From 9e725e467f539cf6e5131a6e6f4be0e9ac5f8e76 Mon Sep 17 00:00:00 2001 From: guanjihuan Date: Sat, 13 Aug 2022 07:39:18 +0800 Subject: [PATCH] update --- ...bution_with_Wilson_loop_(function_form).py | 83 ++++++++++++++ ...oop_for_degenerate_case_(function_form).py | 102 ++++++++++++++++++ ...th_the_efficient_method_(function_form).py | 71 ++++++++++++ 3 files changed, 256 insertions(+) create mode 100644 academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_(function_form).py create mode 100644 academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_for_degenerate_case_(function_form).py create mode 100644 academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_the_efficient_method_(function_form).py diff --git a/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_(function_form).py b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_(function_form).py new file mode 100644 index 0000000..1aaea88 --- /dev/null +++ b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_(function_form).py @@ -0,0 +1,83 @@ +""" +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 +import guan + + +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))*(1+0j) + 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) + # 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 + + +if __name__ == '__main__': + main() \ No newline at end of file diff --git a/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_for_degenerate_case_(function_form).py b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_for_degenerate_case_(function_form).py new file mode 100644 index 0000000..a283928 --- /dev/null +++ b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_Wilson_loop_for_degenerate_case_(function_form).py @@ -0,0 +1,102 @@ +""" +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 +import guan + + +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))*(1+0j) + 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) + # 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 = 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) + # 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 + + +if __name__ == '__main__': + main() \ No newline at end of file diff --git a/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_the_efficient_method_(function_form).py b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_the_efficient_method_(function_form).py new file mode 100644 index 0000000..afedb1f --- /dev/null +++ b/academic_codes/2022_08_13_Berry_curvature_distribution_in_function_form/Berry_curvature_distribution_with_the_efficient_method_(function_form).py @@ -0,0 +1,71 @@ +""" +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 guan +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))*(1+0j) + 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) + # 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) + vector = guan.calculate_eigenvector(H) + H_delta_kx = hamiltonian_function(kx+delta, ky) + vector_delta_kx = guan.calculate_eigenvector(H_delta_kx) + H_delta_ky = hamiltonian_function(kx, ky+delta) + vector_delta_ky = guan.calculate_eigenvector(H_delta_ky) + H_delta_kx_ky = hamiltonian_function(kx+delta, ky+delta) + vector_delta_kx_ky = guan.calculate_eigenvector(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 + + +if __name__ == '__main__': + main() \ No newline at end of file