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2023-11-07 03:38:46 +08:00
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2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_Wilson_loop.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/17984
"""
import numpy as np
import cmath
from math import *
def hamiltonian(kx, ky): # BBH model
# label of atoms in a unit cell
# (2) —— (0)
# | |
# (1) —— (3)
gamma_x = 0.5 # hopping inside one unit cell
lambda_x = 1 # hopping between unit cells
gamma_y = gamma_x
lambda_y = lambda_x
h = np.zeros((4, 4), dtype=complex)
h[0, 2] = gamma_x+lambda_x*cmath.exp(1j*kx)
h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx)
h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky)
h[1, 2] = -gamma_y-lambda_y*cmath.exp(-1j*ky)
h[2, 0] = np.conj(h[0, 2])
h[3, 1] = np.conj(h[1, 3])
h[3, 0] = np.conj(h[0, 3])
h[2, 1] = np.conj(h[1, 2])
return h
def main():
Num_kx = 100
Num_ky = 100
kx_array = np.linspace(-pi, pi, Num_kx)
ky_array = np.linspace(-pi, pi, Num_ky)
nu_x_array = []
for ky in ky_array:
vector1_array = []
vector2_array = []
for kx in kx_array:
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky))
if kx != pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
# 这里是为了-pi和pi有相同的波函数，使得Wilson loop的值与波函数规范无关。
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])
W_x_k = np.eye(2, dtype=complex)
for i0 in range(Num_kx-1):
F = np.zeros((2, 2), dtype=complex)
F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0])
F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0])
F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0])
F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0])
W_x_k = np.dot(F, W_x_k)
eigenvalue, eigenvector = np.linalg.eig(W_x_k)
nu_x = np.log(eigenvalue)/2/pi/1j
for i0 in range(2):
if np.real(nu_x[i0]) < 0:
nu_x[i0] += 1
nu_x = np.sort(nu_x)
nu_x_array.append(nu_x.real)
plot(ky_array, nu_x_array, xlabel='ky', ylabel='nu_x', style='-', y_min=0, y_max=1)
# import guan
# guan.plot(ky_array, nu_x_array, xlabel='ky', ylabel='nu_x', style='-', y_min=0, y_max=1)
def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2):
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
ax.grid()
ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman')
if y_min!=None or y_max!=None:
if y_min==None:
y_min=min(y_array)
if y_max==None:
y_max=max(y_array)
ax.set_ylim(y_min, y_max)
ax.tick_params(labelsize=labelsize)
labels = ax.get_xticklabels() + ax.get_yticklabels()
[label.set_fontname('Times New Roman') for label in labels]
if save == 1:
plt.savefig(filename+file_format, dpi=dpi)
if show == 1:
plt.show()
plt.close('all')
if __name__ == '__main__':
main()

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2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_bands.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/17984
"""
import numpy as np
import cmath
from math import *
import functools
def hamiltonian(kx, ky): # BBH model
# label of atoms in a unit cell
# (2) —— (0)
# | |
# (1) —— (3)
gamma_x = 0.5 # hopping inside one unit cell
lambda_x = 1 # hopping between unit cells
gamma_y = gamma_x
lambda_y = lambda_x
h = np.zeros((4, 4), dtype=complex)
h[0, 2] = gamma_x+lambda_x*cmath.exp(1j*kx)
h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx)
h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky)
h[1, 2] = -gamma_y-lambda_y*cmath.exp(-1j*ky)
h[2, 0] = np.conj(h[0, 2])
h[3, 1] = np.conj(h[1, 3])
h[3, 0] = np.conj(h[0, 3])
h[2, 1] = np.conj(h[1, 2])
return h
def main():
kx = np.arange(-pi, pi, 0.05)
ky = np.arange(-pi, pi, 0.05)
eigenvalue_array = calculate_eigenvalue_with_two_parameters(kx, ky, hamiltonian)
plot_3d_surface(kx, ky, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E', title='BBH bands')
hamiltonian0 = functools.partial(hamiltonian, ky=0)
eigenvalue_array = calculate_eigenvalue_with_one_parameter(kx, hamiltonian0)
plot(kx, eigenvalue_array, xlabel='kx', ylabel='E', title='BBH bands ky=0')
# import guan
# eigenvalue_array = guan.calculate_eigenvalue_with_two_parameters(kx, ky, hamiltonian)
# guan.plot_3d_surface(kx, ky, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E', title='BBH bands')
# hamiltonian0 = functools.partial(hamiltonian, ky=0)
# eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(kx, hamiltonian0)
# guan.plot(kx, eigenvalue_array, xlabel='kx', ylabel='E', title='BBH bands ky=0')
def calculate_eigenvalue_with_one_parameter(x_array, hamiltonian_function, print_show=0):
dim_x = np.array(x_array).shape[0]
i0 = 0
if np.array(hamiltonian_function(0)).shape==():
eigenvalue_array = np.zeros((dim_x, 1))
for x0 in x_array:
hamiltonian = hamiltonian_function(x0)
eigenvalue_array[i0, 0] = np.real(hamiltonian)
i0 += 1
else:
dim = np.array(hamiltonian_function(0)).shape[0]
eigenvalue_array = np.zeros((dim_x, dim))
for x0 in x_array:
if print_show==1:
print(x0)
hamiltonian = hamiltonian_function(x0)
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian)
eigenvalue_array[i0, :] = eigenvalue
i0 += 1
return eigenvalue_array
def calculate_eigenvalue_with_two_parameters(x_array, y_array, hamiltonian_function, print_show=0, print_show_more=0):
dim_x = np.array(x_array).shape[0]
dim_y = np.array(y_array).shape[0]
if np.array(hamiltonian_function(0,0)).shape==():
eigenvalue_array = np.zeros((dim_y, dim_x, 1))
i0 = 0
for y0 in y_array:
j0 = 0
for x0 in x_array:
hamiltonian = hamiltonian_function(x0, y0)
eigenvalue_array[i0, j0, 0] = np.real(hamiltonian)
j0 += 1
i0 += 1
else:
dim = np.array(hamiltonian_function(0, 0)).shape[0]
eigenvalue_array = np.zeros((dim_y, dim_x, dim))
i0 = 0
for y0 in y_array:
j0 = 0
if print_show==1:
print(y0)
for x0 in x_array:
if print_show_more==1:
print(x0)
hamiltonian = hamiltonian_function(x0, y0)
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian)
eigenvalue_array[i0, j0, :] = eigenvalue
j0 += 1
i0 += 1
return eigenvalue_array
def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2):
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
ax.grid()
ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman')
if y_min!=None or y_max!=None:
if y_min==None:
y_min=min(y_array)
if y_max==None:
y_max=max(y_array)
ax.set_ylim(y_min, y_max)
ax.tick_params(labelsize=labelsize)
labels = ax.get_xticklabels() + ax.get_yticklabels()
[label.set_fontname('Times New Roman') for label in labels]
if save == 1:
plt.savefig(filename+file_format, dpi=dpi)
if show == 1:
plt.show()
plt.close('all')
def plot_3d_surface(x_array, y_array, matrix, xlabel='x', ylabel='y', zlabel='z', title='', fontsize=20, labelsize=15, show=1, save=0, filename='a', file_format='.jpg', dpi=300, z_min=None, z_max=None, rcount=100, ccount=100):
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator
matrix = np.array(matrix)
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
plt.subplots_adjust(bottom=0.1, right=0.65)
x_array, y_array = np.meshgrid(x_array, y_array)
if len(matrix.shape) == 2:
surf = ax.plot_surface(x_array, y_array, matrix, rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False)
elif len(matrix.shape) == 3:
for i0 in range(matrix.shape[2]):
surf = ax.plot_surface(x_array, y_array, matrix[:,:,i0], rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_zlabel(zlabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.zaxis.set_major_locator(LinearLocator(5))
ax.zaxis.set_major_formatter('{x:.2f}')
if z_min!=None or z_max!=None:
if z_min==None:
z_min=matrix.min()
if z_max==None:
z_max=matrix.max()
ax.set_zlim(z_min, z_max)
ax.tick_params(labelsize=labelsize)
labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
[label.set_fontname('Times New Roman') for label in labels]
cax = plt.axes([0.8, 0.1, 0.05, 0.8])
cbar = fig.colorbar(surf, cax=cax)
cbar.ax.tick_params(labelsize=labelsize)
for l in cbar.ax.yaxis.get_ticklabels():
l.set_family('Times New Roman')
if save == 1:
plt.savefig(filename+file_format, dpi=dpi)
if show == 1:
plt.show()
plt.close('all')
if __name__ == '__main__':
main()

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2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_nested_Wilson_loop.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/17984
"""
import numpy as np
import cmath
from math import *
def hamiltonian(kx, ky): # BBH model
# label of atoms in a unit cell
# (2) —— (0)
# | |
# (1) —— (3)
gamma_x = 0.5 # hopping inside one unit cell
lambda_x = 1 # hopping between unit cells
gamma_y = gamma_x
lambda_y = lambda_x
x_symmetry_breaking_1 = 0.000000000000 # default (not breaking): zero
x_symmetry_breaking_2 = 1.0000000000001 # default (not breaking): unity
y_symmetry_breaking_1 = 0.000000000000 # default (not breaking): zero
y_symmetry_breaking_2 = 1.000000000000 # default (not breaking): unity
h = np.zeros((4, 4), dtype=complex)
h[0, 0] = x_symmetry_breaking_1
h[1, 1] = y_symmetry_breaking_1
h[2, 2] = y_symmetry_breaking_1
h[3, 3] = x_symmetry_breaking_1
h[0, 2] = (gamma_x+lambda_x*cmath.exp(1j*kx))*y_symmetry_breaking_2
h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx)
h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky)
h[1, 2] = (-gamma_y-lambda_y*cmath.exp(-1j*ky))*x_symmetry_breaking_2
h[2, 0] = np.conj(h[0, 2])
h[3, 1] = np.conj(h[1, 3])
h[3, 0] = np.conj(h[0, 3])
h[2, 1] = np.conj(h[1, 2])
return h
def main():
Num_kx = 30 # for wilson loop and nested wilson loop
Num_ky = 30 # for wilson loop and nested wilson loop
Num_kx2 = 20 # plot precision
Num_ky2 = 20 # plot precision
kx_array = np.linspace(-pi, pi, Num_kx)
ky_array = np.linspace(-pi, pi, Num_ky)
kx2_array = np.linspace(-pi, pi, Num_kx2)
ky2_array = np.linspace(-pi, pi, Num_ky2)
# Part I: calculate p_y_for_nu_x
p_y_for_nu_x_array = []
for kx in kx2_array:
print('kx=', kx)
w_vector_for_nu1_array = []
vector1_array = []
vector2_array = []
i0 = -1
for ky in ky_array:
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky))
if ky != pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])
i0=0
for ky in ky_array:
if ky != pi:
nu_x_vector_1, nu_x_vector_2 = get_nu_x_vector(kx_array, ky)
# the Wannier band subspaces
w_vector_for_nu1 = vector1_array[i0]*nu_x_vector_1[0]+vector2_array[i0]*nu_x_vector_1[1]
w_vector_for_nu1_array.append(w_vector_for_nu1)
else:
w_vector_for_nu1_array.append(w_vector_for_nu1_array[0])
i0 +=1
W_y_k_for_nu_x = 1
for i0 in range(Num_ky-1):
F_for_nu_x = np.dot(w_vector_for_nu1_array[i0+1].transpose().conj(), w_vector_for_nu1_array[i0])
W_y_k_for_nu_x = F_for_nu_x*W_y_k_for_nu_x
p_y_for_nu_x = np.log(W_y_k_for_nu_x)/2/pi/1j
if np.real(p_y_for_nu_x) < 0:
p_y_for_nu_x += 1
p_y_for_nu_x_array.append(p_y_for_nu_x.real)
print('p_y_for_nu_x=', p_y_for_nu_x)
plot(kx2_array, p_y_for_nu_x_array, xlabel='kx', ylabel='p_y_for_nu_x', style='-o', y_min=0, y_max=1)
# import guan
# guan.plot(kx2_array, p_y_for_nu_x_array, xlabel='kx', ylabel='p_y_for_nu_x', style='-o', y_min=0, y_max=1)
# Part II: calculate p_x_for_nu_y
p_x_for_nu_y_array = []
for ky in ky2_array:
w_vector_for_nu1_array = []
vector1_array = []
vector2_array = []
for kx in kx_array:
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky))
if kx != pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])
i0 = 0
for kx in kx_array:
if kx != pi:
nu_y_vector_1, nu_y_vector_2 = get_nu_y_vector(kx, ky_array)
# the Wannier band subspaces
w_vector_for_nu1 = vector1_array[i0]*nu_y_vector_1[0]+vector2_array[i0]*nu_y_vector_1[1]
w_vector_for_nu1_array.append(w_vector_for_nu1)
else:
w_vector_for_nu1_array.append(w_vector_for_nu1_array[0])
i0 += 1
W_x_k_for_nu_y = 1
for i0 in range(Num_ky-1):
F_for_nu_y = np.dot(w_vector_for_nu1_array[i0+1].transpose().conj(), w_vector_for_nu1_array[i0])
W_x_k_for_nu_y = F_for_nu_y*W_x_k_for_nu_y
p_x_for_nu_y = np.log(W_x_k_for_nu_y)/2/pi/1j
if np.real(p_x_for_nu_y) < 0:
p_x_for_nu_y += 1
p_x_for_nu_y_array.append(p_x_for_nu_y.real)
print('p_x_for_nu_y=', p_x_for_nu_y)
# print(sum(p_x_for_nu_y_array)/len(p_x_for_nu_y_array))
plot(ky2_array, p_x_for_nu_y_array, xlabel='ky', ylabel='p_x_for_nu_y', style='-o', y_min=0, y_max=1)
# import guan
# guan.plot(ky2_array, p_x_for_nu_y_array, xlabel='ky', ylabel='p_x_for_nu_y', style='-o', y_min=0, y_max=1)
def get_nu_x_vector(kx_array, ky):
Num_kx = len(kx_array)
vector1_array = []
vector2_array = []
for kx in kx_array:
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky))
if kx != pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])
W_x_k = np.eye(2, dtype=complex)
for i0 in range(Num_kx-1):
F = np.zeros((2, 2), dtype=complex)
F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0])
F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0])
F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0])
F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0])
W_x_k = np.dot(F, W_x_k)
eigenvalue, eigenvector = np.linalg.eig(W_x_k)
nu_x = np.log(eigenvalue)/2/pi/1j
nu_x_vector_1 = eigenvector[:, np.argsort(np.real(nu_x))[0]]
nu_x_vector_2 = eigenvector[:, np.argsort(np.real(nu_x))[1]]
return nu_x_vector_1, nu_x_vector_2
def get_nu_y_vector(kx, ky_array):
Num_ky = len(ky_array)
vector1_array = []
vector2_array = []
for ky in ky_array:
eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky))
if ky != pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])
W_y_k = np.eye(2, dtype=complex)
for i0 in range(Num_ky-1):
F = np.zeros((2, 2), dtype=complex)
F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0])
F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0])
F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0])
F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0])
W_y_k = np.dot(F, W_y_k)
eigenvalue, eigenvector = np.linalg.eig(W_y_k)
nu_y = np.log(eigenvalue)/2/pi/1j
nu_y_vector_1 = eigenvector[:, np.argsort(np.real(nu_y))[0]]
nu_y_vector_2 = eigenvector[:, np.argsort(np.real(nu_y))[1]]
return nu_y_vector_1, nu_y_vector_2
def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2):
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
ax.grid()
ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman')
if y_min!=None or y_max!=None:
if y_min==None:
y_min=min(y_array)
if y_max==None:
y_max=max(y_array)
ax.set_ylim(y_min, y_max)
ax.tick_params(labelsize=labelsize)
labels = ax.get_xticklabels() + ax.get_yticklabels()
[label.set_fontname('Times New Roman') for label in labels]
if save == 1:
plt.savefig(filename+file_format, dpi=dpi)
if show == 1:
plt.show()
plt.close('all')
if __name__ == '__main__':
main()

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2021.12.09_nested_Wilson_loop_of_BBH_model/simple_method_for_degenerate_bands.py Normal file
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if kx == -pi:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
elif kx != pi:
# 这里的判断是为了处理能带简并时最简单情况，只做个对调。
if np.abs(np.dot(vector1_array[-1].transpose().conj(), eigenvector[:, 0]))>0.5:
vector1_array.append(eigenvector[:, 0])
vector2_array.append(eigenvector[:, 1])
else:
vector1_array.append(eigenvector[:, 1])
vector2_array.append(eigenvector[:, 0])
else:
# 这里是为了-pi和pi有相同的波函数，使得Wilson loop的值与波函数规范无关。
vector1_array.append(vector1_array[0])
vector2_array.append(vector2_array[0])