Dear Pablo,
thank You for Your answer - it clarifies a lot, but at the same time a few
questions in my head appear.
(At the beginning I'll mention that I'm more familiar with analytical
calculation thus maybe my questions are trivial for the people familiar with
numerical methodes, but I just want to understand some issues and figure out
what is going in the results which I've obtained here.)
1. Now I have noticed that in Your previous reply You had changed the hoppings
in the system in this way: <code>-t*sigma_0</code> -> <code>-t*sigma_z</code>
and without this change the energy gap does not appear. I don't understand this
change, because for me it creates a different system: this part originates from
the kinetic term of the Hamiltonian and this term is defined with the identity
matrix (sigma_0). Could You explain me why we can introduce that change?
2. Regarding the noises in xy-conductivity outside the gap in the finite
system. You wrote about the tricks how to reduce this noise by changing the
parametres of computation in KPM method - I checked, the noise is in fact
smaller. But I want to find out how the xy-component will behave in the
infinite system using the scattering-matrix method (will I get the results
expected for the infinite system?). Thus, I've added the top and bottom leads
and with the <code>smatrix.transmission()</code> method I've calulated the xx-
and xy-conductance (based on 2.3.1 tutorial). Unfortunately, the results are
unclear for me, because:
a) adding the top and bottom leads to the system cancels the quantization in
xx-conductance - I do not understand why adding leads perpendicular to the
current flow does have the influence on the longitudinal conductance,
b) the energy dependence of the both, the xx- and xy-conductance, increasing
with oscillations - I don't understand this behaviour either,
c) if I replace the hopping terms as You recommended, means
<code>-t*sigma_0</code> -> <code>-t*sigma_z</code>, the results are unexpected
either (0 in conductance till some energy, then random positive values).
3. The third thing, where I have a problem, is about the band structure. Taking
a pen and a piece of paper: if I defined the system describing by the
Hamiltonian which is the matrix 2x2, I will get two eigenvalues corresponding
to two branches of the dispersion relation. Here, by calculating the band
structure of the lead (describing by 2x2 matrix written out as on-site and
hopping parameters), I get a lot of bands (which number increases with the size
of the lead/system). I suppose that in this notation every unit cell/site
produces a vector, and thus eigenvalue, and in that way we obtain so plentiful
band structure, but it's not clear for me.
Below I attach the code for the 2. point.
Best regards,
Anna
-----------------------------------------------------
<code>
import kwant
from matplotlib import pyplot
import tinyarray
# we define the identity matrix and Pauli matrices
sigma_0 = tinyarray.array([[1,0], [0,1]])
sigma_x = tinyarray.array([[0,1], [1,0]])
sigma_y = tinyarray.array([[0,-1j], [1j,0]])
sigma_z = tinyarray.array([[1,0], [0,-1]])
def make_system(t=1.0, alpha=1, e_z=0.5, W=30, L=30, a=1): # for simplicity we
set 'a' equal to 1
lat = kwant.lattice.square()
syst = kwant.Builder()
# Zeeman energy adds to the on-site term
syst[(lat(x,y) for x in range(L) for y in range(W))] = 4*t*sigma_0 +
e_z*sigma_z
# hoppings in x-direction; (a,0) means hopping form (i,j) to (i+1,j)
syst[kwant.builder.HoppingKind((a,0), lat, lat)] = -t*sigma_0 -
1j*alpha*sigma_y/(2*a)
#syst[kwant.builder.HoppingKind((a,0), lat, lat)] = -t*sigma_z+
1j*alpha*sigma_y/(2*a) #change from the Pablo's code
# hoppings in y-direction; (0,a) means hopping form (i,j) to (i,j+1)
syst[kwant.builder.HoppingKind((0,a), lat, lat)] = -t*sigma_0 +
1j*alpha*sigma_x/(2*a)
#syst[kwant.builder.HoppingKind((0,a), lat, lat)] = -t*sigma_z +
1j*alpha*sigma_x/(2*a) #change from the Pablo's code
lead = kwant.Builder(kwant.TranslationalSymmetry((-a,0)))
lead[(lat(0,j) for j in range(W))] = 4*t*sigma_0 + e_z*sigma_z
lead[kwant.builder.HoppingKind((1,0), lat, lat)] = -t*sigma_0 -
1j*alpha*sigma_y/(2*a) # hoppings in x-direction
lead[kwant.builder.HoppingKind((0,1), lat, lat)] = -t*sigma_0 +
1j*alpha*sigma_x/(2*a) # hoppings in y-direction
# attach the left and right leads
syst.attach_lead(lead) # left
syst.attach_lead(lead.reversed()) # right
# create the leads (top and bottom)
lead2 = kwant.Builder(kwant.TranslationalSymmetry((0,a)))
lead2[(lat(i,0) for i in range(L))] = 4*t*sigma_0 + e_z*sigma_z
lead2[kwant.builder.HoppingKind((1,0), lat, lat)] = -t*sigma_0 -
1j*alpha*sigma_y/(2*a) # hoppings in x-direction
lead2[kwant.builder.HoppingKind((0,1), lat, lat)] = -t*sigma_0 +
1j*alpha*sigma_x/(2*a) # hoppings in y-direction
# attach the leads (top and bottom)
syst.attach_lead(lead2) # top - ADDING THIS ELECTRODE changes the
xx-component of conductivity tensor
syst.attach_lead(lead2.reversed()) # bottom
# return the finalized system
return syst
def plot_conductance(syst, energies, firstLead, secondLead):
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
data.append(smatrix.transmission(secondLead,firstLead)) # form lead
'firstLead' to 'secondLead'
pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel(r"conductance [e$^2$/h]")
pyplot.title("form " + str(firstLead) + " lead to " + str(secondLead) + "
lead")
pyplot.show()
def main():
syst = make_system()
kwant.plot(syst); # check that the system looks as intended
syst = syst.finalized()
# calculate the band structure (of the lead)
# the results are the same for each (left,right,top,bottom) lead
kwant.plotter.bands(syst.leads[0])
print('leads:\t 0 - left \n\t 1 - right \n\t 2 - top \n\t 3 - bottom')
# longitudinal conductivity (right-left = top-bottom)
plot_conductance(syst, energies=[0.01*i-0.3 for i in range(100)],
firstLead=0, secondLead=1) # from right to left
#plot_conductance(syst, energies=[0.01*i-0.3 for i in range(100)],
firstLead=2, secondLead=3) # from bottom to top
# transverse conductivity (reversible values, means top-left = left-top)
plot_conductance(syst, energies=[0.01*i-0.3 for i in range(100)],
firstLead=0, secondLead=2) # form top to left
#plot_conductance(syst, energies=[0.01*i-0.3 for i in range(100)],
firstLead=2, secondLead=0) # form left to top
#plot_conductance(syst, energies=[0.01*i-0.3 for i in range(100)],
firstLead=1, secondLead=2) # form top to right
# standard Python construct to execute 'main' if the program is used as a script
if __name__ == '__main__':
main()
</code>
