category

academic_codes/topological_invariant/2022.07.12_gauge_fixing/example_find_vector_with_fixed_gauge_by_making_one_component_real.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/22604
+"""
+
+
+import numpy as np
+import cmath
+
+def find_vector_with_fixed_gauge_by_making_one_component_real(vector, precision=0.005, index=None):
+    vector = np.array(vector)
+    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 
+
+vector_1 = np.array([np.sqrt(0.5), np.sqrt(0.5)])*cmath.exp(np.random.uniform(0, 1)*1j)
+vector_2 = np.array([1, 0])*cmath.exp(np.random.uniform(0, 1)*1j)
+
+print('\n随机规范的原向量：', vector_1)
+vector_1 = find_vector_with_fixed_gauge_by_making_one_component_real(vector_1, precision=0.001)
+print('固定规范后的向量：', vector_1)
+
+print('\n随机规范的原向量：', vector_2)
+vector_2 = find_vector_with_fixed_gauge_by_making_one_component_real(vector_2, precision=0.001)
+print('固定规范后的向量：', vector_2)
+
+
+# # 可直接使用Guan软件包来调用以上函数：https://py.guanjihuan.com。
+# # 安装命令：pip install --upgrade guan。
+
+# import guan
+
+# print('\n随机规范的原向量：', vector_1)
+# vector_1 = guan.find_vector_with_fixed_gauge_by_making_one_component_real(vector_1, precision=0.001)
+# print('固定规范后的向量：', vector_1)
+
+# print('\n随机规范的原向量：', vector_2)
+# vector_2 = guan.find_vector_with_fixed_gauge_by_making_one_component_real(vector_2, precision=0.001)
+# print('固定规范后的向量：', vector_2)

academic_codes/topological_invariant/2022.07.12_gauge_fixing/example_rotation_of_degenerate_vectors.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/22604
+"""
+
+import numpy as np
+import math
+import cmath
+# from numba import jit
+
+# @jit
+def rotation_of_degenerate_vectors(vector1, vector2, index1=None, index2=None, precision=0.01, criterion=0.01, show_theta=0):
+    vector1 = np.array(vector1)
+    vector2 = np.array(vector2)
+    if index1 == None:
+        index1 = np.argmax(np.abs(vector1))
+    if index2 == None:
+        index2 = np.argmax(np.abs(vector2))
+    if np.abs(vector1[index2])>criterion or np.abs(vector2[index1])>criterion:
+        for theta in np.arange(0, 2*math.pi, precision):
+            if show_theta==1:
+                print(theta)
+            for phi1 in np.arange(0, 2*math.pi, precision):
+                for phi2 in np.arange(0, 2*math.pi, precision):
+                    vector1_test = cmath.exp(1j*phi1)*vector1*math.cos(theta)+cmath.exp(1j*phi2)*vector2*math.sin(theta)
+                    vector2_test = -cmath.exp(-1j*phi2)*vector1*math.sin(theta)+cmath.exp(-1j*phi1)*vector2*math.cos(theta)
+                    if np.abs(vector1_test[index2])<criterion and np.abs(vector2_test[index1])<criterion:
+                        vector1 = vector1_test
+                        vector2 = vector2_test
+                        break
+                if np.abs(vector1_test[index2])<criterion and np.abs(vector2_test[index1])<criterion:
+                    break
+            if np.abs(vector1_test[index2])<criterion and np.abs(vector2_test[index1])<criterion:
+                break
+    return vector1, vector2
+
+def hamiltonian_of_bbh_model(kx, ky, gamma_x=0.5, gamma_y=0.5, lambda_x=1, lambda_y=1):
+    # label of atoms in a unit cell
+    # (2) —— (0)
+    #  |      |
+    # (1) —— (3)   
+    hamiltonian = np.zeros((4, 4), dtype=complex)
+    hamiltonian[0, 2] = gamma_x+lambda_x*cmath.exp(1j*kx)
+    hamiltonian[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx)
+    hamiltonian[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky)
+    hamiltonian[1, 2] = -gamma_y-lambda_y*cmath.exp(-1j*ky)
+    hamiltonian[2, 0] = np.conj(hamiltonian[0, 2])
+    hamiltonian[3, 1] = np.conj(hamiltonian[1, 3])
+    hamiltonian[3, 0] = np.conj(hamiltonian[0, 3])
+    hamiltonian[2, 1] = np.conj(hamiltonian[1, 2]) 
+    return hamiltonian
+
+# For kx=0
+print('\nFor kx=0:\n')
+eigenvalue, eigenvector = np.linalg.eigh(hamiltonian_of_bbh_model(kx=0, ky=0))
+print(eigenvalue, '\n')
+print(eigenvector[:, 0])
+print(eigenvector[:, 1], '\n')
+
+# For kx=0.005
+print('\nFor kx=0.005:\n')
+eigenvalue, eigenvector = np.linalg.eigh(hamiltonian_of_bbh_model(kx=0.005, ky=0))
+print(eigenvalue, '\n')
+print(eigenvector[:, 0])
+print(eigenvector[:, 1], '\n\n')
+
+# Rotaion
+vector1, vector2 = rotation_of_degenerate_vectors(eigenvector[:, 0], eigenvector[:, 1], precision=0.01, criterion=0.01, show_theta=1)
+print()
+print(vector1)
+print(vector2, '\n')
+
+
+# # 可直接使用Guan软件包来调用以上函数：https://py.guanjihuan.com。
+# # 安装命令：pip install --upgrade guan。
+# import guan
+# vector1, vector2 = guan.rotation_of_degenerate_vectors(vector1, vector2, index1=None, index2=None, precision=0.01, criterion=0.01, show_theta=0)
+# hamiltonian = guan.hamiltonian_of_bbh_model(kx, ky, gamma_x=0.5, gamma_y=0.5, lambda_x=1, lambda_y=1)