学术, 拓扑不变量

外尔半金属在动量截面上的陈数(附Python代码)

这是之前的一篇:外尔半金属的哈密顿量和费米弧(附Python代码)。本篇计算的是外尔半金属在动量截面上的陈数。

首先把外尔半金属模型的哈密顿量晶格离散化,得到布里渊区的范围在[-\pi, \pi]之间。离散的方法参照:常用的泰勒近似,即

k  \approx \mathrm{sin}(k)

k^2  \approx  2(1-\mathrm{cos}(k))

离散化后,外尔金属计算能带的代码:

"""
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/6896
"""

import numpy as np
from math import *
import matplotlib.pyplot as plt


def main():
    k1 = np.arange(-pi, pi, 0.05)
    k2 = np.arange(-pi, pi, 0.05)
    plot_bands_two_dimension(k1, k2, hamiltonian)


def hamiltonian(kx,kz,ky=0):  # Weyl semimetal
    A = 1
    M0 = 1
    M1 = 1
    H = A*(sin(kx)*sigma_x()+sin(ky)*sigma_y())+(M0-M1*(2*(1-cos(kx))+2*(1-cos(ky))+2*(1-cos(kz))))*sigma_z()
    return H


def sigma_x():
    return np.array([[0, 1],[1, 0]])


def sigma_y():
    return np.array([[0, -1j],[1j, 0]])


def sigma_z():
    return np.array([[1, 0],[0, -1]])


def plot_bands_two_dimension(k1, k2, hamiltonian):
    from mpl_toolkits.mplot3d import Axes3D
    from matplotlib import cm
    from matplotlib.ticker import LinearLocator, FormatStrFormatter
    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.xlabel('kx')
    plt.ylabel('kz') 
    ax.set_zlabel('E')  
    plt.show()


if __name__ == '__main__':
    main()

计算结果为:

把离散化的哈密顿量和之前的这篇代码相结合:陈数Chern number的计算(高效法,附Python/Matlab代码),得到:

"""
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/6896
"""

import numpy as np
from math import *
import matplotlib.pyplot as plt
import time
import cmath


def hamiltonian(kx,ky,kz):  # Weyl semimetal
    A = 1
    M0 = 1
    M1 = 1
    H = A*(sin(kx)*sigma_x()+sin(ky)*sigma_y())+(M0-M1*(2*(1-cos(kx))+2*(1-cos(ky))+2*(1-cos(kz))))*sigma_z()
    return H


def sigma_x():
    return np.array([[0, 1],[1, 0]])


def sigma_y():
    return np.array([[0, -1j],[1j, 0]])


def sigma_z():
    return np.array([[1, 0],[0, -1]])


def main():
    start_time = time.time()
    n = 50 
    delta = 2*pi/n
    kz_array = np.arange(-pi, pi, 0.1)
    chern_number_array = np.zeros(kz_array.shape[0])
    i0 = 0
    for kz in kz_array:
        print('kz=', kz)
        chern_number = 0  # 陈数初始化
        for kx in  np.arange(-pi, pi, 2*pi/n):
            for ky in  np.arange(-pi, pi, 2*pi/n):
                H = hamiltonian(kx, ky, kz)
                eigenvalue, eigenvector = np.linalg.eig(H)
                vector = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]  # 价带波函数
            
                H_delta_kx = hamiltonian(kx+delta, ky, kz) 
                eigenvalue, eigenvector = np.linalg.eig(H_delta_kx)
                vector_delta_kx = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]   # 略偏离kx的波函数

                H_delta_ky = hamiltonian(kx, ky+delta, kz)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta_ky)
                vector_delta_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]  # 略偏离ky的波函数
                
                H_delta_kx_ky = hamiltonian(kx+delta, ky+delta, kz)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta_kx_ky)
                vector_delta_kx_ky = eigenvector[:, np.argsort(np.real(eigenvalue))[0]]  # 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))

                # 陈数(chern number)
                chern_number = chern_number + F
        chern_number = chern_number/(2*pi*1j)
        print('Chern number = ', chern_number, '\n')
        chern_number_array[i0] = np.real(chern_number)
        i0 += 1
    end_time = time.time()
    print('运行时间(min)=', (end_time-start_time)/60)
    plt.plot(kz_array, chern_number_array, 'o-')
    plt.xlabel('kz')
    plt.ylabel('Chern number') 
    plt.show()


if __name__ == '__main__':
    main()

计算结果为:

可以看出:在两个外尔点之间的动量截面上,陈数不为零。

使用Guan开源软件包(https://py.guanjihuan.com),代码如下:

"""
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/6896
"""

import numpy as np
import guan
import functools

def main():
    kz_array = np.arange(-np.pi, np.pi, 0.1)
    chern_number_array = []
    for kz in kz_array:
        print(kz)
        hamiltonian_function = functools.partial(hamiltonian, kz=kz)
        chern_number = guan.calculate_chern_number_for_square_lattice_with_efficient_method(hamiltonian_function)
        chern_number_array.append(chern_number)
    guan.plot(kz_array, chern_number_array, style='-o')


def hamiltonian(kx,ky,kz):  # Weyl semimetal
    A = 1
    M0 = 1
    M1 = 1
    H = A*(np.sin(kx)*guan.sigma_x()+np.sin(ky)*guan.sigma_y())+(M0-M1*(2*(1-np.cos(kx))+2*(1-np.cos(ky))+2*(1-np.cos(kz))))*guan.sigma_z()
    return H


if __name__ == '__main__':
    main()

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5 thoughts on “外尔半金属在动量截面上的陈数(附Python代码)”

    1. 谢谢你的建议。我对这方面还不是很了解,接触的不多,之后如果有学习到,可能会写。

  1. 博主,我看到你今年文章里面有计算Loop Polarization,可以出一篇相关的文章嘛

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