update

2021.03.01_Wilson_loop_in_SSH_model/Wilson_loop_in_SSH_model.py

@@ -0,0 +1,77 @@
+"""
+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/10064
+"""
+
+import numpy as np
+import cmath
+from math import *
+
+
+def hamiltonian(k):  # SSH模型哈密顿量
+    gamma = 0.5
+    lambda0 = 1
+    h = np.zeros((2, 2), dtype=complex)
+    h[0,0] = 0
+    h[1,1] = 0
+    h[0,1] = gamma+lambda0*cmath.exp(-1j*k)
+    h[1,0] = gamma+lambda0*cmath.exp(1j*k)
+    return h
+
+
+def main():
+    Num_k = 100
+    k_array = np.linspace(-pi, pi, Num_k)
+    vector_array = []
+    for k in k_array:
+        vector  = get_occupied_bands_vectors(k, hamiltonian)   
+        vector_array.append(vector)
+        # vector_array.append(vector*cmath.exp(1j*np.random.uniform(0, pi))) # 随机相位测试
+
+    # 波函数固定一个规范
+    vector_sum = 0
+    for i0 in range(Num_k):
+        vector_sum += np.abs(vector_array[i0])
+    index = np.argmax(np.abs(vector_sum))
+    for i0 in range(Num_k):
+        vector_array[i0] = find_vector_with_fixed_gauge_by_making_one_component_real(vector_array[i0], index=index)
+
+    # # 波函数固定一个规范
+    # import guan
+    # vector_array = guan.find_vector_array_with_fixed_gauge_by_making_one_component_real(vector_array, precision=0.005)
+
+    # 计算Wilson loop
+    W_k = 1
+    for i0 in range(Num_k-1):
+        F = np.dot(vector_array[i0+1].transpose().conj(), vector_array[i0])
+        W_k = np.dot(F, W_k)
+    nu = np.log(W_k)/2/pi/1j
+    # if np.real(nu) < 0:
+    #     nu += 1
+    print('p=', nu, '\n')
+    
+
+def get_occupied_bands_vectors(x, matrix):  
+    matrix0 = matrix(x)
+    eigenvalue, eigenvector = np.linalg.eig(matrix0) 
+    vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] 
+    return vector
+
+
+def find_vector_with_fixed_gauge_by_making_one_component_real(vector, precision=0.005, index=None):
+    if index == None:
+        index = np.argmax(np.abs(vector))
+    sign_pre = np.sign(np.imag(vector[index]))
+    for phase in np.arange(0, 2*np.pi, precision):
+        sign =  np.sign(np.imag(vector[index]*cmath.exp(1j*phase)))
+        if np.abs(np.imag(vector[index]*cmath.exp(1j*phase))) < 1e-9 or sign == -sign_pre:
+            break
+        sign_pre = sign
+    vector = vector*cmath.exp(1j*phase)
+    if np.real(vector[index]) < 0:
+        vector = -vector
+    return vector
+
+
+if __name__ == '__main__':
+    main()

2021.03.01_Wilson_loop_in_SSH_model/Wilson_loop_in_SSH_model_with_better_code.py

@@ -0,0 +1,53 @@
+"""
+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/10064
+"""
+
+import numpy as np
+import cmath
+from math import *
+
+
+def hamiltonian(k):
+    gamma = 0.5
+    lambda0 = 1
+    delta = 0
+    h = np.zeros((2, 2), dtype=complex)
+    h[0,0] = delta
+    h[1,1] = -delta
+    h[0,1] = gamma+lambda0*cmath.exp(-1j*k)
+    h[1,0] = gamma+lambda0*cmath.exp(1j*k)
+    return h
+
+
+def main():
+    Num_k = 100
+    k_array = np.linspace(-pi, pi, Num_k)
+    vector_array = []
+    for k in k_array:
+        vector  = get_occupied_bands_vectors(k, hamiltonian)   
+        if k != pi:
+            vector_array.append(vector)
+        else:
+            vector_array.append(vector_array[0])
+
+    # 计算Wilson loop
+    W_k = 1
+    for i0 in range(Num_k-1):
+        F = np.dot(vector_array[i0+1].transpose().conj(), vector_array[i0])
+        W_k = np.dot(F, W_k)
+    nu = np.log(W_k)/2/pi/1j
+    # if np.real(nu) < 0:
+    #     nu += 1
+    print('p=', nu, '\n')
+    
+
+def get_occupied_bands_vectors(x, matrix):  
+    matrix0 = matrix(x)
+    eigenvalue, eigenvector = np.linalg.eig(matrix0) 
+    vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]] 
+    return vector
+
+
+if __name__ == '__main__':
+    main()