Spectrum with OBC and PBC of an amorphous lattice

In some cases, we might want to look at the energy spectrum of a system that is not translationally invariant, such as amorphous solids. From the perspective of topology, to identify whether the solid is topological or not we can compare the energy spectra with both OBC and PBC. In the PBC we expect to see a gap for a topological insulator, while with OBC the gap should be closed because of the presence of edge states.

spectrum_obc_v_pbc.py

from tightbinder.models import AgarwalaChern
from tightbinder.disorder import amorphize
import matplotlib.pyplot as plt
import numpy as np

def main():

    ncells = 6
    disorder = 0.2
    m = -1 # Regulates the topological behaviour
    cutoff = 1.3 # Cutoff distance to identify neighbours
    np.random.seed(1)

    # Initialize model and amorphize the supercell
    model = AgarwalaChern(m=m, r=cutoff).supercell(n1=ncells, n2=ncells)
    model = amorphize(model, disorder)

    # Obtain the spectrum of the model as it is (PBC)
    model.initialize_hamiltonian()
    results_pbc = model.solve()

    # Change the boundary condition and obtain the OBC spectrum
    model.boundary = "OBC"
    model.initialize_hamiltonian()
    results_obc = model.solve()

    # Plot both spectra into the same Axes object
    fig, ax = plt.subplots(1, 1)
    n = range(model.basisdim)
    ax.scatter(n, results_pbc.eigen_energy, marker="o", edgecolors="black", label="PBC", zorder=2.2)
    ax.scatter(n, results_obc.eigen_energy, marker="o", edgecolors="black", label="OBC", zorder=2.1)
    ax.grid("on")
    ax.set_xlabel(r"$n$")
    ax.set_ylabel(r"$E$ (eV)")
    ax.legend()


if __name__ == "__main__":
    main()
    plt.show()

This produces the following plot: