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Title: Twist-tuned quantum criticality in moiré bilayer graphene
Authors: Jan Biedermann and Lukas Janssen
Reference: arXiv:2412.16042
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Description:
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The archive contains the data used to construct Figs. 1-4 of the paper "Twist-tuned quantum criticality in moiré bilayer graphene" by J. Biedermann and L. Janssen [arXiv:2412.16042].

All data is stored in the form of one or two-dimensional arrays in the .txt format. Rows correspond to the first axis. Columns, if they exist, correspond to the second axis. The arrays may be conveniently accessed using the numpy.loadtxt function of Python's NumPy library.

The data is organized in three folders, corresponding to Fig. 1, Fig. 2, and a joint folder for Figs. 3 and 4.


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Parameters:
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Bistritzer-MacDonald Model:
    Interlayer hopping: intrasublattice w0 = 80 meV, intersublattice w1 = 110 meV
    Intralayer hopping: t0 = 2800 meV
    Nearest-neighbor distance: a0 = 0.142 nm

Interaction:
    Gate distance: d = 4 ... 40 nm (unless otherwise specified, a value of d = 20 nm is used)
    Substrate permittivity: epsilon_{substrate} = 5
    Phenomenological parameters for internal screening [Eq. (A8)]: c1 = 0.305, c2 = 3.38, c3 = 39.0, theta_{star} = 1.068° * pi / 180°
    Subtraction scheme: decoupled scheme
    Gate type: double gated
    Number of active bands per spin and valley: N_b = 2 for the insets of Fig. 3, N_b = 6 otherwise

Numerics:
    Grid size: L = 12, 15, 18 (unless otherwise specified, a value of L = 18 is used)
    Bistritzer-MacDonald plane-wave cutoff: |G| <= 3 * |G1| (corresponds to 148 BM bands)
    Coulomb potential cutoff: |G| <= 2 * |G1|
    Hartree-Fock threshold: max_k( ||P_{n + 1}(k) - P_n(k)|| ) <= 1e-4,
        where P is the density matrix, n labels the Hartree-Fock iteration,
        k denotes momentum in the moiré Brillouin Zone and ||.|| is the Frobenius norm


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Fig_1:
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The folder contains the files phis_L.txt and deltas_L.txt, with L = 12, 15, 18. The former contain results for the dimensionless KIVC order parameter, the latter contain results for the band gap in units of eV.

All file contents are in the form of two-dimensional arrays. The first dimension corresponds to the twist angle, the second dimension is the gate distance.


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Fig_2:
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The folder contains the files energies_BM_theta.txt and energies_HF_theta.txt, with theta = 105, 120, 140 corresponding to twist angles 1.05°, 1.2° and 1.4°, respectively. The files energies_BM_theta.txt contain the non-interacting Bistritzer-MacDonald bands. The files energies_HF_theta.txt contain the Hartree-Fock dispersion. 

The energies are stored in units of eV. All files contain two-dimensional arrays, where the first dimension corresponds to the band index and the second dimension is the index of the k-point.


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Fig_3_Fig_4:
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The folder contains the two subfolders "observables" and "density_matrices"

The subfolder observables contains the files phis_L.txt and deltas_L.txt, with L = 12, 15, 18. The former contain results for the dimensionless KIVC order parameter, the latter contain results for the band gap in units of meV. Both are stored as a function of the twist angle in a one-dimensional array.

The subfolder density_matrices contains the files P_KIVC.txt and P_DSM.txt, corresponding to the gamma point density matrices used to generate the insets of Fig. 3. Both files contain complex arrays of shape 4x4, where the array indices correspond to the flavor labels in the insets.