update

2019.10.24_Halmiltonian_and_bands_of_Haldane_model/1D_Haldane_model.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/410
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import *
+import cmath 
+import functools
+
+
+def hamiltonian(k, N, M, t1, t2, phi):  # Haldane哈密顿量（N是条带的宽度参数）
+    # 初始化为零矩阵
+    h00 = np.zeros((4*N, 4*N), dtype=complex)
+    h01 = np.zeros((4*N, 4*N), dtype=complex)
+
+    # 原胞内的跃迁h00
+    for i in range(N):
+        h00[i*4+0, i*4+0] = M
+        h00[i*4+1, i*4+1] = -M
+        h00[i*4+2, i*4+2] = M
+        h00[i*4+3, i*4+3] = -M
+
+        # 最近邻
+        h00[i*4+0, i*4+1] = t1
+        h00[i*4+1, i*4+0] = t1
+        h00[i*4+1, i*4+2] = t1
+        h00[i*4+2, i*4+1] = t1
+        h00[i*4+2, i*4+3] = t1
+        h00[i*4+3, i*4+2] = t1
+
+        # 次近邻
+        h00[i*4+0, i*4+2] = t2*cmath.exp(-1j*phi)
+        h00[i*4+2, i*4+0] = h00[i*4+0, i*4+2].conj()
+        h00[i*4+1, i*4+3] = t2*cmath.exp(-1j*phi)
+        h00[i*4+3, i*4+1] = h00[i*4+1, i*4+3].conj()
+    for i in range(N-1):
+        # 最近邻
+        h00[i*4+3, (i+1)*4+0] = t1
+        h00[(i+1)*4+0, i*4+3] = t1
+
+        # 次近邻
+        h00[i*4+2, (i+1)*4+0] = t2*cmath.exp(1j*phi)
+        h00[(i+1)*4+0, i*4+2] = h00[i*4+2, (i+1)*4+0].conj()
+        h00[i*4+3, (i+1)*4+1] = t2*cmath.exp(1j*phi)
+        h00[(i+1)*4+1, i*4+3] = h00[i*4+3, (i+1)*4+1].conj()
+
+    # 原胞间的跃迁h01
+    for i in range(N):
+        # 最近邻
+        h01[i*4+1, i*4+0] = t1
+        h01[i*4+2, i*4+3] = t1
+
+        # 次近邻
+        h01[i*4+0, i*4+0] = t2*cmath.exp(1j*phi)
+        h01[i*4+1, i*4+1] = t2*cmath.exp(-1j*phi)
+        h01[i*4+2, i*4+2] = t2*cmath.exp(1j*phi)
+        h01[i*4+3, i*4+3] = t2*cmath.exp(-1j*phi)
+
+        h01[i*4+1, i*4+3] = t2*cmath.exp(1j*phi)
+        h01[i*4+2, i*4+0] = t2*cmath.exp(-1j*phi)
+        if i != 0:
+            h01[i*4+1, (i-1)*4+3] = t2*cmath.exp(1j*phi)
+    for i in range(N-1):
+        h01[i*4+2, (i+1)*4+0] = t2*cmath.exp(-1j*phi)
+
+    matrix = h00 + h01*cmath.exp(1j*k) + h01.transpose().conj()*cmath.exp(-1j*k)
+    return matrix
+
+
+def main():
+    hamiltonian0 = functools.partial(hamiltonian, N=40, M=2/3, t1=1, t2=1/3, phi=pi/4)
+    k = np.linspace(-pi, pi, 300)
+    plot_bands_one_dimension(k, hamiltonian0)
+
+
+def plot_bands_one_dimension(k, hamiltonian):
+    dim = hamiltonian(0).shape[0]
+    dim_k = k.shape[0]
+    eigenvalue_k = np.zeros((dim_k, dim))
+    i0 = 0
+    for k0 in k:
+        matrix0 = hamiltonian(k0)
+        eigenvalue, eigenvector = np.linalg.eig(matrix0)
+        eigenvalue_k[i0, :] = np.sort(np.real(eigenvalue[:]))
+        i0 += 1
+    for dim0 in range(dim):
+        plt.plot(k, eigenvalue_k[:, dim0], '-k')
+    plt.show()
+
+
+if __name__ == '__main__':
+    main()

2019.10.24_Halmiltonian_and_bands_of_Haldane_model/2D_Haldane_model.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/410
+"""
+
+import numpy as np
+import matplotlib.pyplot as plt
+from math import * 
+import cmath 
+import functools
+
+
+def hamiltonian(k1, k2, M, t1, t2, phi, a=1/sqrt(3)):  # Haldane哈密顿量(a为原子间距，不赋值的话默认为1/sqrt(3)）
+    # 初始化为零矩阵
+    h0 = np.zeros((2, 2), dtype=complex)
+    h1 = np.zeros((2, 2), dtype=complex)
+    h2 = np.zeros((2, 2), dtype=complex)
+
+    # 质量项(mass term)，用于打开带隙
+    h0[0, 0] = M
+    h0[1, 1] = -M
+
+    # 最近邻项
+    h1[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))
+    h1[0, 1] = h1[1, 0].conj()
+
+    # 最近邻项也可写成这种形式
+    # h1[1, 0] = t1+t1*cmath.exp(1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)+t1*cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a)
+    # h1[0, 1] = h1[1, 0].conj()
+
+    # 次近邻项
+    h2[0, 0] = t2*cmath.exp(-1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
+    h2[1, 1] = t2*cmath.exp(1j*phi)*(cmath.exp(1j*sqrt(3)*k1*a)+cmath.exp(-1j*sqrt(3)/2*k1*a+1j*3/2*k2*a)+cmath.exp(-1j*sqrt(3)/2*k1*a-1j*3/2*k2*a))
+
+    matrix = h0 + h1 + h2 + h2.transpose().conj()
+    return matrix
+
+
+def main():
+    hamiltonian0 = functools.partial(hamiltonian, M=2/3, t1=1, t2=1/3, phi=pi/4, a=1/sqrt(3))
+    k1 = np.linspace(-2*pi, 2*pi, 500)
+    k2 = np.linspace(-2*pi, 2*pi, 500)
+    plot_bands_two_dimension(k1, k2, hamiltonian0)
+
+
+def plot_bands_two_dimension(k1, k2, hamiltonian):
+    from matplotlib import cm
+    dim = hamiltonian(0, 0).shape[0]
+    dim1 = k1.shape[0]
+    dim2 = k2.shape[0]
+    eigenvalue_k = np.zeros((dim2, dim1, dim))
+    i0 = 0
+    for k10 in k1:
+        j0 = 0
+        for k20 in k2:
+            matrix0 = hamiltonian(k10, k20)
+            eigenvalue, eigenvector = np.linalg.eig(matrix0)
+            eigenvalue_k[j0, i0, :] = np.sort(np.real(eigenvalue[:]))
+            j0 += 1
+        i0 += 1
+    fig = plt.figure()
+    ax = fig.gca(projection='3d')
+    k1, k2 = np.meshgrid(k1, k2)
+    for dim0 in range(dim):
+        ax.plot_surface(k1, k2, eigenvalue_k[:, :, dim0], cmap=cm.coolwarm, linewidth=0, antialiased=False)
+    plt.show()
+
+
+if __name__ == '__main__':
+    main()