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2023-11-07 03:38:46 +08:00
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2020.07.13_calculation_of_Chern_number_in_Haldane_model/Chern_number_in_Haldane_model.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/5133
"""
import numpy as np
from math import * # 引入pi, cos等
import cmath
import time
import functools # 使用偏函数functools.partial()
def hamiltonian(k1, k2, M, t1, t2, phi, a=1/sqrt(3)): # Haldane哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
# 初始化为零矩阵
h0 = np.zeros((2, 2), dtype=complex)
h1 = np.zeros((2, 2), dtype=complex)
h2 = np.zeros((2, 2), dtype=complex)
# 质量项(mass term), 用于打开带隙
h0[0, 0] = M
h0[1, 1] = -M
# 最近邻项
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))
h1[0, 1] = h1[1, 0].conj()
# # 最近邻项也可写成这种形式
# h1[1, 0] = t1+t1*cmath.exp(1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)+t1*cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)
# h1[0, 1] = h1[1, 0].conj()
#次近邻项 # 对应陈数为-1
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))
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))
# # 次近邻项 # 对应陈数为1
# 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))
# 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))
matrix = h0 + h1 + h2 + h2.transpose().conj()
return matrix
def main():
start_clock = time.perf_counter()
delta = 0.005
chern_number = 0 # 陈数初始化
# 几个坐标中常出现的项
a = 1/sqrt(3)
aa1 = 4*sqrt(3)*pi/9/a
aa2 = 2*sqrt(3)*pi/9/a
bb1 = 2*pi/3/a
hamiltonian0 = functools.partial(hamiltonian, M=2/3, t1=1, t2=1/3, phi=pi/4, a=a) # 使用偏函数，固定一些参数
for kx in np.arange(-aa1, aa1, delta):
print(kx)
for ky in np.arange(-bb1, bb1, delta):
if (-aa2<=kx<=aa2) or (kx>aa2 and -(aa1-kx)*tan(pi/3)<=ky<=(aa1-kx)*tan(pi/3)) or (kx<-aa2 and -(kx-(-aa1))*tan(pi/3)<=ky<=(kx-(-aa1))*tan(pi/3)): # 限制在六角格子布里渊区内
H = hamiltonian0(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian0(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian0(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian0(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
Ux = np.dot(np.conj(vector), vector_delta_kx)/abs(np.dot(np.conj(vector), vector_delta_kx))
Uy = np.dot(np.conj(vector), vector_delta_ky)/abs(np.dot(np.conj(vector), vector_delta_ky))
Ux_y = np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky))
Uy_x = np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))
# 陈数(chern number)
chern_number = chern_number + F
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_clock = time.perf_counter()
print('CPU执行时间(min)=', (end_clock-start_clock)/60)
if __name__ == '__main__':
main()

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2020.07.13_calculation_of_Chern_number_in_Haldane_model/Chern_number_in_Haldane_model_by_integration_method_2.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/5133
"""
import numpy as np
import matplotlib.pyplot as plt
from math import * # 引入pi, cos等
import cmath
import time
import functools # 使用偏函数functools.partial()
def hamiltonian(k1, k2, M, t1, t2, phi, a=1/sqrt(3)): # Haldane哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
# 初始化为零矩阵
h0 = np.zeros((2, 2), dtype=complex)
h1 = np.zeros((2, 2), dtype=complex)
h2 = np.zeros((2, 2), dtype=complex)
# 质量项(mass term), 用于打开带隙
h0[0, 0] = M
h0[1, 1] = -M
# 最近邻项
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))
h1[0, 1] = h1[1, 0].conj()
# # 最近邻项也可写成这种形式
# h1[1, 0] = t1+t1*cmath.exp(1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)+t1*cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)
# h1[0, 1] = h1[1, 0].conj()
#次近邻项 # 对应陈数为-1
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))
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))
# # 次近邻项 # 对应陈数为1
# 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))
# 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))
matrix = h0 + h1 + h2 + h2.transpose().conj()
return matrix
def main():
start_clock = time.perf_counter()
delta = 0.005
chern_number = 0 # 陈数初始化
# 常出现的项
a = 1/sqrt(3)
bb1 = 2*sqrt(3)*pi/3/a
bb2 = 2*pi/3/a
hamiltonian0 = functools.partial(hamiltonian, M=2/3, t1=1, t2=1/3, phi=pi/4, a=a) # 使用偏函数，固定一些参数
for kx in np.arange(0 , bb1, delta):
print(kx)
for ky in np.arange(0, 2*bb2, delta):
H = hamiltonian0(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian0(kx+delta, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian0(kx, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian0(kx+delta, ky+delta)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
Ux = np.dot(np.conj(vector), vector_delta_kx)/abs(np.dot(np.conj(vector), vector_delta_kx))
Uy = np.dot(np.conj(vector), vector_delta_ky)/abs(np.dot(np.conj(vector), vector_delta_ky))
Ux_y = np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky))
Uy_x = np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))
# 陈数(chern number)
chern_number = chern_number + F
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_clock = time.perf_counter()
print('CPU执行时间(min)=', (end_clock-start_clock)/60)
if __name__ == '__main__':
main()

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2020.07.13_calculation_of_Chern_number_in_Haldane_model/Chern_number_in_Haldane_model_by_integration_method_3.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/5133
"""
import numpy as np
import matplotlib.pyplot as plt
from math import * # 引入pi, cos等
import cmath
import time
import functools # 使用偏函数functools.partial()
def hamiltonian(k1, k2, M, t1, t2, phi, a=1/sqrt(3)): # Haldane哈密顿量# Haldane哈密顿量（a为原子间距，不赋值的话默认为1/sqrt(3)）
# 初始化为零矩阵
h0 = np.zeros((2, 2), dtype=complex)
h1 = np.zeros((2, 2), dtype=complex)
h2 = np.zeros((2, 2), dtype=complex)
# 质量项(mass term), 用于打开带隙
h0[0, 0] = M
h0[1, 1] = -M
# 最近邻项
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))
h1[0, 1] = h1[1, 0].conj()
# # 最近邻项也可写成这种形式
# h1[1, 0] = t1+t1*cmath.exp(1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)+t1*cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)
# h1[0, 1] = h1[1, 0].conj()
#次近邻项 # 对应陈数为-1
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))
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))
# # 次近邻项 # 对应陈数为1
# 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))
# 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))
matrix = h0 + h1 + h2 + h2.transpose().conj()
return matrix
def main():
start_clock = time.perf_counter()
delta = 0.005
chern_number = 0 # 陈数初始化
# 常出现的项
a = 1/sqrt(3)
bb1 = 2*sqrt(3)*pi/3/a
bb2 = 2*pi/3/a
hamiltonian0 = functools.partial(hamiltonian, M=2/3, t1=1, t2=1/3, phi=pi/4, a=a) # 使用偏函数，固定一些参数
for k1 in np.arange(0 , 1, delta):
print(k1)
for k2 in np.arange(0, 1, delta):
# 坐标变换
kx = (k1-k2)*bb1
ky = (k1+k2)*bb2
# 这里乘2或除以2是为了保证“步长与积分个数的乘积刚好为布里渊区面积”
delta_x = delta*bb1*2
delta_y = delta*bb2*2/2
H = hamiltonian0(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian0(kx+delta_x, ky)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx的波函数
H_delta_ky = hamiltonian0(kx, ky+delta_y)
eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离ky的波函数
H_delta_kx_ky = hamiltonian0(kx+delta_x, ky+delta_y)
eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 略偏离kx和ky的波函数
Ux = np.dot(np.conj(vector), vector_delta_kx)/abs(np.dot(np.conj(vector), vector_delta_kx))
Uy = np.dot(np.conj(vector), vector_delta_ky)/abs(np.dot(np.conj(vector), vector_delta_ky))
Ux_y = np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_ky), vector_delta_kx_ky))
Uy_x = np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky)/abs(np.dot(np.conj(vector_delta_kx), vector_delta_kx_ky))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))
# 陈数(chern number)
chern_number = chern_number + F
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_clock = time.perf_counter()
print('CPU执行时间(min)=', (end_clock-start_clock)/60)
if __name__ == '__main__':
main()