update

2020.06.25_numerically_verify_Green_functions_in_Dyson_equations/numerically_verify_Green_functions_in_Dyson_equations.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/4396
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+import copy
+import time
+
+
+def matrix_00(width):    # 一个切片（slide)内的哈密顿量
+    h00 = np.zeros((width, width))
+    for width0 in range(width-1):
+        h00[width0, width0+1] = 1
+        h00[width0+1, width0] = 1
+    return h00
+
+
+def matrix_01(width):    # 切片之间的跃迁项（hopping）
+    h01 = np.identity(width)
+    return h01
+
+
+def matrix_whole(width, length):   # 方格子整体的哈密顿量，宽度为width，长度为length
+    hamiltonian = np.zeros((width*length, width*length))
+    for x in range(length):
+        for y in range(width-1):
+            hamiltonian[x*width+y, x*width+y+1] = 1
+            hamiltonian[x*width+y+1, x*width+y] = 1
+    for x in range(length-1):
+        for y in range(width):
+            hamiltonian[x*width+y, (x+1)*width+y] = 1
+            hamiltonian[(x+1)*width+y, x*width+y] = 1
+    return hamiltonian
+
+
+def main():
+    width =4   # 方格子的宽度
+    length = 200  # 方格子的长度
+    h00 = matrix_00(width)  # 一个切片（slide)内的哈密顿量
+    h01 = matrix_01(width)   # 切片之间的跃迁项（hopping）
+    hamiltonian = matrix_whole(width, length)  # 方格子整体的哈密顿量，宽度为width，长度为length
+    fermi_energy = 0.1   # 费米能取为0.1为例。按理来说计算格林函数时，这里需要加上一个无穷小的虚数，但Python中好像不加也不会有什么问题。
+
+    start_1= time.perf_counter()
+    green = General_way(fermi_energy, hamiltonian)  # 利用通常直接求逆的方法得到整体的格林函数green
+    end_1 = time.perf_counter()
+    start_2= time.perf_counter()
+    green_0n_n = Dyson_way(fermi_energy, h00, h01, length)  # 利用Dyson方程得到的格林函数green_0n
+    end_2 = time.perf_counter()
+
+    # print(green)
+    print('\n整体格林函数中的一个分块矩阵green_0n：\n', green[0:width, (length-1)*width+0:(length-1)*width+width])  # a:b代表 a <= x < b，左闭右开
+    print('Dyson方程得到的格林函数green_0n：\n', green_0n_n)
+    print('观察以上两个矩阵，可以直接看出两个矩阵完全相同。\n')
+
+    print('General_way执行时间=', end_1-start_1) 
+    print('Dyson_way执行时间=', end_2-start_2)
+
+
+def General_way(fermi_energy, hamiltonian):
+    dim_hamiltonian = hamiltonian.shape[0]
+    green = np.linalg.inv((fermi_energy)*np.eye(dim_hamiltonian)-hamiltonian)
+    return green
+
+
+def Dyson_way(fermi_energy, h00, h01, length):
+    dim = h00.shape[0]
+    for ix in range(length):
+        if ix == 0:
+            green_nn_n = np.linalg.inv(fermi_energy*np.identity(dim)-h00)   # 如果有左电极，还需要减去左电极的自能left_self_energy
+            green_0n_n = copy.deepcopy(green_nn_n)  # 如果直接用等于，两个变量会指向相同的id，改变一个值，另外一个值可能会发生改变，容易出错，所以要用上这个COPY
+        elif ix != length-1:
+            green_nn_n = np.linalg.inv(fermi_energy*np.identity(dim)-h00-np.dot(np.dot(h01.transpose().conj(), green_nn_n), h01))
+            green_0n_n = np.dot(np.dot(green_0n_n, h01), green_nn_n)
+        else:  # 这里和（elif ix != length-1）中的内容完全一样，但如果有右电极，这里是还需要减去右电极的自能right_self_energy
+            green_nn_n = np.linalg.inv(fermi_energy*np.identity(dim)-h00-np.dot(np.dot(h01.transpose().conj(), green_nn_n), h01))  
+            green_0n_n = np.dot(np.dot(green_0n_n, h01), green_nn_n)
+    return green_0n_n
+
+
+if __name__ == '__main__':
+    main()