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"""
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This code is supported by the website: https://www.guanjihuan.com
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The newest version of this code is on the web page: https://www.guanjihuan.com/archives/3785
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"""
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import numpy as np
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from math import *
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import matplotlib.pyplot as plt
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from matplotlib.colors import ListedColormap
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import time
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def green_function(fermi_energy, k1, k2, hamiltonian): # 计算格林函数
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matrix0 = hamiltonian(k1, k2)
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dim = matrix0.shape[0]
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green = np.linalg.inv(fermi_energy * np.identity(dim) - matrix0)
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return green
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def spectral_function(fermi_energy, k1, k2, hamiltonian): # 计算谱函数
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dim1 = k1.shape[0]
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dim2 = k2.shape[0]
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spectrum = np.zeros((dim1, dim2))
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i0 = 0
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for k10 in k1:
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j0 = 0
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for k20 in k2:
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green = green_function(fermi_energy, k10, k20, hamiltonian)
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spectrum[i0, j0] = (np.imag(green[0,0])+np.imag(green[2,2]))/(-pi)
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j0 += 1
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i0 += 1
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# print(spectrum)
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print()
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print('Spectral function显示的网格点 =', k1.shape[0], '*', k1.shape[0], '; 步长 =', k1[1] - k1[0])
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print()
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return spectrum
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def qpi(fermi_energy, q1, q2, hamiltonian, potential_i): # 计算QPI
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dim = hamiltonian(0, 0).shape[0]
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ki1 = np.arange(-pi, pi, 0.01) # 计算gamma_0时，k的积分密度
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ki2 = np.arange(-pi, pi, 0.01)
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print('gamma_0的积分网格点 =', ki1.shape[0], '*', ki1.shape[0], '; 步长 =', ki1[1] - ki1[0])
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gamma_0 = integral_of_green(fermi_energy, ki1, ki2, hamiltonian)/np.square(2*pi)
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t_matrix = np.dot(np.linalg.inv(np.identity(dim)-np.dot(potential_i, gamma_0)), potential_i)
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ki1 = np.arange(-pi, pi, 0.06) # 计算induced_local_density时，k的积分密度
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ki2 = np.arange(-pi, pi, 0.06)
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print('局域态密度变化的积分网格点 =', ki1.shape[0], '*', ki1.shape[0], '; 步长 =', ki1[1] - ki1[0])
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print('QPI显示的网格点 =', q1.shape[0], '*', q1.shape[0], '; 步长 =', q1[1] - q1[0])
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step_length = ki1[1] - ki1[0]
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induced_local_density = np.zeros((q1.shape[0], q2.shape[0]))*(1+0j)
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print()
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i0 = 0
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for q10 in q1:
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print('i0=', i0)
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j0 = 0
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for q20 in q2:
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for ki10 in ki1:
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for ki20 in ki2:
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green_01 = green_function(fermi_energy, ki10, ki20, hamiltonian)
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green_02 = green_function(fermi_energy, ki10+q10, ki20+q20, hamiltonian)
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induced_green = np.dot(np.dot(green_01, t_matrix), green_02)
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temp = induced_green[0, 0]-induced_green[0, 0].conj()+induced_green[2, 2]-induced_green[2, 2].conj()
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induced_local_density[i0, j0] = induced_local_density[i0, j0]+temp*np.square(step_length)
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j0 += 1
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i0 += 1
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write_matrix_k1_k2(q1, q2, np.real(induced_local_density*1j/np.square(2*pi)/(2*pi)), 'QPI') # 数据写入文件（临时写入，会被多次替代）
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induced_local_density = np.real(induced_local_density*1j/np.square(2*pi)/(2*pi))
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return induced_local_density
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def integral_of_green(fermi_energy, ki1, ki2, hamiltonian): # 在计算QPI时需要对格林函数积分
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dim = hamiltonian(0, 0).shape[0]
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integral_value = np.zeros((dim, dim))*(1+0j)
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step_length = ki1[1]-ki1[0]
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for ki10 in ki1:
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for ki20 in ki2:
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green = green_function(fermi_energy, ki10, ki20, hamiltonian)
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integral_value = integral_value+green*np.square(step_length)
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return integral_value
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def write_matrix_k1_k2(x1, x2, value, filename='matrix_k1_k2'): # 把矩阵数据写入文件（格式化输出）
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with open(filename+'.txt', 'w') as f:
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np.set_printoptions(suppress=True) # 取消输出科学记数法
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f.write('0 ')
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for x10 in x1:
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f.write(str(x10)+' ')
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f.write('\n')
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i0 = 0
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for x20 in x2:
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f.write(str(x20))
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for j0 in range(x1.shape[0]):
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f.write(' '+str(value[i0, j0])+' ')
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f.write('\n')
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i0 += 1
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def plot_contour(x1, x2, value, filename='contour'): # 直接画出contour图像（保存图像）
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plt.contourf(x1, x2, value) #, cmap=plt.cm.hot)
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plt.savefig(filename+'.eps')
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# plt.show()
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def hamiltonian(kx, ky): # 体系的哈密顿量
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t1 = -1; t2 = 1.3; t3 = -0.85; t4 = -0.85; delta_0 = 0.1; mu = 1.54
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epsilon_x = -2*t1*cos(kx)-2*t2*cos(ky)-4*t3*cos(kx)*cos(ky)
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epsilon_y = -2*t1*cos(ky)-2*t2*cos(kx)-4*t3*cos(kx)*cos(ky)
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epsilon_xy = -4*t4*sin(kx)*sin(ky)
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delta_1 = delta_0*cos(kx)*cos(ky)
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delta_2 = delta_0*cos(kx)*cos(ky)
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h = np.zeros((4, 4))
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h[0, 0] = epsilon_x-mu
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h[1, 1] = -epsilon_x+mu
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h[2, 2] = epsilon_y-mu
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h[3, 3] = -epsilon_y+mu
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h[0, 1] = delta_1
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h[1, 0] = delta_1
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h[0, 2] = epsilon_xy
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h[2, 0] = epsilon_xy
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h[0, 3] = 0
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h[3, 0] = 0
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h[1, 2] = 0
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h[2, 1] = 0
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h[1, 3] = -epsilon_xy
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h[3, 1] = -epsilon_xy
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h[2, 3] = delta_2
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h[3, 2] = delta_2
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return h
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def main(): # 主程序
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start_clock = time.perf_counter()
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fermi_energy = 0.07 # 费米能
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energy_broadening_width = 0.005 # 展宽
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k1 = np.arange(-pi, pi, 0.01) # 谱函数的图像精度
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k2 = np.arange(-pi, pi, 0.01)
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spectrum = spectral_function(fermi_energy+energy_broadening_width*1j, k1, k2, hamiltonian) # 调用谱函数子程序
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write_matrix_k1_k2(k1, k2, spectrum, 'Spectral_function') # 把谱函数的数据写入文件
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# plot_contour(k1, k2, spectrum, 'Spectral_function') # 直接显示谱函数的图像（保存图像）
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q1 = np.arange(-pi, pi, 0.01) # QPI数的图像精度
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q2 = np.arange(-pi, pi, 0.01)
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potential_i = (0.4+0j)*np.identity(hamiltonian(0, 0).shape[0]) # 杂质势
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potential_i[1, 1] = - potential_i[1, 1] # for nonmagnetic
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potential_i[3, 3] = - potential_i[3, 3]
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induced_local_density = qpi(fermi_energy+energy_broadening_width*1j, q1, q2, hamiltonian, potential_i) # 调用QPI子程序
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write_matrix_k1_k2(q1, q2, induced_local_density, 'QPI') # 把QPI数据写入文件（这里用的方法是计算结束后一次性把数据写入）
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# plot_contour(q1, q2, induced_local_density, 'QPI') # 直接显示QPI图像（保存图像）
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end_clock = time.perf_counter()
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print('CPU执行时间=', end_clock - start_clock)
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if __name__ == '__main__':
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main()
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