From 9f268861250e76244254415f36e111f68ce8b46a Mon Sep 17 00:00:00 2001 From: guanjihuan Date: Thu, 9 Dec 2021 11:17:34 +0800 Subject: [PATCH] update --- .../flat_bands_of_Kagome lattice.nb} | 0 .../BBH_Wilson_loop.py | 83 +++++++++ .../BBH_bands.py | 44 +++++ .../BBH_nested_Wilson_loop.py | 175 ++++++++++++++++++ .../simple_method_for_degenerate_bands.py | 15 ++ 5 files changed, 317 insertions(+) rename academic_codes/{2021.08.09_flat_bands_of_kagome lattice/flat_bands_of_kagome lattice_with_sympy.nb => 2021.08.09_flat_bands_of_Kagome lattice/flat_bands_of_Kagome lattice.nb} (100%) create mode 100644 academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_Wilson_loop.py create mode 100644 academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_bands.py create mode 100644 academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_nested_Wilson_loop.py create mode 100644 academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/simple_method_for_degenerate_bands.py diff --git a/academic_codes/2021.08.09_flat_bands_of_kagome lattice/flat_bands_of_kagome lattice_with_sympy.nb b/academic_codes/2021.08.09_flat_bands_of_Kagome lattice/flat_bands_of_Kagome lattice.nb similarity index 100% rename from academic_codes/2021.08.09_flat_bands_of_kagome lattice/flat_bands_of_kagome lattice_with_sympy.nb rename to academic_codes/2021.08.09_flat_bands_of_Kagome lattice/flat_bands_of_Kagome lattice.nb diff --git a/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_Wilson_loop.py b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_Wilson_loop.py new file mode 100644 index 0000000..cfd79ea --- /dev/null +++ b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_Wilson_loop.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/17984 +""" + +import numpy as np +import cmath +from math import * + +def hamiltonian(kx, ky): # BBH model + # label of atoms in a unit cell + # (2) —— (0) + # | | + # (1) —— (3) + gamma_x = 0.5 # hopping inside one unit cell + lambda_x = 1 # hopping between unit cells + gamma_y = gamma_x + lambda_y = lambda_x + h = np.zeros((4, 4), dtype=complex) + h[0, 2] = gamma_x+lambda_x*cmath.exp(1j*kx) + h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx) + h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky) + h[1, 2] = -gamma_y-lambda_y*cmath.exp(-1j*ky) + h[2, 0] = np.conj(h[0, 2]) + h[3, 1] = np.conj(h[1, 3]) + h[3, 0] = np.conj(h[0, 3]) + h[2, 1] = np.conj(h[1, 2]) + return h + +def main(): + Num_kx = 100 + Num_ky = 100 + kx_array = np.linspace(-pi, pi, Num_kx) + ky_array = np.linspace(-pi, pi, Num_ky) + nu_x_array = [] + for ky in ky_array: + vector1_array = [] + vector2_array = [] + for kx in kx_array: + eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky)) + if kx != pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + # 这里是为了-pi和pi有相同的波函数，使得Wilson loop的值与波函数规范无关。 + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) + W_x_k = np.eye(2, dtype=complex) + for i0 in range(Num_kx-1): + F = np.zeros((2, 2), dtype=complex) + F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0]) + F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0]) + F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0]) + F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0]) + W_x_k = np.dot(F, W_x_k) + eigenvalue, eigenvector = np.linalg.eig(W_x_k) + nu_x = np.log(eigenvalue)/2/pi/1j + for i0 in range(2): + if np.real(nu_x[i0]) < 0: + nu_x[i0] += 1 + nu_x = np.sort(nu_x) + nu_x_array.append(nu_x.real) + + import guan + guan.plot(ky_array, nu_x_array, xlabel='ky', ylabel='nu_x', type='-', y_min=0, y_max=1) + + # # Guan安装方法：https://py.guanjihuan.com/installation + # # 不安装Guan开源软件包的，可把上面两行注释，用以下代码代替。 + # import matplotlib.pyplot as plt + # fig, ax = plt.subplots() + # plt.subplots_adjust(bottom=0.20, left=0.18) + # ax.grid() + # ax.plot(ky_array, nu_x_array, '-') + # ax.set_xlabel('ky', fontsize=20, fontfamily='Times New Roman') + # ax.set_ylabel('nu_x', fontsize=20, fontfamily='Times New Roman') + # ax.tick_params(labelsize=20) + # labels = ax.get_xticklabels() + ax.get_yticklabels() + # [label.set_fontname('Times New Roman') for label in labels] + # ax.set_ylim(0, 1) + # plt.show() + +if __name__ == '__main__': + main() \ No newline at end of file diff --git a/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_bands.py b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_bands.py new file mode 100644 index 0000000..5ee20b7 --- /dev/null +++ b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_bands.py @@ -0,0 +1,44 @@ +""" +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/17984 +""" + +import numpy as np +import cmath +from math import * +import functools +import guan + +def hamiltonian(kx, ky): # BBH model + # label of atoms in a unit cell + # (2) —— (0) + # | | + # (1) —— (3) + gamma_x = 0.5 # hopping inside one unit cell + lambda_x = 1 # hopping between unit cells + gamma_y = gamma_x + lambda_y = lambda_x + h = np.zeros((4, 4), dtype=complex) + h[0, 2] = gamma_x+lambda_x*cmath.exp(1j*kx) + h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx) + h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky) + h[1, 2] = -gamma_y-lambda_y*cmath.exp(-1j*ky) + h[2, 0] = np.conj(h[0, 2]) + h[3, 1] = np.conj(h[1, 3]) + h[3, 0] = np.conj(h[0, 3]) + h[2, 1] = np.conj(h[1, 2]) + return h + +def main(): + kx = np.arange(-pi, pi, 0.05) + ky = np.arange(-pi, pi, 0.05) + + eigenvalue_array = guan.calculate_eigenvalue_with_two_parameters(kx, ky, hamiltonian) + guan.plot_3d_surface(kx, ky, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E', title='BBH bands') + + hamiltonian0 = functools.partial(hamiltonian, ky=0) + eigenvalue_array = guan.calculate_eigenvalue_with_one_parameter(kx, hamiltonian0) + guan.plot(kx, eigenvalue_array, xlabel='kx', ylabel='E', title='BBH bands ky=0') + +if __name__ == '__main__': + main() \ No newline at end of file diff --git a/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_nested_Wilson_loop.py b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_nested_Wilson_loop.py new file mode 100644 index 0000000..b623db0 --- /dev/null +++ b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/BBH_nested_Wilson_loop.py @@ -0,0 +1,175 @@ +""" +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/17984 +""" + +import numpy as np +import cmath +from math import * +import guan + +def hamiltonian(kx, ky): # BBH model + # label of atoms in a unit cell + # (2) —— (0) + # | | + # (1) —— (3) + gamma_x = 0.5 # hopping inside one unit cell + lambda_x = 1 # hopping between unit cells + gamma_y = gamma_x + lambda_y = lambda_x + x_symmetry_breaking_1 = 0.000000000000 # default (not breaking): zero + x_symmetry_breaking_2 = 1.0000000000001 # default (not breaking): unity + y_symmetry_breaking_1 = 0.000000000000 # default (not breaking): zero + y_symmetry_breaking_2 = 1.000000000000 # default (not breaking): unity + h = np.zeros((4, 4), dtype=complex) + h[0, 0] = x_symmetry_breaking_1 + h[1, 1] = y_symmetry_breaking_1 + h[2, 2] = y_symmetry_breaking_1 + h[3, 3] = x_symmetry_breaking_1 + h[0, 2] = (gamma_x+lambda_x*cmath.exp(1j*kx))*y_symmetry_breaking_2 + h[1, 3] = gamma_x+lambda_x*cmath.exp(-1j*kx) + h[0, 3] = gamma_y+lambda_y*cmath.exp(1j*ky) + h[1, 2] = (-gamma_y-lambda_y*cmath.exp(-1j*ky))*x_symmetry_breaking_2 + h[2, 0] = np.conj(h[0, 2]) + h[3, 1] = np.conj(h[1, 3]) + h[3, 0] = np.conj(h[0, 3]) + h[2, 1] = np.conj(h[1, 2]) + return h + +def main(): + Num_kx = 30 # for wilson loop and nested wilson loop + Num_ky = 30 # for wilson loop and nested wilson loop + Num_kx2 = 20 # plot precision + Num_ky2 = 20 # plot precision + kx_array = np.linspace(-pi, pi, Num_kx) + ky_array = np.linspace(-pi, pi, Num_ky) + kx2_array = np.linspace(-pi, pi, Num_kx2) + ky2_array = np.linspace(-pi, pi, Num_ky2) + + # Part I: calculate p_y_for_nu_x + p_y_for_nu_x_array = [] + for kx in kx2_array: + print('kx=', kx) + w_vector_for_nu1_array = [] + vector1_array = [] + vector2_array = [] + i0 = -1 + for ky in ky_array: + eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky)) + if ky != pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) + i0=0 + for ky in ky_array: + if ky != pi: + nu_x_vector_1, nu_x_vector_2 = get_nu_x_vector(kx_array, ky) + # the Wannier band subspaces + w_vector_for_nu1 = vector1_array[i0]*nu_x_vector_1[0]+vector2_array[i0]*nu_x_vector_1[1] + w_vector_for_nu1_array.append(w_vector_for_nu1) + else: + w_vector_for_nu1_array.append(w_vector_for_nu1_array[0]) + i0 +=1 + W_y_k_for_nu_x = 1 + for i0 in range(Num_ky-1): + F_for_nu_x = np.dot(w_vector_for_nu1_array[i0+1].transpose().conj(), w_vector_for_nu1_array[i0]) + W_y_k_for_nu_x = F_for_nu_x*W_y_k_for_nu_x + p_y_for_nu_x = np.log(W_y_k_for_nu_x)/2/pi/1j + if np.real(p_y_for_nu_x) < 0: + p_y_for_nu_x += 1 + p_y_for_nu_x_array.append(p_y_for_nu_x.real) + print('p_y_for_nu_x=', p_y_for_nu_x) + guan.plot(kx2_array, p_y_for_nu_x_array, xlabel='kx', ylabel='p_y_for_nu_x', type='-o', y_min=0, y_max=1) + + # Part II: calculate p_x_for_nu_y + p_x_for_nu_y_array = [] + for ky in ky2_array: + w_vector_for_nu1_array = [] + vector1_array = [] + vector2_array = [] + for kx in kx_array: + eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky)) + if kx != pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) + i0 = 0 + for kx in kx_array: + if kx != pi: + nu_y_vector_1, nu_y_vector_2 = get_nu_y_vector(kx, ky_array) + # the Wannier band subspaces + w_vector_for_nu1 = vector1_array[i0]*nu_y_vector_1[0]+vector2_array[i0]*nu_y_vector_1[1] + w_vector_for_nu1_array.append(w_vector_for_nu1) + else: + w_vector_for_nu1_array.append(w_vector_for_nu1_array[0]) + i0 += 1 + W_x_k_for_nu_y = 1 + for i0 in range(Num_ky-1): + F_for_nu_y = np.dot(w_vector_for_nu1_array[i0+1].transpose().conj(), w_vector_for_nu1_array[i0]) + W_x_k_for_nu_y = F_for_nu_y*W_x_k_for_nu_y + p_x_for_nu_y = np.log(W_x_k_for_nu_y)/2/pi/1j + if np.real(p_x_for_nu_y) < 0: + p_x_for_nu_y += 1 + p_x_for_nu_y_array.append(p_x_for_nu_y.real) + print('p_x_for_nu_y=', p_x_for_nu_y) + # print(sum(p_x_for_nu_y_array)/len(p_x_for_nu_y_array)) + guan.plot(ky2_array, p_x_for_nu_y_array, xlabel='ky', ylabel='p_x_for_nu_y', type='-o', y_min=0, y_max=1) + +def get_nu_x_vector(kx_array, ky): + Num_kx = len(kx_array) + vector1_array = [] + vector2_array = [] + for kx in kx_array: + eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky)) + if kx != pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) + W_x_k = np.eye(2, dtype=complex) + for i0 in range(Num_kx-1): + F = np.zeros((2, 2), dtype=complex) + F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0]) + F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0]) + F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0]) + F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0]) + W_x_k = np.dot(F, W_x_k) + eigenvalue, eigenvector = np.linalg.eig(W_x_k) + nu_x = np.log(eigenvalue)/2/pi/1j + nu_x_vector_1 = eigenvector[:, np.argsort(np.real(nu_x))[0]] + nu_x_vector_2 = eigenvector[:, np.argsort(np.real(nu_x))[1]] + return nu_x_vector_1, nu_x_vector_2 + +def get_nu_y_vector(kx, ky_array): + Num_ky = len(ky_array) + vector1_array = [] + vector2_array = [] + for ky in ky_array: + eigenvalue, eigenvector = np.linalg.eigh(hamiltonian(kx, ky)) + if ky != pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) + W_y_k = np.eye(2, dtype=complex) + for i0 in range(Num_ky-1): + F = np.zeros((2, 2), dtype=complex) + F[0, 0] = np.dot(vector1_array[i0+1].transpose().conj(), vector1_array[i0]) + F[1, 1] = np.dot(vector2_array[i0+1].transpose().conj(), vector2_array[i0]) + F[0, 1] = np.dot(vector1_array[i0+1].transpose().conj(), vector2_array[i0]) + F[1, 0] = np.dot(vector2_array[i0+1].transpose().conj(), vector1_array[i0]) + W_y_k = np.dot(F, W_y_k) + eigenvalue, eigenvector = np.linalg.eig(W_y_k) + nu_y = np.log(eigenvalue)/2/pi/1j + nu_y_vector_1 = eigenvector[:, np.argsort(np.real(nu_y))[0]] + nu_y_vector_2 = eigenvector[:, np.argsort(np.real(nu_y))[1]] + return nu_y_vector_1, nu_y_vector_2 + +if __name__ == '__main__': + main() diff --git a/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/simple_method_for_degenerate_bands.py b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/simple_method_for_degenerate_bands.py new file mode 100644 index 0000000..8d09a0c --- /dev/null +++ b/academic_codes/2021.12.09_nested_Wilson_loop_of_BBH_model/simple_method_for_degenerate_bands.py @@ -0,0 +1,15 @@ +if kx == -pi: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) +elif kx != pi: + # 这里的判断是为了处理能带简并时最简单情况，只做个对调。 + if np.abs(np.dot(vector1_array[-1].transpose().conj(), eigenvector[:, 0]))>0.5: + vector1_array.append(eigenvector[:, 0]) + vector2_array.append(eigenvector[:, 1]) + else: + vector1_array.append(eigenvector[:, 1]) + vector2_array.append(eigenvector[:, 0]) +else: + # 这里是为了-pi和pi有相同的波函数，使得Wilson loop的值与波函数规范无关。 + vector1_array.append(vector1_array[0]) + vector2_array.append(vector2_array[0]) \ No newline at end of file