update

2020.02.26_calculation_of_Chern_number_by_efficient_method/calculation_of_Chern_number_by_efficient_method.m

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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/4179
+
+% 陈数高效法
+clear;clc;
+n=1000 % 积分密度
+delta=2*pi/n;
+C=0; 
+for kx=-pi:(2*pi/n):pi
+    for ky=-pi:(2*pi/n):pi
+        VV=get_vector(HH(kx,ky));
+        Vkx=get_vector(HH(kx+delta,ky)); % 略偏离kx的波函数
+        Vky=get_vector(HH(kx,ky+delta)); % 略偏离ky的波函数
+        Vkxky=get_vector(HH(kx+delta,ky+delta)); % 略偏离kx，ky的波函数
+        
+        Ux = VV'*Vkx/abs(VV'*Vkx);
+        Uy = VV'*Vky/abs(VV'*Vky);
+        Ux_y = Vky'*Vkxky/abs(Vky'*Vkxky);
+        Uy_x = Vkx'*Vkxky/abs(Vkx'*Vkxky);
+        
+        % berry curvature
+        F=log(Ux*Uy_x*(1/Ux_y)*(1/Uy));
+        % chern number
+        C=C+F;  
+    end
+end
+C=C/(2*pi*1i)
+
+function vector_new = get_vector(H)
+[vector,eigenvalue] = eig(H);
+[eigenvalue, index]=sort(diag(eigenvalue), 'descend');
+vector_new = vector(:, index(2));
+end
+
+function H=HH(kx,ky)
+H(1,2)=2*cos(kx)-1i*2*cos(ky);
+H(2,1)=2*cos(kx)+1i*2*cos(ky);
+H(1,1)=-1+2*0.5*sin(kx)+2*0.5*sin(ky)+2*cos(kx+ky);
+H(2,2)=-(-1+2*0.5*sin(kx)+2*0.5*sin(ky)+2*cos(kx+ky));
+end

2020.02.26_calculation_of_Chern_number_by_efficient_method/calculation_of_Chern_number_by_efficient_method.py

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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/4179
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+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 
+    delta = 2*pi/n
+    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
+            
+            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_time = time.time()
+    print('运行时间(min)=', (end_time-start_time)/60)
+
+
+if __name__ == '__main__':
+    main()