (particle-hole, translation, parity-time, etc), and plots them

effpots_against_d.jl creates the figures for a study of effective potentials quantities against d, corresponding to the article "A simple derivation of moiré-scale continuous models for twisted bilayer graphene"

Scripts to compute bands

band_diagrams_bm_like.jl assembles the effective Hamiltonian and computes the band diagrams

apply_bands.jl applies the previous script functions. It will either create and plot the band diagram for some definite value of angle theta or do a study of the bandwidth, band gap and Fermi velocity with respect to theta

Secondary scripts

misc/lobpcg.jl is the LOBPCG solver taken from DFTK

misc/create_bm_pot.jl enables to create the true Bistritzer-MacDonald potential, for comparisions purposes

common_functions.jl are low-level functions shared by all the main scripts

Details

misc/numerical_details.pdf contains more information about the implementation

Contact

If you have a question, for instance to run the code, it will be a pleasure to answer you, so please contact me at louis.garrigue@gmail.com