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import numpy as np
import cmath
from math import *
import copy
# test
def test():
print('\nSuccess in the installation of GJH package!\n')
# basic functions
def sigma_0():
return np.eye(2)
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]])
# hermitian hamiltonian of tight binding model
def finite_size_along_one_direction(N, on_site=0, hopping=1, period=0):
on_site = np.array(on_site)
hopping = np.array(hopping)
if on_site.shape==():
dim = 1
else:
dim = on_site.shape[0]
hamiltonian = np.zeros((N*dim, N*dim), dtype=complex)
for i0 in range(N):
hamiltonian[i0*dim+0:i0*dim+dim, i0*dim+0:i0*dim+dim] = on_site
for i0 in range(N-1):
hamiltonian[i0*dim+0:i0*dim+dim, (i0+1)*dim+0:(i0+1)*dim+dim] = hopping
hamiltonian[(i0+1)*dim+0:(i0+1)*dim+dim, i0*dim+0:i0*dim+dim] = hopping.transpose().conj()
if period == 1:
hamiltonian[(N-1)*dim+0:(N-1)*dim+dim, 0:dim] = hopping
hamiltonian[0:dim, (N-1)*dim+0:(N-1)*dim+dim] = hopping.transpose().conj()
return hamiltonian
def finite_size_along_two_directions_for_square_lattice(N1, N2, on_site=0, hopping_1=1, hopping_2=1, period_1=0, period_2=0):
on_site = np.array(on_site)
hopping_1 = np.array(hopping_1)
hopping_2 = np.array(hopping_2)
if on_site.shape==():
dim = 1
else:
dim = on_site.shape[0]
hamiltonian = np.zeros((N1*N2*dim, N1*N2*dim), dtype=complex)
for i1 in range(N1):
for i2 in range(N2):
hamiltonian[i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim, i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim] = on_site
for i1 in range(N1-1):
for i2 in range(N2):
hamiltonian[i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim, (i1+1)*N2*dim+i2*dim+0:(i1+1)*N2*dim+i2*dim+dim] = hopping_1
hamiltonian[(i1+1)*N2*dim+i2*dim+0:(i1+1)*N2*dim+i2*dim+dim, i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim] = hopping_1.transpose().conj()
for i1 in range(N1):
for i2 in range(N2-1):
hamiltonian[i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim, i1*N2*dim+(i2+1)*dim+0:i1*N2*dim+(i2+1)*dim+dim] = hopping_2
hamiltonian[i1*N2*dim+(i2+1)*dim+0:i1*N2*dim+(i2+1)*dim+dim, i1*N2*dim+i2*dim+0:i1*N2*dim+i2*dim+dim] = hopping_2.transpose().conj()
if period_1 == 1:
for i2 in range(N2):
hamiltonian[(N1-1)*N2*dim+i2*dim+0:(N1-1)*N2*dim+i2*dim+dim, i2*dim+0:i2*dim+dim] = hopping_1
hamiltonian[i2*dim+0:i2*dim+dim, (N1-1)*N2*dim+i2*dim+0:(N1-1)*N2*dim+i2*dim+dim] = hopping_1.transpose().conj()
if period_2 == 1:
for i1 in range(N1):
hamiltonian[i1*N2*dim+(N2-1)*dim+0:i1*N2*dim+(N2-1)*dim+dim, i1*N2*dim+0:i1*N2*dim+dim] = hopping_2
hamiltonian[i1*N2*dim+0:i1*N2*dim+dim, i1*N2*dim+(N2-1)*dim+0:i1*N2*dim+(N2-1)*dim+dim] = hopping_2.transpose().conj()
return hamiltonian
def finite_size_along_three_directions_for_cubic_lattice(N1, N2, N3, on_site=0, hopping_1=1, hopping_2=1, hopping_3=1, period_1=0, period_2=0, period_3=0):
on_site = np.array(on_site)
hopping_1 = np.array(hopping_1)
hopping_2 = np.array(hopping_2)
hopping_3 = np.array(hopping_3)
if on_site.shape==():
dim = 1
else:
dim = on_site.shape[0]
hamiltonian = np.zeros((N1*N2*N3*dim, N1*N2*N3*dim), dtype=complex)
for i1 in range(N1):
for i2 in range(N2):
for i3 in range(N3):
hamiltonian[i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim, i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim] = on_site
for i1 in range(N1-1):
for i2 in range(N2):
for i3 in range(N3):
hamiltonian[i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim, (i1+1)*N2*N3*dim+i2*N3*dim+i3*dim+0:(i1+1)*N2*N3*dim+i2*N3*dim+i3*dim+dim] = hopping_1
hamiltonian[(i1+1)*N2*N3*dim+i2*N3*dim+i3*dim+0:(i1+1)*N2*N3*dim+i2*N3*dim+i3*dim+dim, i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim] = hopping_1.transpose().conj()
for i1 in range(N1):
for i2 in range(N2-1):
for i3 in range(N3):
hamiltonian[i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim, i1*N2*N3*dim+(i2+1)*N3*dim+i3*dim+0:i1*N2*N3*dim+(i2+1)*N3*dim+i3*dim+dim] = hopping_2
hamiltonian[i1*N2*N3*dim+(i2+1)*N3*dim+i3*dim+0:i1*N2*N3*dim+(i2+1)*N3*dim+i3*dim+dim, i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim] = hopping_2.transpose().conj()
for i1 in range(N1):
for i2 in range(N2):
for i3 in range(N3-1):
hamiltonian[i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim, i1*N2*N3*dim+i2*N3*dim+(i3+1)*dim+0:i1*N2*N3*dim+i2*N3*dim+(i3+1)*dim+dim] = hopping_3
hamiltonian[i1*N2*N3*dim+i2*N3*dim+(i3+1)*dim+0:i1*N2*N3*dim+i2*N3*dim+(i3+1)*dim+dim, i1*N2*N3*dim+i2*N3*dim+i3*dim+0:i1*N2*N3*dim+i2*N3*dim+i3*dim+dim] = hopping_3.transpose().conj()
if period_1 == 1:
for i2 in range(N2):
for i3 in range(N3):
hamiltonian[(N1-1)*N2*N3*dim+i2*N3*dim+i3*dim+0:(N1-1)*N2*N3*dim+i2*N3*dim+i3*dim+dim, i2*N3*dim+i3*dim+0:i2*N3*dim+i3*dim+dim] = hopping_1
hamiltonian[i2*N3*dim+i3*dim+0:i2*N3*dim+i3*dim+dim, (N1-1)*N2*N3*dim+i2*N3*dim+i3*dim+0:(N1-1)*N2*N3*dim+i2*N3*dim+i3*dim+dim] = hopping_1.transpose().conj()
if period_2 == 1:
for i1 in range(N1):
for i3 in range(N3):
hamiltonian[i1*N2*N3*dim+(N2-1)*N3*dim+i3*dim+0:i1*N2*N3*dim+(N2-1)*N3*dim+i3*dim+dim, i1*N2*N3*dim+i3*dim+0:i1*N2*N3*dim+i3*dim+dim] = hopping_2
hamiltonian[i1*N2*N3*dim+i3*dim+0:i1*N2*N3*dim+i3*dim+dim, i1*N2*N3*dim+(N2-1)*N3*dim+i3*dim+0:i1*N2*N3*dim+(N2-1)*N3*dim+i3*dim+dim] = hopping_2.transpose().conj()
if period_3 == 1:
for i1 in range(N1):
for i2 in range(N2):
hamiltonian[i1*N2*N3*dim+i2*N3*dim+(N3-1)*dim+0:i1*N2*N3*dim+i2*N3*dim+(N3-1)*dim+dim, i1*N2*N3*dim+i2*N3*dim+0:i1*N2*N3*dim+i2*N3*dim+dim] = hopping_3
hamiltonian[i1*N2*N3*dim+i2*N3*dim+0:i1*N2*N3*dim+i2*N3*dim+dim, i1*N2*N3*dim+i2*N3*dim+(N3-1)*dim+0:i1*N2*N3*dim+i2*N3*dim+(N3-1)*dim+dim] = hopping_3.transpose().conj()
return hamiltonian
def one_dimensional_fourier_transform(k, unit_cell, hopping):
unit_cell = np.array(unit_cell)
hopping = np.array(hopping)
hamiltonian = unit_cell+hopping*cmath.exp(1j*k)+hopping.transpose().conj()*cmath.exp(-1j*k)
return hamiltonian
def two_dimensional_fourier_transform_for_square_lattice(k1, k2, unit_cell, hopping_1, hopping_2):
unit_cell = np.array(unit_cell)
hopping_1 = np.array(hopping_1)
hopping_2 = np.array(hopping_2)
hamiltonian = unit_cell+hopping_1*cmath.exp(1j*k1)+hopping_1.transpose().conj()*cmath.exp(-1j*k1)+hopping_2*cmath.exp(1j*k2)+hopping_2.transpose().conj()*cmath.exp(-1j*k2)
return hamiltonian
def three_dimensional_fourier_transform_for_cubic_lattice(k1, k2, k3, unit_cell, hopping_1, hopping_2, hopping_3):
unit_cell = np.array(unit_cell)
hopping_1 = np.array(hopping_1)
hopping_2 = np.array(hopping_2)
hopping_3 = np.array(hopping_3)
hamiltonian = unit_cell+hopping_1*cmath.exp(1j*k1)+hopping_1.transpose().conj()*cmath.exp(-1j*k1)+hopping_2*cmath.exp(1j*k2)+hopping_2.transpose().conj()*cmath.exp(-1j*k2)+hopping_3*cmath.exp(1j*k3)+hopping_3.transpose().conj()*cmath.exp(-1j*k3)
return hamiltonian
# hamiltonian of graphene lattice
def hopping_along_zigzag_direction_for_graphene(N):
hopping = np.zeros((4*N, 4*N), dtype=complex)
for i0 in range(N):
hopping[4*i0+1, 4*i0+0] = 1
hopping[4*i0+2, 4*i0+3] = 1
return hopping
def finite_size_along_two_directions_for_graphene(N1, N2, period_1=0, period_2=0):
on_site = finite_size_along_one_direction(4)
hopping_1 = hopping_along_zigzag_direction_for_graphene(1)
hopping_2 = np.zeros((4, 4), dtype=complex)
hopping_2[3, 0] = 1
hamiltonian = finite_size_along_two_directions_for_square_lattice(N1, N2, on_site, hopping_1, hopping_2, period_1, period_2)
return hamiltonian
# calculate band structures
def calculate_eigenvalue(hamiltonian):
if np.array(hamiltonian).shape==():
eigenvalue = np.real(hamiltonian)
else:
eigenvalue, eigenvector = np.linalg.eig(hamiltonian)
eigenvalue = np.sort(np.real(eigenvalue))
return eigenvalue
def calculate_eigenvalue_with_one_parameter(x, hamiltonian_function):
dim_x = np.array(x).shape[0]
i0 = 0
if np.array(hamiltonian_function(0)).shape==():
eigenvalue_array = np.zeros((dim_x, 1))
for x0 in x:
hamiltonian = hamiltonian_function(x0)
eigenvalue_array[i0, 0] = np.real(hamiltonian)
i0 += 1
else:
dim = np.array(hamiltonian_function(0)).shape[0]
eigenvalue_array = np.zeros((dim_x, dim))
for x0 in x:
hamiltonian = hamiltonian_function(x0)
eigenvalue, eigenvector = np.linalg.eig(hamiltonian)
eigenvalue_array[i0, :] = np.sort(np.real(eigenvalue[:]))
i0 += 1
return eigenvalue_array
def calculate_eigenvalue_with_two_parameters(x, y, hamiltonian_function):
dim_x = np.array(x).shape[0]
dim_y = np.array(y).shape[0]
if np.array(hamiltonian_function(0,0)).shape==():
eigenvalue_array = np.zeros((dim_y, dim_x, 1))
i0 = 0
for y0 in y:
j0 = 0
for x0 in x:
hamiltonian = hamiltonian_function(x0, y0)
eigenvalue_array[i0, j0, 0] = np.real(hamiltonian)
j0 += 1
i0 += 1
else:
dim = np.array(hamiltonian_function(0, 0)).shape[0]
eigenvalue_array = np.zeros((dim_y, dim_x, dim))
i0 = 0
for y0 in y:
j0 = 0
for x0 in x:
hamiltonian = hamiltonian_function(x0, y0)
eigenvalue, eigenvector = np.linalg.eig(hamiltonian)
eigenvalue_array[i0, j0, :] = np.sort(np.real(eigenvalue[:]))
j0 += 1
i0 += 1
return eigenvalue_array
# calculate wave functions
def calculate_eigenvector(hamiltonian):
eigenvalue, eigenvector = np.linalg.eig(hamiltonian)
eigenvector = eigenvector[:, np.argsort(np.real(eigenvalue))]
return eigenvector
# calculate green functions
def green_function(fermi_energy, hamiltonian, broadening, self_energy=0):
if np.array(hamiltonian).shape==():
dim = 1
else:
dim = np.array(hamiltonian).shape[0]
green = np.linalg.inv((fermi_energy+broadening*1j)*np.eye(dim)-hamiltonian-self_energy)
return green
def green_function_nn_n(fermi_energy, h00, h01, green_nn_n_minus, broadening, self_energy=0):
h01 = np.array(h01)
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
green_nn_n = np.linalg.inv((fermi_energy+broadening*1j)*np.identity(dim)-h00-np.dot(np.dot(h01.transpose().conj(), green_nn_n_minus), h01)-self_energy)
return green_nn_n
def green_function_in_n(green_in_n_minus, h01, green_nn_n):
green_in_n = np.dot(np.dot(green_in_n_minus, h01), green_nn_n)
return green_in_n
def green_function_ni_n(green_nn_n, h01, green_ni_n_minus):
h01 = np.array(h01)
green_ni_n = np.dot(np.dot(green_nn_n, h01.transpose().conj()), green_ni_n_minus)
return green_ni_n
def green_function_ii_n(green_ii_n_minus, green_in_n_minus, h01, green_nn_n, green_ni_n_minus):
green_ii_n = green_ii_n_minus+np.dot(np.dot(np.dot(np.dot(green_in_n_minus, h01), green_nn_n), h01.transpose().conj()),green_ni_n_minus)
return green_ii_n
# calculate density of states
def total_density_of_states(fermi_energy, hamiltonian, broadening=0.01):
green = green_function(fermi_energy, hamiltonian, broadening)
total_dos = -np.trace(np.imag(green))/pi
return total_dos
def total_density_of_states_with_fermi_energy_array(fermi_energy_array, hamiltonian, broadening=0.01):
dim = np.array(fermi_energy_array).shape[0]
total_dos_array = np.zeros(dim)
i0 = 0
for fermi_energy in fermi_energy_array:
total_dos_array[i0] = total_density_of_states(fermi_energy, hamiltonian, broadening)
i0 += 1
return total_dos_array
def local_density_of_states_for_square_lattice(fermi_energy, hamiltonian, N1, N2, internal_degree=1, broadening=0.01):
# dim_hamiltonian = N1*N2*internal_degree
green = green_function(fermi_energy, hamiltonian, broadening)
local_dos = np.zeros((N2, N1))
for i1 in range(N1):
for i2 in range(N2):
for i in range(internal_degree):
local_dos[i2, i1] = local_dos[i2, i1]-np.imag(green[i1*N2*internal_degree+i2*internal_degree+i, i1*N2*internal_degree+i2*internal_degree+i])/pi
return local_dos
def local_density_of_states_for_cubic_lattice(fermi_energy, hamiltonian, N1, N2, N3, internal_degree=1, broadening=0.01):
# dim_hamiltonian = N1*N2*N3*internal_degree
green = green_function(fermi_energy, hamiltonian, broadening)
local_dos = np.zeros((N3, N2, N1))
for i1 in range(N1):
for i2 in range(N2):
for i3 in range(N3):
for i in range(internal_degree):
local_dos[i3, i2, i1] = local_dos[i3, i2, i1]-np.imag(green[i1*N2*N3*internal_degree+i2*N3*internal_degree+i3*internal_degree+i, i1*N2*N3*internal_degree+i2*N3*internal_degree+i3*internal_degree+i])/pi
return local_dos
def local_density_of_states_for_square_lattice_using_dyson_equation(fermi_energy, h00, h01, N2, N1, internal_degree=1, broadening=0.01):
# dim_h00 = N2*internal_degree
local_dos = np.zeros((N2, N1))
green_11_1 = green_function(fermi_energy, h00, broadening)
for i1 in range(N1):
green_nn_n_minus = green_11_1
green_in_n_minus = green_11_1
green_ni_n_minus = green_11_1
green_ii_n_minus = green_11_1
for i2_0 in range(i1):
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n_minus, broadening)
green_nn_n_minus = green_nn_n
if i1!=0:
green_in_n_minus = green_nn_n
green_ni_n_minus = green_nn_n
green_ii_n_minus = green_nn_n
for size_0 in range(N1-1-i1):
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n_minus, broadening)
green_nn_n_minus = green_nn_n
green_ii_n = green_function_ii_n(green_ii_n_minus, green_in_n_minus, h01, green_nn_n, green_ni_n_minus)
green_ii_n_minus = green_ii_n
green_in_n = green_function_in_n(green_in_n_minus, h01, green_nn_n)
green_in_n_minus = green_in_n
green_ni_n = green_function_ni_n(green_nn_n, h01, green_ni_n_minus)
green_ni_n_minus = green_ni_n
for i2 in range(N2):
for i in range(internal_degree):
local_dos[i2, i1] = local_dos[i2, i1] - np.imag(green_ii_n_minus[i2*internal_degree+i, i2*internal_degree+i])/pi
return local_dos
def local_density_of_states_for_cubic_lattice_using_dyson_equation(fermi_energy, h00, h01, N3, N2, N1, internal_degree=1, broadening=0.01):
# dim_h00 = N2*N3*internal_degree
local_dos = np.zeros((N3, N2, N1))
green_11_1 = green_function(fermi_energy, h00, broadening)
for i1 in range(N1):
green_nn_n_minus = green_11_1
green_in_n_minus = green_11_1
green_ni_n_minus = green_11_1
green_ii_n_minus = green_11_1
for i1_0 in range(i1):
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n_minus, broadening)
green_nn_n_minus = green_nn_n
if i1!=0:
green_in_n_minus = green_nn_n
green_ni_n_minus = green_nn_n
green_ii_n_minus = green_nn_n
for size_0 in range(N1-1-i1):
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n_minus, broadening)
green_nn_n_minus = green_nn_n
green_ii_n = green_function_ii_n(green_ii_n_minus, green_in_n_minus, h01, green_nn_n, green_ni_n_minus)
green_ii_n_minus = green_ii_n
green_in_n = green_function_in_n(green_in_n_minus, h01, green_nn_n)
green_in_n_minus = green_in_n
green_ni_n = green_function_ni_n(green_nn_n, h01, green_ni_n_minus)
green_ni_n_minus = green_ni_n
for i2 in range(N2):
for i3 in range(N3):
for i in range(internal_degree):
local_dos[i3, i2, i1] = local_dos[i3, i2, i1] -np.imag(green_ii_n_minus[i2*N3*internal_degree+i3*internal_degree+i, i2*N3*internal_degree+i3*internal_degree+i])/pi
return local_dos
# calculate conductance
def transfer_matrix(fermi_energy, h00, h01):
h01 = np.array(h01)
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
transfer = np.zeros((2*dim, 2*dim), dtype=complex)
transfer[0:dim, 0:dim] = np.dot(np.linalg.inv(h01), fermi_energy*np.identity(dim)-h00)
transfer[0:dim, dim:2*dim] = np.dot(-1*np.linalg.inv(h01), h01.transpose().conj())
transfer[dim:2*dim, 0:dim] = np.identity(dim)
transfer[dim:2*dim, dim:2*dim] = 0
return transfer
def surface_green_function_of_lead(fermi_energy, h00, h01):
h01 = np.array(h01)
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
fermi_energy = fermi_energy+1e-9*1j
transfer = transfer_matrix(fermi_energy, h00, h01)
eigenvalue, eigenvector = np.linalg.eig(transfer)
ind = np.argsort(np.abs(eigenvalue))
temp = np.zeros((2*dim, 2*dim), dtype=complex)
i0 = 0
for ind0 in ind:
temp[:, i0] = eigenvector[:, ind0]
i0 += 1
s1 = temp[dim:2*dim, 0:dim]
s2 = temp[0:dim, 0:dim]
s3 = temp[dim:2*dim, dim:2*dim]
s4 = temp[0:dim, dim:2*dim]
right_lead_surface = np.linalg.inv(fermi_energy*np.identity(dim)-h00-np.dot(np.dot(h01, s2), np.linalg.inv(s1)))
left_lead_surface = np.linalg.inv(fermi_energy*np.identity(dim)-h00-np.dot(np.dot(h01.transpose().conj(), s3), np.linalg.inv(s4)))
return right_lead_surface, left_lead_surface
def self_energy_of_lead(fermi_energy, h00, h01):
h01 = np.array(h01)
right_lead_surface, left_lead_surface = surface_green_function_of_lead(fermi_energy, h00, h01)
right_self_energy = np.dot(np.dot(h01, right_lead_surface), h01.transpose().conj())
left_self_energy = np.dot(np.dot(h01.transpose().conj(), left_lead_surface), h01)
return right_self_energy, left_self_energy
def calculate_conductance(fermi_energy, h00, h01, length=100):
right_self_energy, left_self_energy = self_energy_of_lead(fermi_energy, h00, h01)
for ix in range(length):
if ix == 0:
green_nn_n = green_function(fermi_energy, h00, broadening=0, self_energy=left_self_energy)
green_0n_n = copy.deepcopy(green_nn_n)
elif ix != length-1:
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n, broadening=0)
green_0n_n = green_function_in_n(green_0n_n, h01, green_nn_n)
else:
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n, broadening=0, self_energy=right_self_energy)
green_0n_n = green_function_in_n(green_0n_n, h01, green_nn_n)
right_self_energy = (right_self_energy - right_self_energy.transpose().conj())*(0+1j)
left_self_energy = (left_self_energy - left_self_energy.transpose().conj())*(0+1j)
conductance = np.trace(np.dot(np.dot(np.dot(left_self_energy, green_0n_n), right_self_energy), green_0n_n.transpose().conj()))
return conductance
def calculate_conductance_with_fermi_energy_array(fermi_energy_array, h00, h01, length=100):
dim = np.array(fermi_energy_array).shape[0]
conductance_array = np.zeros(dim)
i0 = 0
for fermi_energy_0 in fermi_energy_array:
conductance_array[i0] = np.real(calculate_conductance(fermi_energy_0, h00, h01, length))
i0 += 1
return conductance_array
# calculate scattering matrix
def if_active_channel(k_of_channel):
if np.abs(np.imag(k_of_channel))<1e-6:
if_active = 1
else:
if_active = 0
return if_active
def get_k_and_velocity_of_channel(fermi_energy, h00, h01):
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
transfer = transfer_matrix(fermi_energy, h00, h01)
eigenvalue, eigenvector = np.linalg.eig(transfer)
k_of_channel = np.log(eigenvalue)/1j
ind = np.argsort(np.real(k_of_channel))
k_of_channel = np.sort(k_of_channel)
temp = np.zeros((2*dim, 2*dim), dtype=complex)
temp2 = np.zeros((2*dim), dtype=complex)
i0 = 0
for ind0 in ind:
temp[:, i0] = eigenvector[:, ind0]
temp2[i0] = eigenvalue[ind0]
i0 += 1
eigenvalue = copy.deepcopy(temp2)
temp = temp[0:dim, :]
factor = np.zeros(2*dim, dtype=complex)
for dim0 in range(dim):
factor = factor+np.square(np.abs(temp[dim0, :]))
for dim0 in range(2*dim):
temp[:, dim0] = temp[:, dim0]/np.sqrt(factor[dim0])
velocity_of_channel = np.zeros((2*dim), dtype=complex)
for dim0 in range(2*dim):
velocity_of_channel[dim0] = eigenvalue[dim0]*np.dot(np.dot(temp[0:dim, :].transpose().conj(), h01),temp[0:dim, :])[dim0, dim0]
velocity_of_channel = -2*np.imag(velocity_of_channel)
eigenvector = copy.deepcopy(temp)
return k_of_channel, velocity_of_channel, eigenvalue, eigenvector
def get_classified_k_velocity_u_and_f(fermi_energy, h00, h01):
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
k_of_channel, velocity_of_channel, eigenvalue, eigenvector = get_k_and_velocity_of_channel(fermi_energy, h00, h01)
ind_right_active = 0; ind_right_evanescent = 0; ind_left_active = 0; ind_left_evanescent = 0
k_right = np.zeros(dim, dtype=complex); k_left = np.zeros(dim, dtype=complex)
velocity_right = np.zeros(dim, dtype=complex); velocity_left = np.zeros(dim, dtype=complex)
lambda_right = np.zeros(dim, dtype=complex); lambda_left = np.zeros(dim, dtype=complex)
u_right = np.zeros((dim, dim), dtype=complex); u_left = np.zeros((dim, dim), dtype=complex)
for dim0 in range(2*dim):
if_active = if_active_channel(k_of_channel[dim0])
if if_active_channel(k_of_channel[dim0]) == 1:
direction = np.sign(velocity_of_channel[dim0])
else:
direction = np.sign(np.imag(k_of_channel[dim0]))
if direction == 1:
if if_active == 1: # right-moving active channel
k_right[ind_right_active] = k_of_channel[dim0]
velocity_right[ind_right_active] = velocity_of_channel[dim0]
lambda_right[ind_right_active] = eigenvalue[dim0]
u_right[:, ind_right_active] = eigenvector[:, dim0]
ind_right_active += 1
else: # right-moving evanescent channel
k_right[dim-1-ind_right_evanescent] = k_of_channel[dim0]
velocity_right[dim-1-ind_right_evanescent] = velocity_of_channel[dim0]
lambda_right[dim-1-ind_right_evanescent] = eigenvalue[dim0]
u_right[:, dim-1-ind_right_evanescent] = eigenvector[:, dim0]
ind_right_evanescent += 1
else:
if if_active == 1: # left-moving active channel
k_left[ind_left_active] = k_of_channel[dim0]
velocity_left[ind_left_active] = velocity_of_channel[dim0]
lambda_left[ind_left_active] = eigenvalue[dim0]
u_left[:, ind_left_active] = eigenvector[:, dim0]
ind_left_active += 1
else: # left-moving evanescent channel
k_left[dim-1-ind_left_evanescent] = k_of_channel[dim0]
velocity_left[dim-1-ind_left_evanescent] = velocity_of_channel[dim0]
lambda_left[dim-1-ind_left_evanescent] = eigenvalue[dim0]
u_left[:, dim-1-ind_left_evanescent] = eigenvector[:, dim0]
ind_left_evanescent += 1
lambda_matrix_right = np.diag(lambda_right)
lambda_matrix_left = np.diag(lambda_left)
f_right = np.dot(np.dot(u_right, lambda_matrix_right), np.linalg.inv(u_right))
f_left = np.dot(np.dot(u_left, lambda_matrix_left), np.linalg.inv(u_left))
return k_right, k_left, velocity_right, velocity_left, f_right, f_left, u_right, u_left, ind_right_active
def calculate_scattering_matrix(fermi_energy, h00, h01, length=100):
h01 = np.array(h01)
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
k_right, k_left, velocity_right, velocity_left, f_right, f_left, u_right, u_left, ind_right_active = get_classified_k_velocity_u_and_f(fermi_energy, h00, h01)
right_self_energy = np.dot(h01, f_right)
left_self_energy = np.dot(h01.transpose().conj(), np.linalg.inv(f_left))
for i0 in range(length):
if i0 == 0:
green_nn_n = green_function(fermi_energy, h00, broadening=0, self_energy=left_self_energy)
green_00_n = copy.deepcopy(green_nn_n)
green_0n_n = copy.deepcopy(green_nn_n)
green_n0_n = copy.deepcopy(green_nn_n)
elif i0 != length-1:
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n, broadening=0)
else:
green_nn_n = green_function_nn_n(fermi_energy, h00, h01, green_nn_n, broadening=0, self_energy=right_self_energy)
green_00_n = green_function_ii_n(green_00_n, green_0n_n, h01, green_nn_n, green_n0_n)
green_0n_n = green_function_in_n(green_0n_n, h01, green_nn_n)
green_n0_n = green_function_ni_n(green_nn_n, h01, green_n0_n)
temp = np.dot(h01.transpose().conj(), np.linalg.inv(f_right)-np.linalg.inv(f_left))
transmission_matrix = np.dot(np.dot(np.linalg.inv(u_right), np.dot(green_n0_n, temp)), u_right)
reflection_matrix = np.dot(np.dot(np.linalg.inv(u_left), np.dot(green_00_n, temp)-np.identity(dim)), u_right)
for dim0 in range(dim):
for dim1 in range(dim):
if_active = if_active_channel(k_right[dim0])*if_active_channel(k_right[dim1])
if if_active == 1:
transmission_matrix[dim0, dim1] = np.sqrt(np.abs(velocity_right[dim0]/velocity_right[dim1])) * transmission_matrix[dim0, dim1]
reflection_matrix[dim0, dim1] = np.sqrt(np.abs(velocity_left[dim0]/velocity_right[dim1]))*reflection_matrix[dim0, dim1]
else:
transmission_matrix[dim0, dim1] = 0
reflection_matrix[dim0, dim1] = 0
sum_of_tran_refl_array = np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active])), axis=0)+np.sum(np.square(np.abs(reflection_matrix[0:ind_right_active, 0:ind_right_active])), axis=0)
for sum_of_tran_refl in sum_of_tran_refl_array:
if sum_of_tran_refl > 1.001:
print('Error Alert: scattering matrix is not normalized!')
return transmission_matrix, reflection_matrix, k_right, k_left, velocity_right, velocity_left, ind_right_active
def print_or_write_scattering_matrix(fermi_energy, h00, h01, length=100, on_print=1, on_write=0):
if np.array(h00).shape==():
dim = 1
else:
dim = np.array(h00).shape[0]
transmission_matrix, reflection_matrix, k_right, k_left, velocity_right, velocity_left, ind_right_active = calculate_scattering_matrix(fermi_energy, h00, h01, length)
if on_print == 1:
print('\nActive channel (left or right) = ', ind_right_active)
print('Evanescent channel (left or right) = ', dim-ind_right_active, '\n')
print('K of right-moving active channels:\n', np.real(k_right[0:ind_right_active]))
print('K of left-moving active channels:\n', np.real(k_left[0:ind_right_active]), '\n')
print('Velocity of right-moving active channels:\n', np.real(velocity_right[0:ind_right_active]))
print('Velocity of left-moving active channels:\n', np.real(velocity_left[0:ind_right_active]), '\n')
print('Transmission matrix:\n', np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active])))
print('Reflection matrix:\n', np.square(np.abs(reflection_matrix[0:ind_right_active, 0:ind_right_active])), '\n')
print('Total transmission of channels:\n', np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active])), axis=0))
print('Total reflection of channels:\n',np.sum(np.square(np.abs(reflection_matrix[0:ind_right_active, 0:ind_right_active])), axis=0))
print('Sum of transmission and reflection of channels:\n', np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active])), axis=0) + np.sum(np.square(np.abs(reflection_matrix[0:ind_right_active, 0:ind_right_active])), axis=0))
print('Total conductance = ', np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active]))), '\n')
if on_write == 1:
with open('a.txt', 'w') as f:
f.write('Active channel (left or right) = ' + str(ind_right_active) + '\n')
f.write('Evanescent channel (left or right) = ' + str(dim - ind_right_active) + '\n\n')
f.write('Channel K Velocity\n')
for ind0 in range(ind_right_active):
f.write(' '+str(ind0 + 1) + ' | '+str(np.real(k_right[ind0]))+' ' + str(np.real(velocity_right[ind0]))+'\n')
f.write('\n')
for ind0 in range(ind_right_active):
f.write(' -' + str(ind0 + 1) + ' | ' + str(np.real(k_left[ind0])) + ' ' + str(np.real(velocity_left[ind0])) + '\n')
f.write('\nScattering matrix:\n ')
for ind0 in range(ind_right_active):
f.write(str(ind0+1)+' ')
f.write('\n')
for ind1 in range(ind_right_active):
f.write(' '+str(ind1+1)+' ')
for ind2 in range(ind_right_active):
f.write('%f' % np.square(np.abs(transmission_matrix[ind1, ind2]))+' ')
f.write('\n')
f.write('\n')
for ind1 in range(ind_right_active):
f.write(' -'+str(ind1+1)+' ')
for ind2 in range(ind_right_active):
f.write('%f' % np.square(np.abs(reflection_matrix[ind1, ind2]))+' ')
f.write('\n')
f.write('\n')
f.write('Total transmission of channels:\n'+str(np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active])), axis=0))+'\n')
f.write('Total conductance = '+str(np.sum(np.square(np.abs(transmission_matrix[0:ind_right_active, 0:ind_right_active]))))+'\n')
# calculate chern number
def calculate_chern_number_for_square_lattice(hamiltonian_function, precision=100):
if np.array(hamiltonian_function(0, 0)).shape==():
dim = 1
else:
dim = np.array(hamiltonian_function(0, 0)).shape[0]
delta = 2*pi/precision
chern_number = np.zeros(dim, dtype=complex)
for kx in np.arange(-pi, pi, delta):
for ky in np.arange(-pi, pi, delta):
H = hamiltonian_function(kx, ky)
vector = calculate_eigenvector(H)
H_delta_kx = hamiltonian_function(kx+delta, ky)
vector_delta_kx = calculate_eigenvector(H_delta_kx)
H_delta_ky = hamiltonian_function(kx, ky+delta)
vector_delta_ky = calculate_eigenvector(H_delta_ky)
H_delta_kx_ky = hamiltonian_function(kx+delta, ky+delta)
vector_delta_kx_ky = 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))
F = cmath.log(Ux*Uy_x*(1/Ux_y)*(1/Uy))
chern_number[i] = chern_number[i] + F
chern_number = chern_number/(2*pi*1j)
return chern_number
# calculate wilson loop
def calculate_wilson_loop(hamiltonian_function, k_min=-pi, k_max=pi, precision=100):
k_array = np.linspace(k_min, k_max, precision)
dim = np.array(hamiltonian_function(0)).shape[0]
wilson_loop_array = np.ones(dim, dtype=complex)
for i in range(dim):
eigenvector_array = []
for k in k_array:
eigenvector = calculate_eigenvector(hamiltonian_function(k))
if k != k_max:
eigenvector_array.append(eigenvector[:, i])
else:
eigenvector_array.append(eigenvector_array[0])
for i0 in range(precision-1):
F = np.dot(eigenvector_array[i0+1].transpose().conj(), eigenvector_array[i0])
wilson_loop_array[i] = np.dot(F, wilson_loop_array[i])
return wilson_loop_array
# read and write
def read_one_dimensional_data(filename='a'):
f = open(filename+'.txt', 'r')
text = f.read()
f.close()
row_list = np.array(text.split('\n'))
dim_column = np.array(row_list[0].split()).shape[0]
x = np.array([])
y = np.array([])
for row in row_list:
column = np.array(row.split())
if column.shape[0] != 0:
x = np.append(x, [float(column[0])], axis=0)
y_row = np.zeros(dim_column-1)
for dim0 in range(dim_column-1):
y_row[dim0] = float(column[dim0+1])
if np.array(y).shape[0] == 0:
y = [y_row]
else:
y = np.append(y, [y_row], axis=0)
return x, y
def read_two_dimensional_data(filename='a'):
f = open(filename+'.txt', 'r')
text = f.read()
f.close()
row_list = np.array(text.split('\n'))
dim_column = np.array(row_list[0].split()).shape[0]
x = np.array([])
y = np.array([])
matrix = np.array([])
for i0 in range(row_list.shape[0]):
column = np.array(row_list[i0].split())
if i0 == 0:
x_str = column[1::]
x = np.zeros(x_str.shape[0])
for i00 in range(x_str.shape[0]):
x[i00] = float(x_str[i00])
elif column.shape[0] != 0:
y = np.append(y, [float(column[0])], axis=0)
matrix_row = np.zeros(dim_column-1)
for dim0 in range(dim_column-1):
matrix_row[dim0] = float(column[dim0+1])
if np.array(matrix).shape[0] == 0:
matrix = [matrix_row]
else:
matrix = np.append(matrix, [matrix_row], axis=0)
return x, y, matrix
def write_one_dimensional_data(x, y, filename='a'):
with open(filename+'.txt', 'w') as f:
i0 = 0
for x0 in x:
f.write(str(x0)+' ')
if len(y.shape) == 1:
f.write(str(y[i0])+'\n')
elif len(y.shape) == 2:
for j0 in range(y.shape[1]):
f.write(str(y[i0, j0])+' ')
f.write('\n')
i0 += 1
def write_two_dimensional_data(x, y, matrix, filename='a'):
with open(filename+'.txt', 'w') as f:
f.write('0 ')
for x0 in x:
f.write(str(x0)+' ')
f.write('\n')
i0 = 0
for y0 in y:
f.write(str(y0))
j0 = 0
for x0 in x:
f.write(' '+str(matrix[i0, j0])+' ')
j0 += 1
f.write('\n')
i0 += 1
# plot figures
def plot(x, y, xlabel='x', ylabel='y', title='', filename='a', show=1, save=0, type=''):
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
plt.subplots_adjust(bottom=0.20, left=0.18)
ax.plot(x, y, type)
ax.grid()
ax.set_title(title, fontsize=20, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=20, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, 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]
if save == 1:
plt.savefig(filename+'.jpg', dpi=300)
if show == 1:
plt.show()
plt.close('all')
def plot_3d_surface(x, y, matrix, xlabel='x', ylabel='y', zlabel='z', title='', filename='a', show=1, save=0):
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator
fig, ax = plt.subplots(subplot_kw={"projection": "3d"})
plt.subplots_adjust(bottom=0.1, right=0.65)
x, y = np.meshgrid(x, y)
if len(matrix.shape) == 2:
surf = ax.plot_surface(x, y, matrix, cmap=cm.coolwarm, linewidth=0, antialiased=False)
elif len(matrix.shape) == 3:
for i0 in range(matrix.shape[2]):
surf = ax.plot_surface(x, y, matrix[:,:,i0], cmap=cm.coolwarm, linewidth=0, antialiased=False)
ax.set_title(title, fontsize=20, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=20, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=20, fontfamily='Times New Roman')
ax.set_zlabel(zlabel, fontsize=20, fontfamily='Times New Roman')
ax.zaxis.set_major_locator(LinearLocator(5))
ax.zaxis.set_major_formatter('{x:.2f}')
ax.tick_params(labelsize=15)
labels = ax.get_xticklabels() + ax.get_yticklabels() + ax.get_zticklabels()
[label.set_fontname('Times New Roman') for label in labels]
cax = plt.axes([0.80, 0.15, 0.05, 0.75])
cbar = fig.colorbar(surf, cax=cax)
cbar.ax.tick_params(labelsize=15)
for l in cbar.ax.yaxis.get_ticklabels():
l.set_family('Times New Roman')
if save == 1:
plt.savefig(filename+'.jpg', dpi=300)
if show == 1:
plt.show()
plt.close('all')
def plot_contour(x, y, matrix, xlabel='x', ylabel='y', title='', filename='a', show=1, save=0):
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
plt.subplots_adjust(bottom=0.2, right=0.75, left = 0.16)
x, y = np.meshgrid(x, y)
contour = ax.contourf(x,y,matrix,cmap='jet')
ax.set_title(title, fontsize=20, fontfamily='Times New Roman')
ax.set_xlabel(xlabel, fontsize=20, fontfamily='Times New Roman')
ax.set_ylabel(ylabel, fontsize=20, fontfamily='Times New Roman')
ax.tick_params(labelsize=15)
labels = ax.get_xticklabels() + ax.get_yticklabels()
[label.set_fontname('Times New Roman') for label in labels]
cax = plt.axes([0.78, 0.17, 0.08, 0.71])
cbar = fig.colorbar(contour, cax=cax)
cbar.ax.tick_params(labelsize=15)
for l in cbar.ax.yaxis.get_ticklabels():
l.set_family('Times New Roman')
if save == 1:
plt.savefig(filename+'.jpg', dpi=300)
if show == 1:
plt.show()
plt.close('all')

5
README.md Executable file
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### GJH project: an open source python package. The official website is https://py.guanjihuan.com
### Installation: pip install --upgrade gjh
### Usage: import gjh

18
Tutorial/Chern_number.py Executable file
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import gjh
import numpy as np
from math import *
def hamiltonian_function(kx, ky): # one QAH model with chern number 2
t1 = 1.0
t2 = 1.0
t3 = 0.5
m = -1.0
hamiltonian = np.zeros((2, 2), dtype=complex)
hamiltonian[0, 1] = 2*t1*cos(kx)-1j*2*t1*cos(ky)
hamiltonian[1, 0] = 2*t1*cos(kx)+1j*2*t1*cos(ky)
hamiltonian[0, 0] = m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky)
hamiltonian[1, 1] = -(m+2*t3*sin(kx)+2*t3*sin(ky)+2*t2*cos(kx+ky))
return hamiltonian
chern_number = gjh.calculate_chern_number_for_square_lattice(hamiltonian_function, precision=100)
print(chern_number)

19
Tutorial/Fourier_transform_and_band_structures.py Executable file
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import gjh
import numpy as np
from math import *
import functools
x = np.linspace(-pi, pi, 100)
y = np.linspace(-pi, pi, 100)
hamiltonian_function = functools.partial(gjh.one_dimensional_fourier_transform, unit_cell=0, hopping=1)
eigenvalue_array = gjh.calculate_eigenvalue_with_one_parameter(x, hamiltonian_function)
gjh.plot(x, eigenvalue_array, xlabel='k', ylabel='E', type='-o')
hamiltonian_function = functools.partial(gjh.two_dimensional_fourier_transform_for_square_lattice, unit_cell=0, hopping_1=1, hopping_2=1)
eigenvalue_array = gjh.calculate_eigenvalue_with_two_parameters(x, y, hamiltonian_function)
gjh.plot_3d_surface(x, y, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E')
hamiltonian_function = functools.partial(gjh.three_dimensional_fourier_transform_for_cubic_lattice, k3=0, unit_cell=0, hopping_1=1, hopping_2=1, hopping_3=1)
eigenvalue_array = gjh.calculate_eigenvalue_with_two_parameters(x, y, hamiltonian_function)
gjh.plot_3d_surface(x, y, eigenvalue_array, xlabel='kx', ylabel='ky', zlabel='E')

5
Tutorial/Hamiltonian_of_finite_size.py Executable file
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import gjh
print(gjh.finite_size_along_one_direction(3), '\n')
print(gjh.finite_size_along_two_directions_for_square_lattice(2, 2), '\n')
print(gjh.finite_size_along_three_directions_for_cubic_lattice(2, 2, 2), '\n')

6
Tutorial/Pauli_matrix.py Executable file
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import gjh
print('sigma_0:\n', gjh.sigma_0(), '\n')
print('sigma_x:\n', gjh.sigma_x(), '\n')
print('sigma_y:\n', gjh.sigma_y(), '\n')
print('sigma_z:\n', gjh.sigma_z(), '\n')

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Tutorial/Wilson_loop.py Executable file
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import gjh
import numpy as np
import cmath
from math import *
def hamiltonian_function(k): # SSH model
gamma = 0.5
lambda0 = 1
delta = 0
hamiltonian = np.zeros((2, 2), dtype=complex)
hamiltonian[0,0] = delta
hamiltonian[1,1] = -delta
hamiltonian[0,1] = gamma+lambda0*cmath.exp(-1j*k)
hamiltonian[1,0] = gamma+lambda0*cmath.exp(1j*k)
return hamiltonian
wilson_loop_array = gjh.calculate_wilson_loop(hamiltonian_function)
print('wilson loop =', wilson_loop_array)
p = np.log(wilson_loop_array)/2/pi/1j
print('p =', p, '\n')

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Tutorial/bands_of_zigzag_graphene.py Executable file
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import gjh
import numpy as np
from math import *
import functools
x = np.linspace(-pi, pi, 100)
Ny = 10
unit_cell = gjh.finite_size_along_two_directions_for_graphene(1, Ny)
hopping = gjh.hopping_along_zigzag_direction_for_graphene(Ny)
hamiltonian_function = functools.partial(gjh.one_dimensional_fourier_transform, unit_cell=unit_cell, hopping=hopping)
eigenvalue_array = gjh.calculate_eigenvalue_with_one_parameter(x, hamiltonian_function)
gjh.plot(x, eigenvalue_array, xlabel='k', ylabel='E')

8
Tutorial/conductance.py Executable file
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import gjh
import numpy as np
fermi_energy_array = np.linspace(-5, 5, 400)
h00 = gjh.finite_size_along_one_direction(4)
h01 = np.identity(4)
conductance_array = gjh.calculate_conductance_with_fermi_energy_array(fermi_energy_array, h00, h01)
gjh.plot(fermi_energy_array, conductance_array, xlabel='E', ylabel='Conductance', type='-o')

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Tutorial/local_density_of_states.py Executable file
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import gjh
import numpy as np
fermi_energy = 0
N1 = 3
N2 = 4
hamiltonian = gjh.finite_size_along_two_directions_for_square_lattice(N1,N2)
LDOS = gjh.local_density_of_states_for_square_lattice(fermi_energy, hamiltonian, N1=N1, N2=N2)
print('square lattice:\n', LDOS, '\n')
h00 = gjh.finite_size_along_one_direction(N2)
h01 = np.identity(N2)
LDOS = gjh.local_density_of_states_for_square_lattice_using_dyson_equation(fermi_energy, h00=h00, h01=h01, N2=N2, N1=N1)
print(LDOS, '\n\n')
gjh.plot_contour(range(N1), range(N2), LDOS)
N1 = 3
N2 = 4
N3 = 5
hamiltonian = gjh.finite_size_along_three_directions_for_cubic_lattice(N1, N2, N3)
LDOS = gjh.local_density_of_states_for_cubic_lattice(fermi_energy, hamiltonian, N1=N1, N2=N2, N3=N3)
print('cubic lattice:\n', LDOS, '\n')
h00 = gjh.finite_size_along_two_directions_for_square_lattice(N2, N3)
h01 = np.identity(N2*N3)
LDOS = gjh.local_density_of_states_for_cubic_lattice_using_dyson_equation(fermi_energy, h00, h01, N3=N3, N2=N2, N1=N1)
print(LDOS)

20
Tutorial/read_and_write.py Executable file
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import gjh
import numpy as np
x = np.array([1, 2, 3])
y = np.array([5, 6, 7])
gjh.write_one_dimensional_data(x, y, filename='one_dimensional_data')
matrix = np.zeros((3, 3))
matrix[0, 1] = 11
gjh.write_two_dimensional_data(x, y, matrix, filename='two_dimensional_data')
x_read, y_read = gjh.read_one_dimensional_data('one_dimensional_data')
print(x_read, '\n')
print(y_read, '\n\n')
x_read, y_read, matrix_read = gjh.read_two_dimensional_data('two_dimensional_data')
print(x_read, '\n')
print(y_read, '\n')
print(matrix_read)

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Tutorial/scattering_matrix.py Executable file
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import gjh
import numpy as np
fermi_energy = 0
h00 = gjh.finite_size_along_one_direction(4)
h01 = np.identity(4)
gjh.print_or_write_scattering_matrix(fermi_energy, h00, h01)

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Tutorial/test.py Executable file
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import gjh
gjh.test()

7
Tutorial/total_density_of_states.py Executable file
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import gjh
import numpy as np
hamiltonian = gjh.finite_size_along_two_directions_for_square_lattice(2,2)
fermi_energy_array = np.linspace(-4, 4, 400)
total_dos_array = gjh.total_density_of_states_with_fermi_energy_array(fermi_energy_array, hamiltonian, broadening=0.1)
gjh.plot(fermi_energy_array, total_dos_array, xlabel='E', ylabel='Total DOS', type='-o')