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# marekyggdrasil/run.py

Last active November 24, 2022 17:29
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Simulation of the 1D Tight-Binding Model, full tutorial https://mareknarozniak.com/2020/05/07/tight-binding/
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 import matplotlib.pyplot as plt import numpy as np from qutip import basis def _ket(n, N): return basis(N, n) def _bra(n, N): return basis(N, n).dag() # spacing between atoms a = 2 # number of atoms N = 23 # total length of the 1D chain L = N*a # onsite energy eps = 0.25 # hopping matrix element t = 0.5 # construct the Hamiltonian H = sum([eps*_ket(n, L)*_bra(n, L) for n in range(0, L)]) H -= sum([t*_ket(n, L)*_bra(n + 1, L) for n in range(0, L - 1)]) H -= sum([t*_ket(n, L)*_bra(n - 1, L) for n in range(1, L)]) if N <= 8: print('Hamiltonian') print(H) evals, ekets = H.eigenstates() if N <= 8: print() print('Eigenvalues') print(evals) # satisfy periodic boundary conditions we need E(-k) = E(k) numerical = np.concatenate((np.flip(evals, 0), evals), axis=0) k = np.linspace(-np.pi/a, np.pi/a, 2*L) exact = eps - 2.*t*np.cos(k*a) xticks = np.linspace(-np.pi/a, np.pi/a, 9) xlabels = ['' for k in xticks] xlabels[0] = '$-\\frac{\pi}{a}$' xlabels[-1] = '$\\frac{\pi}{a}$' fig, axs = plt.subplots() axs.set_xlim(-np.pi/a, np.pi/a) axs.set_title('Tight-Binding Model, $N='+str(N)+', a='+str(a)+'$') axs.set_ylabel('$E$') axs.set_xlabel('$ka$') axs.axvline(x=0., color='k') axs.plot(k, numerical, 'ro', label='Eigenvalues of H') axs.plot(k, exact, label='$E(k) = \epsilon_0 - 2 t \cos(ka)$') axs.set_yticks([eps-2.*t, eps-t, eps, eps+t, eps+2.*t]) axs.set_yticklabels(['$\epsilon_0-2t$', '$\epsilon_0-t$', '$\epsilon_0$', '$\epsilon_0+t$', '$\epsilon_0+2t$']) axs.set_xticks(xticks) axs.set_xticklabels(xlabels) axs.legend() axs.grid(True) # fig.savefig('tight_binding_theory.png') fig.savefig('tight_binding_diag.png')