category

academic_codes/topological_invariant/2022.04.21_Berry_curvature/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_Kubo_formula.py

@@ -0,0 +1,55 @@
+"""
+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/20869
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import *  
+import cmath
+
+
+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), dtype=complex)
+    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():
+    n = 2000  # 取点密度
+    delta = 1e-9  # 求导的偏离量
+    for band in range(2):
+        F_all = []  # 贝里曲率
+        for kx in np.linspace(-2*pi, 2*pi, n):
+            for ky in [0]: # 这里只考虑ky=0对称轴上的情况
+                H = hamiltonian(kx, ky)
+                eigenvalue, eigenvector = np.linalg.eig(H)
+                if band==0:
+                    vector_0 = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]
+                    vector_1 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]]
+                elif band==1:
+                    vector_0 = eigenvector[:, np.argsort(np.real(eigenvalue))[1]]
+                    vector_1 = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]
+                eigenvalue = np.sort(np.real(eigenvalue))
+
+                H_delta_kx = hamiltonian(kx+delta, ky)-hamiltonian(kx, ky) 
+                H_delta_ky = hamiltonian(kx, ky+delta)-hamiltonian(kx, ky)
+
+                berry_curvature = 1j*(np.dot(np.dot(np.dot(np.dot(np.dot(vector_0.transpose().conj(), H_delta_kx/delta), vector_1), vector_1.transpose().conj()), H_delta_ky/delta), vector_0)- np.dot(np.dot(np.dot(np.dot(np.dot(vector_0.transpose().conj(), H_delta_ky/delta), vector_1), vector_1.transpose().conj()), H_delta_kx/delta), vector_0))/(eigenvalue[0]-eigenvalue[1])**2
+
+                F_all = np.append(F_all,[berry_curvature], axis=0) 
+        plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all))
+        plt.xlabel('k_x (pi)')
+        plt.ylabel('Berry curvature')
+        if band==0:
+            plt.title('Valence Band')
+        else:
+            plt.title('Conductance Band')
+        plt.show()
+
+
+if __name__ == '__main__':
+    main()

academic_codes/topological_invariant/2022.04.21_Berry_curvature/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_Wilson_loop.py

@@ -0,0 +1,72 @@
+"""
+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/20869
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import *  
+import cmath
+
+
+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), dtype=complex)
+    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():
+    n1 = 1000 # small plaquettes精度
+    n2 = 10 # Wilson loop精度
+    delta = 2*pi/n1
+    for band in range(2):
+        F_all = []  # 贝里曲率
+        for kx in np.linspace(-2*pi, 2*pi, n1):
+            for ky in [0]: # 这里只考虑ky=0对称轴上的情况
+                vector_array = []
+                # line_1
+                for i2 in range(n2+1):
+                    H_delta = hamiltonian(kx+delta/n2*i2, ky) 
+                    eigenvalue, eigenvector = np.linalg.eig(H_delta)
+                    vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))[band]]
+                    vector_array.append(vector_delta)
+                # line_2
+                for i2 in range(n2):
+                    H_delta = hamiltonian(kx+delta, ky+delta/n2*(i2+1))  
+                    eigenvalue, eigenvector = np.linalg.eig(H_delta)
+                    vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))[band]]
+                    vector_array.append(vector_delta)
+                # line_3
+                for i2 in range(n2):
+                    H_delta = hamiltonian(kx+delta-delta/n2*(i2+1), ky+delta)  
+                    eigenvalue, eigenvector = np.linalg.eig(H_delta)
+                    vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))[band]]
+                    vector_array.append(vector_delta)
+                # line_4
+                for i2 in range(n2-1):
+                    H_delta = hamiltonian(kx, ky+delta-delta/n2*(i2+1))  
+                    eigenvalue, eigenvector = np.linalg.eig(H_delta)
+                    vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))[band]]
+                    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])
+                arg = np.log(Wilson_loop)/delta/delta*1j
+
+                F_all = np.append(F_all,[arg], axis=0) 
+        plt.plot(np.linspace(-2*pi, 2*pi, n1)/pi, np.real(F_all))
+        plt.xlabel('k_x (pi)')
+        plt.ylabel('Berry curvature')
+        if band==0:
+            plt.title('Valence Band')
+        else:
+            plt.title('Conductance Band')
+        plt.show()
+
+
+if __name__ == '__main__':
+    main()

academic_codes/topological_invariant/2022.04.21_Berry_curvature/Berry_curvature_distribution_of_graphene_under_broken_inversion_symmery_with_the_efficient_method.py

@@ -0,0 +1,63 @@
+"""
+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/20869
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import *  
+import cmath
+
+
+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), dtype=complex)
+    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():
+    n = 2000  # 取点密度
+    delta = 4*pi/n  # 求导的偏离量
+    for band in range(2):
+        F_all = []  # 贝里曲率
+        for kx in np.linspace(-2*pi, 2*pi, n):
+            for ky in [0]: # 这里只考虑ky=0对称轴上的情况
+                H = hamiltonian(kx, ky)
+                eigenvalue, eigenvector = np.linalg.eig(H)
+                vector = eigenvector[:, np.argsort(np.real(eigenvalue))[band]]  # 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))[band]]   # 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))[band]]  # 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))[band]]  # 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))/delta/delta*1j
+
+                F_all = np.append(F_all,[F], axis=0) 
+        plt.plot(np.linspace(-2*pi, 2*pi, n)/pi, np.real(F_all))
+        plt.xlabel('k_x (pi)')
+        plt.ylabel('Berry curvature')
+        if band==0:
+            plt.title('Valence Band')
+        else:
+            plt.title('Conductance Band')
+        plt.show()
+
+
+if __name__ == '__main__':
+    main()