拓扑不变量, 学术

贝里曲率的计算(写成函数形式,附Python代码)

这是之前的一篇:贝里曲率的计算(附Python代码)。本篇把其中的“高效法”和“Wilson loop方法”的贝里曲率计算代码改写成函数形式。

此外加上这两篇文章的内容:陈数Chern number的计算(多条能带的高效法,附Python代码)陈数Chern number的计算(多条能带的Wilson loop方法,附Python代码),多条能带同时计算可以支持能带交叉或简并的情况。

还是以这篇为例子:空间反演对称性破缺的石墨烯的贝里曲率分布(附Python代码)。计算结果和之前的相同。

本篇代码目录:

  1. 贝里曲率 高效法(写成函数形式)
  2. 贝里曲率 多条能带的高效法(写成函数形式)
  3. 贝里曲率 Wilson loop方法(写成函数形式)
  4. 贝里曲率 多条能带的Wilson loop方法(写成函数形式)

部分函数是复制自Guan软件包的源码。为了使代码更加简洁,可以删除函数代码,直接调用Guan函数。官网:https://py.guanjihuan.com。安装方法:pip install --upgrade guan

补充:Kubo公式也可以推广到简并的情况,计算方法可以看这篇综述文献的公式(73):First-principle calculations of the Berry curvature of Bloch states for charge and spin transport of electrons。该文献由评论区提供。这里暂时没考虑这个公式。

一、贝里曲率 高效法(写成函数形式)

"""
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/24059
"""

import numpy as np
from math import *  
import cmath
import math

def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)):  # 石墨烯哈密顿量(a为原子间距,不赋值的话默认为1/sqrt(3))
    h = np.zeros((2, 2), dtype=complex)
    h[0, 0] = 0.28/2
    h[1, 1] = -0.28/2
    h[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))
    h[0, 1] = h[1, 0].conj()
    return h


def main():
    k_array, berry_curvature_array = calculate_berry_curvature_with_efficient_method(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision=500, print_show=0)
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    dim = berry_curvature_array.shape
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0

    # import guan
    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_efficient_method(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision=500, print_show=0)
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # dim = berry_curvature_array.shape
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0


def calculate_berry_curvature_with_efficient_method(hamiltonian_function, k_min=-math.pi, k_max=math.pi, precision=100, print_show=0):
    if np.array(hamiltonian_function(0, 0)).shape==():
        dim = 1
    else:
        dim = np.array(hamiltonian_function(0, 0)).shape[0]   
    delta = (k_max-k_min)/precision
    k_array = np.arange(k_min, k_max, delta)
    berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0], dim), dtype=complex)
    i0 = 0
    for kx in k_array:
        if print_show == 1:
            print(kx)
        j0 = 0
        for ky in k_array:
            H = hamiltonian_function(kx, ky)
            eigenvalue, vector = np.linalg.eigh(H) 
            H_delta_kx = hamiltonian_function(kx+delta, ky) 
            eigenvalue, vector_delta_kx = np.linalg.eigh(H_delta_kx) 
            H_delta_ky = hamiltonian_function(kx, ky+delta)
            eigenvalue, vector_delta_ky = np.linalg.eigh(H_delta_ky) 
            H_delta_kx_ky = hamiltonian_function(kx+delta, ky+delta)
            eigenvalue, vector_delta_kx_ky = np.linalg.eigh(H_delta_kx_ky) 
            for i in range(dim):
                vector_i = vector[:, i]
                vector_delta_kx_i = vector_delta_kx[:, i]
                vector_delta_ky_i = vector_delta_ky[:, i]
                vector_delta_kx_ky_i = vector_delta_kx_ky[:, i]
                Ux = np.dot(np.conj(vector_i), vector_delta_kx_i)/abs(np.dot(np.conj(vector_i), vector_delta_kx_i))
                Uy = np.dot(np.conj(vector_i), vector_delta_ky_i)/abs(np.dot(np.conj(vector_i), vector_delta_ky_i))
                Ux_y = np.dot(np.conj(vector_delta_ky_i), vector_delta_kx_ky_i)/abs(np.dot(np.conj(vector_delta_ky_i), vector_delta_kx_ky_i))
                Uy_x = np.dot(np.conj(vector_delta_kx_i), vector_delta_kx_ky_i)/abs(np.dot(np.conj(vector_delta_kx_i), vector_delta_kx_ky_i))
                berry_curvature = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))/delta/delta*1j
                berry_curvature_array[j0, i0, i] = berry_curvature
            j0 += 1
        i0 += 1
    return k_array, berry_curvature_array


def plot_3d_surface(x_array, y_array, matrix, xlabel='x', ylabel='y', zlabel='z', title='', fontsize=20, labelsize=15, show=1, save=0, filename='a', file_format='.jpg', dpi=300, z_min=None, z_max=None, rcount=100, ccount=100): 
    import matplotlib.pyplot as plt
    from matplotlib import cm
    from matplotlib.ticker import LinearLocator
    matrix = np.array(matrix)
    fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
    plt.subplots_adjust(bottom=0.1, right=0.65) 
    x_array, y_array = np.meshgrid(x_array, y_array)
    if len(matrix.shape) == 2:
        surf = ax.plot_surface(x_array, y_array, matrix, rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    elif len(matrix.shape) == 3:
        for i0 in range(matrix.shape[2]):
            surf = ax.plot_surface(x_array, y_array, matrix[:,:,i0], rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_zlabel(zlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.zaxis.set_major_locator(LinearLocator(5)) 
    ax.zaxis.set_major_formatter('{x:.2f}')  
    if z_min!=None or z_max!=None:
        if z_min==None:
            z_min=matrix.min()
        if z_max==None:
            z_max=matrix.max()
        ax.set_zlim(z_min, z_max)
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
    [label.set_fontname('Times New Roman') for label in labels] 
    cax = plt.axes([0.8, 0.1, 0.05, 0.8]) 
    cbar = fig.colorbar(surf, cax=cax)  
    cbar.ax.tick_params(labelsize=labelsize)
    for l in cbar.ax.yaxis.get_ticklabels():
        l.set_family('Times New Roman')
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2): 
    import matplotlib.pyplot as plt
    fig, ax = plt.subplots()
    plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
    ax.grid()
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels()
    [label.set_fontname('Times New Roman') for label in labels]
    ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    if y_min!=None or y_max!=None:
        if y_min==None:
            y_min=min(y_array)
        if y_max==None:
            y_max=max(y_array)
        ax.set_ylim(y_min, y_max)
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


if __name__ == '__main__':
    main()

二、贝里曲率 多条能带的高效法(写成函数形式)

"""
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/24059
"""

import numpy as np
from math import *  
import cmath
import math


def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)):  # 石墨烯哈密顿量(a为原子间距,不赋值的话默认为1/sqrt(3))
    h = np.zeros((2, 2), dtype=complex)
    h[0, 0] = 0.28/2
    h[1, 1] = -0.28/2
    h[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))
    h[0, 1] = h[1, 0].conj()
    return h


def main():
    k_array, berry_curvature_array = calculate_berry_curvature_with_efficient_method_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision=500)
    dim = berry_curvature_array.shape
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0

    k_array, berry_curvature_array = calculate_berry_curvature_with_efficient_method_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision=500)
    dim = berry_curvature_array.shape
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='All Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='All Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0


    # import guan
    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_efficient_method_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision=500)
    # dim = berry_curvature_array.shape
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0

    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_efficient_method_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision=500)
    # dim = berry_curvature_array.shape
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='All Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='All Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0



def calculate_berry_curvature_with_efficient_method_for_degenerate_case(hamiltonian_function, index_of_bands=[0, 1], k_min=-math.pi, k_max=math.pi, precision=100, print_show=0):
    delta = (k_max-k_min)/precision
    k_array = np.arange(k_min, k_max, delta)
    berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0]), dtype=complex)
    i00 = 0
    for kx in np.arange(k_min, k_max, delta):
        if print_show == 1:
            print(kx)
        j00 = 0
        for ky in np.arange(k_min, k_max, delta):
            H = hamiltonian_function(kx, ky)
            eigenvalue, vector = np.linalg.eigh(H) 
            H_delta_kx = hamiltonian_function(kx+delta, ky) 
            eigenvalue, vector_delta_kx = np.linalg.eigh(H_delta_kx) 
            H_delta_ky = hamiltonian_function(kx, ky+delta)
            eigenvalue, vector_delta_ky = np.linalg.eigh(H_delta_ky) 
            H_delta_kx_ky = hamiltonian_function(kx+delta, ky+delta)
            eigenvalue, vector_delta_kx_ky = np.linalg.eigh(H_delta_kx_ky)
            dim = len(index_of_bands)
            det_value = 1
            # first dot product
            dot_matrix = np.zeros((dim , dim), dtype=complex)
            i0 = 0
            for dim1 in index_of_bands:
                j0 = 0
                for dim2 in index_of_bands:
                    dot_matrix[i0, j0] = np.dot(np.conj(vector[:, dim1]), vector_delta_kx[:, dim2])
                    j0 += 1
                i0 += 1
            dot_matrix = np.linalg.det(dot_matrix)/abs(np.linalg.det(dot_matrix))
            det_value = det_value*dot_matrix
            # second dot product
            dot_matrix = np.zeros((dim , dim), dtype=complex)
            i0 = 0
            for dim1 in index_of_bands:
                j0 = 0
                for dim2 in index_of_bands:
                    dot_matrix[i0, j0] = np.dot(np.conj(vector_delta_kx[:, dim1]), vector_delta_kx_ky[:, dim2])
                    j0 += 1
                i0 += 1
            dot_matrix = np.linalg.det(dot_matrix)/abs(np.linalg.det(dot_matrix))
            det_value = det_value*dot_matrix
            # third dot product
            dot_matrix = np.zeros((dim , dim), dtype=complex)
            i0 = 0
            for dim1 in index_of_bands:
                j0 = 0
                for dim2 in index_of_bands:
                    dot_matrix[i0, j0] = np.dot(np.conj(vector_delta_kx_ky[:, dim1]), vector_delta_ky[:, dim2])
                    j0 += 1
                i0 += 1
            dot_matrix = np.linalg.det(dot_matrix)/abs(np.linalg.det(dot_matrix))
            det_value = det_value*dot_matrix
            # four dot product
            dot_matrix = np.zeros((dim , dim), dtype=complex)
            i0 = 0
            for dim1 in index_of_bands:
                j0 = 0
                for dim2 in index_of_bands:
                    dot_matrix[i0, j0] = np.dot(np.conj(vector_delta_ky[:, dim1]), vector[:, dim2])
                    j0 += 1
                i0 += 1
            dot_matrix = np.linalg.det(dot_matrix)/abs(np.linalg.det(dot_matrix))
            det_value= det_value*dot_matrix
            berry_curvature = cmath.log(det_value)/delta/delta*1j
            berry_curvature_array[j00, i00] = berry_curvature
            j00 += 1
        i00 += 1
    return k_array, berry_curvature_array


def plot_3d_surface(x_array, y_array, matrix, xlabel='x', ylabel='y', zlabel='z', title='', fontsize=20, labelsize=15, show=1, save=0, filename='a', file_format='.jpg', dpi=300, z_min=None, z_max=None, rcount=100, ccount=100): 
    import matplotlib.pyplot as plt
    from matplotlib import cm
    from matplotlib.ticker import LinearLocator
    matrix = np.array(matrix)
    fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
    plt.subplots_adjust(bottom=0.1, right=0.65) 
    x_array, y_array = np.meshgrid(x_array, y_array)
    if len(matrix.shape) == 2:
        surf = ax.plot_surface(x_array, y_array, matrix, rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    elif len(matrix.shape) == 3:
        for i0 in range(matrix.shape[2]):
            surf = ax.plot_surface(x_array, y_array, matrix[:,:,i0], rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_zlabel(zlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.zaxis.set_major_locator(LinearLocator(5)) 
    ax.zaxis.set_major_formatter('{x:.2f}')  
    if z_min!=None or z_max!=None:
        if z_min==None:
            z_min=matrix.min()
        if z_max==None:
            z_max=matrix.max()
        ax.set_zlim(z_min, z_max)
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
    [label.set_fontname('Times New Roman') for label in labels] 
    cax = plt.axes([0.8, 0.1, 0.05, 0.8]) 
    cbar = fig.colorbar(surf, cax=cax)  
    cbar.ax.tick_params(labelsize=labelsize)
    for l in cbar.ax.yaxis.get_ticklabels():
        l.set_family('Times New Roman')
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2): 
    import matplotlib.pyplot as plt
    fig, ax = plt.subplots()
    plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
    ax.grid()
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels()
    [label.set_fontname('Times New Roman') for label in labels]
    ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    if y_min!=None or y_max!=None:
        if y_min==None:
            y_min=min(y_array)
        if y_max==None:
            y_max=max(y_array)
        ax.set_ylim(y_min, y_max)
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


if __name__ == '__main__':
    main()

三、贝里曲率 Wilson loop方法(写成函数形式)

"""
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/24059
"""

import numpy as np
from math import *  
import cmath
import math


def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)):  # 石墨烯哈密顿量(a为原子间距,不赋值的话默认为1/sqrt(3))
    h = np.zeros((2, 2), dtype=complex)
    h[0, 0] = 0.28/2
    h[1, 1] = -0.28/2
    h[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))
    h[0, 1] = h[1, 0].conj()
    return h


def main():
    k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    dim = berry_curvature_array.shape
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0

    # import guan
    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop(hamiltonian_function=hamiltonian, k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 0]), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array[:, :, 1]), title='Conductance Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # dim = berry_curvature_array.shape
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 0]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :, 1]), title='Conductance Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0


def calculate_berry_curvature_with_wilson_loop(hamiltonian_function, k_min=-math.pi, k_max=math.pi, precision_of_plaquettes=20, precision_of_wilson_loop=5, print_show=0):
    if np.array(hamiltonian_function(0, 0)).shape==():
        dim = 1
    else:
        dim = np.array(hamiltonian_function(0, 0)).shape[0]   
    delta = (k_max-k_min)/precision_of_plaquettes
    k_array = np.arange(k_min, k_max, delta)
    berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0], dim), dtype=complex)
    i00 = 0
    for kx in k_array:
        if print_show == 1:
            print(kx)
        j00 = 0
        for ky in k_array:
            vector_array = []
            # line_1
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta/precision_of_wilson_loop*i0, ky) 
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_2
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta, ky+delta/precision_of_wilson_loop*i0)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_3
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta-delta/precision_of_wilson_loop*i0, ky+delta)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_4
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx, ky+delta-delta/precision_of_wilson_loop*i0)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            wilson_loop = 1
            for i0 in range(len(vector_array)-1):
                wilson_loop = wilson_loop*np.dot(vector_array[i0].transpose().conj(), vector_array[i0+1])
            wilson_loop = wilson_loop*np.dot(vector_array[len(vector_array)-1].transpose().conj(), vector_array[0])
            berry_curvature = np.log(np.diagonal(wilson_loop))/delta/delta*1j
            berry_curvature_array[j00, i00, :]=berry_curvature
            j00 += 1
        i00 += 1
    return k_array, berry_curvature_array


def plot_3d_surface(x_array, y_array, matrix, xlabel='x', ylabel='y', zlabel='z', title='', fontsize=20, labelsize=15, show=1, save=0, filename='a', file_format='.jpg', dpi=300, z_min=None, z_max=None, rcount=100, ccount=100): 
    import matplotlib.pyplot as plt
    from matplotlib import cm
    from matplotlib.ticker import LinearLocator
    matrix = np.array(matrix)
    fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
    plt.subplots_adjust(bottom=0.1, right=0.65) 
    x_array, y_array = np.meshgrid(x_array, y_array)
    if len(matrix.shape) == 2:
        surf = ax.plot_surface(x_array, y_array, matrix, rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    elif len(matrix.shape) == 3:
        for i0 in range(matrix.shape[2]):
            surf = ax.plot_surface(x_array, y_array, matrix[:,:,i0], rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_zlabel(zlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.zaxis.set_major_locator(LinearLocator(5)) 
    ax.zaxis.set_major_formatter('{x:.2f}')  
    if z_min!=None or z_max!=None:
        if z_min==None:
            z_min=matrix.min()
        if z_max==None:
            z_max=matrix.max()
        ax.set_zlim(z_min, z_max)
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
    [label.set_fontname('Times New Roman') for label in labels] 
    cax = plt.axes([0.8, 0.1, 0.05, 0.8]) 
    cbar = fig.colorbar(surf, cax=cax)  
    cbar.ax.tick_params(labelsize=labelsize)
    for l in cbar.ax.yaxis.get_ticklabels():
        l.set_family('Times New Roman')
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2): 
    import matplotlib.pyplot as plt
    fig, ax = plt.subplots()
    plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
    ax.grid()
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels()
    [label.set_fontname('Times New Roman') for label in labels]
    ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    if y_min!=None or y_max!=None:
        if y_min==None:
            y_min=min(y_array)
        if y_max==None:
            y_max=max(y_array)
        ax.set_ylim(y_min, y_max)
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


if __name__ == '__main__':
    main()

四、贝里曲率 多条能带的Wilson loop方法(写成函数形式)

"""
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/24059
"""

import numpy as np
from math import *  
import cmath
import math


def hamiltonian(k1, k2, t1=2.82, a=1/sqrt(3)):  # 石墨烯哈密顿量(a为原子间距,不赋值的话默认为1/sqrt(3))
    h = np.zeros((2, 2), dtype=complex)
    h[0, 0] = 0.28/2
    h[1, 1] = -0.28/2
    h[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))
    h[0, 1] = h[1, 0].conj()
    return h


def main():
    k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    dim = berry_curvature_array.shape
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0

    k_array, berry_curvature_array = calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    dim = berry_curvature_array.shape
    plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='All Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='All Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0


    # import guan
    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    # dim = berry_curvature_array.shape
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='Valence Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='Valence Band  ky=0', xlabel='kx', ylabel='Berry curvature')  # ky=0

    # k_array, berry_curvature_array = guan.calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function=hamiltonian, index_of_bands=[0, 1], k_min=-2*math.pi, k_max=2*math.pi, precision_of_plaquettes=500, precision_of_wilson_loop=1)
    # dim = berry_curvature_array.shape
    # guan.plot_3d_surface(k_array, k_array, np.real(berry_curvature_array), title='All Band', xlabel='kx', ylabel='ky', zlabel='Berry curvature')
    # guan.plot(k_array, np.real(berry_curvature_array[int(dim[0]/2), :]), title='All Band  ky=0', xlabel='kx', ylabel='Berry curvature') # ky=0



def calculate_berry_curvature_with_wilson_loop_for_degenerate_case(hamiltonian_function, index_of_bands=[0, 1], k_min=-math.pi, k_max=math.pi, precision_of_plaquettes=20, precision_of_wilson_loop=5, print_show=0):
    delta = (k_max-k_min)/precision_of_plaquettes
    k_array = np.arange(k_min, k_max, delta)
    berry_curvature_array = np.zeros((k_array.shape[0], k_array.shape[0]), dtype=complex)
    i000 = 0
    for kx in k_array:
        if print_show == 1:
            print(kx)
        j000 = 0
        for ky in k_array:
            vector_array = []
            # line_1
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta/precision_of_wilson_loop*i0, ky) 
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_2
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta, ky+delta/precision_of_wilson_loop*i0)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_3
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx+delta-delta/precision_of_wilson_loop*i0, ky+delta)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)
            # line_4
            for i0 in range(precision_of_wilson_loop):
                H_delta = hamiltonian_function(kx, ky+delta-delta/precision_of_wilson_loop*i0)  
                eigenvalue, eigenvector = np.linalg.eig(H_delta)
                vector_delta = eigenvector[:, np.argsort(np.real(eigenvalue))]
                vector_array.append(vector_delta)           
            wilson_loop = 1
            dim = len(index_of_bands)
            for i0 in range(len(vector_array)-1):
                dot_matrix = np.zeros((dim , dim), dtype=complex)
                i01 = 0
                for dim1 in index_of_bands:
                    i02 = 0
                    for dim2 in index_of_bands:
                        dot_matrix[i01, i02] = np.dot(vector_array[i0][:, dim1].transpose().conj(), vector_array[i0+1][:, dim2])
                        i02 += 1
                    i01 += 1
                det_value = np.linalg.det(dot_matrix)
                wilson_loop = wilson_loop*det_value
            dot_matrix_plus = np.zeros((dim , dim), dtype=complex)
            i01 = 0
            for dim1 in index_of_bands:
                i02 = 0
                for dim2 in index_of_bands:
                    dot_matrix_plus[i01, i02] = np.dot(vector_array[len(vector_array)-1][:, dim1].transpose().conj(), vector_array[0][:, dim2])
                    i02 += 1
                i01 += 1
            det_value = np.linalg.det(dot_matrix_plus)
            wilson_loop = wilson_loop*det_value
            berry_curvature = np.log(wilson_loop)/delta/delta*1j
            berry_curvature_array[j000, i000]=berry_curvature
            j000 += 1
        i000 += 1
    return k_array, berry_curvature_array


def plot_3d_surface(x_array, y_array, matrix, xlabel='x', ylabel='y', zlabel='z', title='', fontsize=20, labelsize=15, show=1, save=0, filename='a', file_format='.jpg', dpi=300, z_min=None, z_max=None, rcount=100, ccount=100): 
    import matplotlib.pyplot as plt
    from matplotlib import cm
    from matplotlib.ticker import LinearLocator
    matrix = np.array(matrix)
    fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
    plt.subplots_adjust(bottom=0.1, right=0.65) 
    x_array, y_array = np.meshgrid(x_array, y_array)
    if len(matrix.shape) == 2:
        surf = ax.plot_surface(x_array, y_array, matrix, rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    elif len(matrix.shape) == 3:
        for i0 in range(matrix.shape[2]):
            surf = ax.plot_surface(x_array, y_array, matrix[:,:,i0], rcount=rcount, ccount=ccount, cmap=cm.coolwarm, linewidth=0, antialiased=False) 
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_zlabel(zlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.zaxis.set_major_locator(LinearLocator(5)) 
    ax.zaxis.set_major_formatter('{x:.2f}')  
    if z_min!=None or z_max!=None:
        if z_min==None:
            z_min=matrix.min()
        if z_max==None:
            z_max=matrix.max()
        ax.set_zlim(z_min, z_max)
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
    [label.set_fontname('Times New Roman') for label in labels] 
    cax = plt.axes([0.8, 0.1, 0.05, 0.8]) 
    cbar = fig.colorbar(surf, cax=cax)  
    cbar.ax.tick_params(labelsize=labelsize)
    for l in cbar.ax.yaxis.get_ticklabels():
        l.set_family('Times New Roman')
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


def plot(x_array, y_array, xlabel='x', ylabel='y', title='', fontsize=20, labelsize=20, show=1, save=0, filename='a', file_format='.jpg', dpi=300, style='', y_min=None, y_max=None, linewidth=None, markersize=None, adjust_bottom=0.2, adjust_left=0.2): 
    import matplotlib.pyplot as plt
    fig, ax = plt.subplots()
    plt.subplots_adjust(bottom=adjust_bottom, left=adjust_left)
    ax.grid()
    ax.tick_params(labelsize=labelsize) 
    labels = ax.get_xticklabels() + ax.get_yticklabels()
    [label.set_fontname('Times New Roman') for label in labels]
    ax.plot(x_array, y_array, style, linewidth=linewidth, markersize=markersize)
    ax.set_title(title, fontsize=fontsize, fontfamily='Times New Roman')
    ax.set_xlabel(xlabel, fontsize=fontsize, fontfamily='Times New Roman') 
    ax.set_ylabel(ylabel, fontsize=fontsize, fontfamily='Times New Roman') 
    if y_min!=None or y_max!=None:
        if y_min==None:
            y_min=min(y_array)
        if y_max==None:
            y_max=max(y_array)
        ax.set_ylim(y_min, y_max)
    if save == 1:
        plt.savefig(filename+file_format, dpi=dpi) 
    if show == 1:
        plt.show()
    plt.close('all')


if __name__ == '__main__':
    main()
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6 thoughts on “贝里曲率的计算(写成函数形式,附Python代码)”

    1. 如果不是简并的情况,可以把直接多个带的贝里曲率求和。如果是简并的情况,用Kubo公式好像会出一些问题,我暂时还不知道怎么推广这个公式。

      1. 感谢回复,我想计算三维情况下的berry curvature。似乎kubo公式比较容易推广, 用高效法或wilson loop可以实现吗?

        1. 都是可以算的吧。我的理解是:在一个截面上算的贝里曲率只是三维贝里曲率的一个分量。

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