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83
academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_(function_form).py Normal file
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"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/24059
"""
import numpy as np
from math import *
import cmath
import math
import guan
def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2
h[1, 1] = -0.28/2
h[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))
h[0, 1] = h[1, 0].conj()
return h
def main():
k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
# k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
dim = berry_curvature_array.shape
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
def calculate_berry_curvature_with_wilson_loop(hamiltonian_function, k_min=-math.pi, k_max=math.pi, precision_of_plaquettes=20, precision_of_wilson_loop=5, print_show=0):
if np.array(hamiltonian_function(0, 0)).shape==():
dim = 1
else:
dim = np.array(hamiltonian_function(0, 0)).shape[0]
delta = (k_max-k_min)/precision_of_plaquettes
k_array = np.arange(k_min, k_max, delta)
berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0], dim), dtype=complex)
i00 = 0
for kx in k_array:
if print_show == 1:
print(kx)
j00 = 0
for ky in k_array:
vector_array = []
# line_1
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta/precision_of_wilson_loop*i0, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_2
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta, ky+delta/precision_of_wilson_loop*i0)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_3
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta-delta/precision_of_wilson_loop*i0, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_4
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx, ky+delta-delta/precision_of_wilson_loop*i0)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
wilson_loop = 1
for i0 in range(len(vector_array)-1):
wilson_loop = wilson_loop*np.dot(vector_array[i0].transpose().conj(), vector_array[i0+1])
wilson_loop = wilson_loop*np.dot(vector_array[len(vector_array)-1].transpose().conj(), vector_array[0])
berry_curvature = np.log(np.diagonal(wilson_loop))/delta/delta*1j
berry_curvature_array[j00, i00, :]=berry_curvature
j00 += 1
i00 += 1
return k_array, berry_curvature_array
if __name__ == '__main__':
main()

102
academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_for_degenerate_case_(function_form).py Normal file
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"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/24059
"""
import numpy as np
from math import *
import cmath
import math
import guan
def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2
h[1, 1] = -0.28/2
h[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))
h[0, 1] = h[1, 0].conj()
return h
def main():
k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
# k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
dim = berry_curvature_array.shape
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='Valence Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
# k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
dim = berry_curvature_array.shape
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='All Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='All Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
def calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function, index_of_bands=[0, 1], k_min=-math.pi, k_max=math.pi, precision_of_plaquettes=20, precision_of_wilson_loop=5, print_show=0):
delta = (k_max-k_min)/precision_of_plaquettes
k_array = np.arange(k_min, k_max, delta)
berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0]), dtype=complex)
i000 = 0
for kx in k_array:
if print_show == 1:
print(kx)
j000 = 0
for ky in k_array:
vector_array = []
# line_1
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta/precision_of_wilson_loop*i0, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_2
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta, ky+delta/precision_of_wilson_loop*i0)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_3
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx+delta-delta/precision_of_wilson_loop*i0, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
# line_4
for i0 in range(precision_of_wilson_loop):
H_delta = hamiltonian_function(kx, ky+delta-delta/precision_of_wilson_loop*i0)
eigenvalue, eigenvector = np.linalg.eig(H_delta)
vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
vector_array.append(vector_delta)
wilson_loop = 1
dim = len(index_of_bands)
for i0 in range(len(vector_array)-1):
dot_matrix = np.zeros((dim , dim), dtype=complex)
i01 = 0
for dim1 in index_of_bands:
i02 = 0
for dim2 in index_of_bands:
dot_matrix[i01, i02] = np.dot(vector_array[i0][:, dim1].transpose().conj(), vector_array[i0+1][:, dim2])
i02 += 1
i01 += 1
det_value = np.linalg.det(dot_matrix)
wilson_loop = wilson_loop*det_value
dot_matrix_plus = np.zeros((dim , dim), dtype=complex)
i01 = 0
for dim1 in index_of_bands:
i02 = 0
for dim2 in index_of_bands:
dot_matrix_plus[i01, i02] = np.dot(vector_array[len(vector_array)-1][:, dim1].transpose().conj(), vector_array[0][:, dim2])
i02 += 1
i01 += 1
det_value = np.linalg.det(dot_matrix_plus)
wilson_loop = wilson_loop*det_value
berry_curvature = np.log(wilson_loop)/delta/delta*1j
berry_curvature_array[j000, i000]=berry_curvature
j000 += 1
i000 += 1
return k_array, berry_curvature_array
if __name__ == '__main__':
main()

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academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_the_efficient_method_(function_form).py Normal file
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"""
This code is supported by the website: https://www.guanjihuan.com
The newest version of this code is on the web page: https://www.guanjihuan.com/archives/24059
"""
import numpy as np
from math import *
import cmath
import guan
import math
def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)): # 石墨烯哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
h = np.zeros((2, 2))*(1+0j)
h[0, 0] = 0.28/2
h[1, 1] = -0.28/2
h[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))
h[0, 1] = h[1, 0].conj()
return h
def main():
k_array, berry_curvature_array = calculate_berry_curvature_with_efficient_method(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision=500, print_show=0)
# k_array, berry_curvature_array = guan.calculate_berry_curvature_with_efficient_method(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision=500, print_show=0)
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
dim = berry_curvature_array.shape
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0
def calculate_berry_curvature_with_efficient_method(hamiltonian_function, k_min=-math.pi, k_max=math.pi, precision=100, print_show=0):
if np.array(hamiltonian_function(0, 0)).shape==():
dim = 1
else:
dim = np.array(hamiltonian_function(0, 0)).shape[0]
delta = (k_max-k_min)/precision
k_array = np.arange(k_min, k_max, delta)
berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0], dim), dtype=complex)
i0 = 0
for kx in k_array:
if print_show == 1:
print(kx)
j0 = 0
for ky in k_array:
H = hamiltonian_function(kx, ky)
vector = guan.calculate_eigenvector(H)
H_delta_kx = hamiltonian_function(kx+delta, ky)
vector_delta_kx = guan.calculate_eigenvector(H_delta_kx)
H_delta_ky = hamiltonian_function(kx, ky+delta)
vector_delta_ky = guan.calculate_eigenvector(H_delta_ky)
H_delta_kx_ky = hamiltonian_function(kx+delta, ky+delta)
vector_delta_kx_ky = guan.calculate_eigenvector(H_delta_kx_ky)
for i in range(dim):
vector_i = vector[:, i]
vector_delta_kx_i = vector_delta_kx[:, i]
vector_delta_ky_i = vector_delta_ky[:, i]
vector_delta_kx_ky_i = vector_delta_kx_ky[:, i]
Ux = np.dot(np.conj(vector_i), vector_delta_kx_i)/abs(np.dot(np.conj(vector_i), vector_delta_kx_i))
Uy = np.dot(np.conj(vector_i), vector_delta_ky_i)/abs(np.dot(np.conj(vector_i), vector_delta_ky_i))
Ux_y = np.dot(np.conj(vector_delta_ky_i), vector_delta_kx_ky_i)/abs(np.dot(np.conj(vector_delta_ky_i), vector_delta_kx_ky_i))
Uy_x = np.dot(np.conj(vector_delta_kx_i), vector_delta_kx_ky_i)/abs(np.dot(np.conj(vector_delta_kx_i), vector_delta_kx_ky_i))
berry_curvature = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))/delta/delta*1j
berry_curvature_array[j0, i0, i] = berry_curvature
j0 += 1
i0 += 1
return k_array, berry_curvature_array
if __name__ == '__main__':
main()