94 lines
4.1 KiB
Python
94 lines
4.1 KiB
Python
import numpy as np
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from math import *
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import cmath
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import functools
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import guan
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# Installation: pip install --upgrade guan
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def main():
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M = 0
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t1 = 1
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phi= pi/2
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t2 = 0.03
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tc = 0.31
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bilayer_bands(M, t1, t2, phi, tc)
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chern_number(M, t1, t2, phi)
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def hamiltonian_of_Haldane(k1, k2, M, t1, t2, phi, a=1/sqrt(3)):
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h0 = np.zeros((2, 2), dtype=complex)
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h1 = np.zeros((2, 2), dtype=complex)
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h2 = np.zeros((2, 2), dtype=complex)
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h1[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a))
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h1[0, 1] = h1[0, 1].conj()
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h2[0, 0] = t2*cmath.exp(1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
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h2[1, 1] = t2*cmath.exp(-1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
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matrix = h0 + h1 + h2 + h2.transpose().conj()
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return matrix
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def hamiltonian_of_modified_Haldane(k1, k2, M, t1, t2, phi, a=1/sqrt(3)):
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h0 = np.zeros((2, 2), dtype=complex)
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h1 = np.zeros((2, 2), dtype=complex)
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h2 = np.zeros((2, 2), dtype=complex)
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k1 = -k1 # kx # Note that to get the unit cell the bilayer honeycomb lattice containing all possible layer hoppings, here one layer (HM layer or mHM layer） needs to be applied with its counterpart of mirror symmetry along x direction.
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h1[1, 0] = t1*(cmath.exp(1j*k2*a)+cmath.exp(1j*sqrt(3)/2*k1*a-1j/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j/2*k2*a))
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h1[0, 1] = h1[1, 0].conj()
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h2[0, 0] = t2*cmath.exp(1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
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h2[1, 1] = t2*cmath.exp(1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
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matrix = h0 + h1 + h2 + h2.transpose().conj()
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return matrix
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def hamiltonian_of_bilayer(k1, k2, M, t1, t2, phi, tc):
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hamiltonian1 = hamiltonian_of_Haldane(k1, k2, M, t1, t2, phi, a=1/sqrt(3))
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hamiltonian2 = hamiltonian_of_modified_Haldane(k1, k2, M, t1, t2, phi, a=1/sqrt(3))
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hamiltonian = np.zeros((4, 4), dtype=complex)
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hamiltonian[0:2, 0:2] = hamiltonian1
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hamiltonian[2:4, 2:4] = hamiltonian2
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# AB stacking
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hamiltonian[1, 2] = tc # B on HM， A on mHM
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hamiltonian[2, 1] = tc
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# BA stacking
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# hamiltonian[0, 3] = tc # A on HM， B on mHM
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# hamiltonian[3, 0] = tc
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# AA stacking
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# hamiltonian[0, 2] = tc
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# hamiltonian[2, 0] = tc
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# hamiltonian[1, 3] = tc
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# hamiltonian[3, 1] = tc
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return hamiltonian
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def bilayer_bands(M, t1, t2, phi, tc):
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k1_array = np.linspace(0, 4*pi, 3000)
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k2 = 0
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hamiltonian = functools.partial(hamiltonian_of_bilayer, k2=k2, M=M, t1=t1, t2=t2, phi=phi, tc=tc)
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eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(k1_array, hamiltonian)
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plt, fig, ax = guan.import_plt_and_start_fig_ax(labelsize=25)
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guan.plot_without_starting_fig_ax(plt, fig, ax, k1_array, eigenvalue_array[:, 0], linewidth=2)
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guan.plot_without_starting_fig_ax(plt, fig, ax, k1_array, eigenvalue_array[:, 1], linewidth=2)
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guan.plot_without_starting_fig_ax(plt, fig, ax, k1_array, eigenvalue_array[:, 2], linewidth=2, color='k')
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guan.plot_without_starting_fig_ax(plt, fig, ax, k1_array, eigenvalue_array[:, 3], linewidth=2, color='k')
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guan.plot_without_starting_fig_ax(plt, fig, ax, [], [], fontsize=30, y_max=1.5, y_min=-1.5, xlabel='$\mathrm{k_x}$', ylabel='$\mathrm{E}$')
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plt.show()
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def chern_number(M, t1, t2, phi):
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tc_array = np.arange(0.05, 0.8, 0.05)
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chern_number_array = []
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for tc in tc_array:
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print(tc)
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hamiltonian = functools.partial(hamiltonian_of_bilayer, M=M, t1=t1, t2=t2, phi=phi, tc=tc)
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chern_number = guan.calculate_chern_number_for_honeycomb_lattice(hamiltonian, a=1/sqrt(3), precision=500, print_show=0)
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print(chern_number, '\n')
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chern_number_array.append(chern_number[0:2])
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guan.plot(tc_array, np.real(chern_number_array), xlabel='$\mathrm{t_c}$', ylabel='$\mathrm{C}$', style='o-')
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if __name__ == '__main__':
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main() |