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guanjihuan
2023-11-07 03:38:46 +08:00
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2020.11.27_find_the_same_gauge_numerically_with_binary_search/find_the_same_gauge_numerically_with_binary_search.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/7516
"""
import numpy as np
from math import * # 引入pi, cos等
import cmath
import time
def hamiltonian(kx, ky): # 量子反常霍尔QAH模型（该参数对应的陈数为2）
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
matrix = np.zeros((2, 2))*(1+0j)
matrix[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
matrix[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
matrix[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
matrix[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return matrix
def main():
start_time = time.time()
n = 100 # 积分密度
delta = 1e-9 # 求导的偏离量
chern_number = 0 # 陈数初始化
for kx in np.arange(-pi, pi, 2*pi/n):
for ky in np.arange(-pi, pi, 2*pi/n):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian(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 = hamiltonian(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 = hamiltonian(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
# vector = vector*cmath.exp(-1j*1)
# vector_delta_kx = vector_delta_kx*cmath.exp(-1j*1)
# vector_delta_ky = vector_delta_ky*cmath.exp(-1j*1)
# vector_delta_kx_ky = vector_delta_kx_ky*cmath.exp(-1j*(1+1e-8))
rand = np.random.uniform(-pi, pi)
vector_delta_kx_ky = vector_delta_kx_ky*cmath.exp(-1j*rand)
# 寻找近似的同一的规范
phase_1_pre = 0
phase_2_pre = pi
n_test = 10001
for i0 in range(n_test):
test_1 = np.sum(np.abs(vector_delta_kx_ky*cmath.exp(1j*phase_1_pre) - vector))
test_2 = np.sum(np.abs(vector_delta_kx_ky*cmath.exp(1j*phase_2_pre) - vector))
if test_1 < 1e-6:
phase = phase_1_pre
print('Done with i0=', i0)
break
if i0 == n_test-1:
phase = phase_1_pre
print('Not Found with i0=', i0)
if test_1 < test_2:
if i0 == 0:
phase_1 = phase_1_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else:
phase_1 = phase_1_pre
phase_2 = phase_1_pre+(phase_2_pre-phase_1_pre)/2
else:
if i0 == 0:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre+(phase_2_pre-phase_1_pre)/2
else:
phase_1 = phase_2_pre-(phase_2_pre-phase_1_pre)/2
phase_2 = phase_2_pre
phase_1_pre = phase_1
phase_2_pre = phase_2
vector_delta_kx_ky = vector_delta_kx_ky*cmath.exp(1j*phase)
print('随机的规范=', rand) # 可注释掉
print('二分查找找到的规范=', phase) # 可注释掉
print() # 可注释掉
# 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature)
F = (A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta
# 陈数(chern number)
chern_number = chern_number + F*(2*pi/n)**2
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60)
if __name__ == '__main__':
main()

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2020.11.27_find_the_same_gauge_numerically_with_binary_search/find_the_same_gauge_numerically_with_guan.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/7516
"""
import numpy as np
from math import *
import cmath
import time
import guan
def hamiltonian(kx, ky): # 量子反常霍尔QAH模型（该参数对应的陈数为2）
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
matrix = np.zeros((2, 2))*(1+0j)
matrix[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
matrix[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
matrix[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
matrix[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return matrix
def main():
start_time = time.time()
n = 100 # 积分密度
delta = 1e-9 # 求导的偏离量
chern_number = 0 # 陈数初始化
for kx in np.arange(-pi, pi, 2*pi/n):
for ky in np.arange(-pi, pi, 2*pi/n):
H = hamiltonian(kx, ky)
eigenvalue, eigenvector = np.linalg.eig(H)
vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] # 价带波函数
H_delta_kx = hamiltonian(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 = hamiltonian(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 = hamiltonian(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
# vector = vector*cmath.exp(-1j*1)
# vector_delta_kx = vector_delta_kx*cmath.exp(-1j*1)
# vector_delta_ky = vector_delta_ky*cmath.exp(-1j*1)
# vector_delta_kx_ky = vector_delta_kx_ky*cmath.exp(-1j*(1+1e-8))
rand = np.random.uniform(-pi, pi)
vector_delta_kx_ky = vector_delta_kx_ky*cmath.exp(-1j*rand)
vector_delta_kx_ky = guan.find_vector_with_the_same_gauge_with_binary_search(vector_delta_kx_ky, vector)
# 价带的波函数的贝里联络(berry connection) # 求导后内积
A_x = np.dot(vector.transpose().conj(), (vector_delta_kx-vector)/delta) # 贝里联络Ax（x分量）
A_y = np.dot(vector.transpose().conj(), (vector_delta_ky-vector)/delta) # 贝里联络Ay（y分量）
A_x_delta_ky = np.dot(vector_delta_ky.transpose().conj(), (vector_delta_kx_ky-vector_delta_ky)/delta) # 略偏离ky的贝里联络Ax
A_y_delta_kx = np.dot(vector_delta_kx.transpose().conj(), (vector_delta_kx_ky-vector_delta_kx)/delta) # 略偏离kx的贝里联络Ay
# 贝里曲率(berry curvature)
F = (A_y_delta_kx-A_y)/delta-(A_x_delta_ky-A_x)/delta
# 陈数(chern number)
chern_number = chern_number + F*(2*pi/n)**2
chern_number = chern_number/(2*pi*1j)
print('Chern number = ', chern_number)
end_time = time.time()
print('运行时间(min)=', (end_time-start_time)/60)
if __name__ == '__main__':
main()