Dear Henry,
Your code is full of parts with potential mistakes. I mean by that, using
scipy.sparse.linalg.*eigsh *to get the eigenvalues and eigenvectors might
be a source of error because you only calculate part of the eigenvalues,
and then need to reorder them in the same way linalg.eigh computes all the
eigenvalues. What you should notice is that you have many degenerate
eigenvalues and reordering them will not give you the requested order for
the eigenvectors.
Another point to add concerning the degenerate eigenvalues: the eigenvector
is any combination of linearly independent eigenvectors which means it is
not unique.
The comparison between the eigenvectors is not safe.
(even a simple global phase can make it not safe!)
I hope this helps ,
Adel
On Tue, Sep 28, 2021 at 3:07 PM Henry Axt <henry....@gmail.com> wrote:
> This is related to my other post, but as I see them as different (maybe
> not?) problems, I will post a second thread.
>
> The situation is, again, a graphene strip with periodic boundary
> conditions in one direction and open boundaries in the other.
>
> Using the sparse solver from linalg, I have solved for energies and gotten
> reasonable results for the energies, except for the case when the length
> (along the axis of the open boundary condition) is metallic. Attached is a
> code that shows it for length = 17, which is a metallic length for graphene
> (has energies at zero), but it does not produce messy energy calculation
> for when it is not a metallic length, i.e. something like length = 16.
>
> """
> 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 = 17.0,
> 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 (between sparse and dense): ")
> for i_ in range(len(eigen_val)):
> print(eigen_val[i_] - eigen_val2[i_] )
>
>
>
> return
>
> if __name__ == "__main__":
> main()
>
--
Abbout Adel
