Hello,
I am not too sure where my problem lies so, I will post my question here, since
I might suspect it has something to do with the kwant interface.
I have a simple graphene strip with periodic boundaries in the vertical
direction and open boundaries in the other direction. To calculate the energies
and states of this system, I feed the hamiltonian from kwant into either the
dense or the sparse solver from linalg. I do not understand why I am getting
significant difference in the values of the eigenstates using sparse vs using
density, especially considering the small size of my system. Attached is a code
that reproduces the problem:
"""
Graphene wire with periodic boundary conditions in the vertical direction.
"""
import kwant
from math import pi, sqrt, tanh, cos, ceil, floor, atan, acos, asin
from cmath import exp
import numpy as np
import scipy
import scipy.linalg as lina
sin_30, cos_30, tan_30 = (1 / 2, sqrt(3) / 2, 1 / sqrt(3))
def create_closed_system(length,
width, lattice_spacing,
onsite_potential,
hopping_parameter, boundary_hopping):
padding = 0.5*lattice_spacing*tan_30
def calc_total_length(length):
total_length = length
N = total_length//lattice_spacing # Number of times a graphene hexagon
fits in horizontially fully
new_length = N*lattice_spacing + lattice_spacing*0.5
diff = total_length - new_length
if diff != 0:
length = length - diff
total_length = new_length
return total_length
def calc_width(width,lattice_spacing, padding):
stacking_width = lattice_spacing*((tan_30/2)+(1/(2*cos_30)))
N = width//stacking_width
if N % 2 == 0.0: # Making sure that N is odd.
N = N-1
new_width = N*stacking_width + padding
width = new_width
return width, int(N)
def rectangle(pos):
x,y = pos
if (0 < x <= total_length) and (-padding <= y <= width -padding):
return True
return False
def lead_shape(pos):
x, y = pos
return 0 - padding <= y <= width
def tag_site_calc(x):
return int(-1*(x*0.5+0.5))
#Initation of geometrical limits of the lattice of the system
total_length = calc_total_length(length)
width, N = calc_width(width, lattice_spacing, padding)
# The definition of the potential over the entire system
def potential_e(site,pot):
return pot
### Definig the lattices ###
graphene_e = kwant.lattice.honeycomb(a=lattice_spacing,name='e')
a, b = graphene_e.sublattices
sys = kwant.Builder()
# The following functions are required for input in the
def onsite_shift_e(site, pot):
return potential_e(site,pot)
sys[graphene_e.shape(rectangle, (0.5*lattice_spacing, 0))] = onsite_shift_e
sys[graphene_e.neighbors()] = -hopping_parameter
### Boundary conditions for scattering region ###
for site in sys.sites():
(x,y) = site.tag
if float(site.pos[1]) < 0:
if str(site.family) == "<Monatomic lattice e1>":
sys[b(x,y),a(int(x+tag_site_calc(N)),N)] = -boundary_hopping
kwant.plot(sys)
return sys
def eigen_vectors_and_values(sys,sparse_dense,k): #sparse = 0, dense = 1
if sparse_dense == 0:
ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=True)
eigen_val, eigen_vec = scipy.sparse.linalg.eigsh(ham_mat.tocsc(), k=k,
sigma=0,
return_eigenvectors=True)
if sparse_dense == 1:
ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=False)
eigen_val, eigen_vec = lina.eigh(ham_mat)
#sort the ee and ev in ascending order
idx = eigen_val.argsort()
eigen_val= eigen_val[idx]
eigen_vec = eigen_vec[:,idx]
if sparse_dense == 1:
eigen_val = eigen_val[int(len(eigen_val)*0.5-k*0.5):] # Remove first
part
eigen_val = eigen_val[:k] # Take first k-values
eigen_vec = eigen_vec[:,int(len(eigen_val)*0.5-k*0.5):]
eigen_vec = eigen_vec[:,:k]
return eigen_vec, eigen_val
def main():
sys = create_closed_system(length = 16.0,
width = 20.0, lattice_spacing = 1.0,
onsite_potential = 0.0,
hopping_parameter = 1.0, boundary_hopping = 1.0)
sys = sys.finalized()
#sparse = 0, dense = 1
eigen_vec, eigen_val = eigen_vectors_and_values(sys,sparse_dense=0,k=50)
eigen_vec2, eigen_val2 = eigen_vectors_and_values(sys,sparse_dense=1,k=50)
print("Eigenenergy difference: ")
for i_ in range(len(eigen_val)):
print(eigen_val[i_] - eigen_val2[i_] )
new_m = eigen_vec - eigen_vec2
new_m = abs(new_m)
print("Max difference in eigenvector values: ", np.amax(new_m))
return new_m
if __name__ == "__main__":
new_m = main()
