I searched a lot and got the answer! Here it is some links that helped me Some exemples links: https://kwant-project.org/doc/1/tutorial/spin_potential_shape.html?highlight... https://kwant-project.org/doc/1/_downloads/spin_orbit.py https://kwant-project.org/doc/1/tutorial/superconductors.html?highlight=norb... Some forum threads links: https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01353.html https://kwant-discuss.kwant-project.narkive.com/96MScaQw/kwant-spin-currents...

Summarizing: To obtain spin up-up, up-down, down-up and down-down transmissions, it is required use norbs = 2 in a lattice construction and make the scattering matrix ordered. Attention in lat and lead declaration Code exemple:

import kwant import numpy as np

# For plotting from matplotlib import pyplot

# For matrix support import tinyarray

# define Pauli-matrices for convenience 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=0.5, e_z=0.08, W=10, L=30): # Start with an empty tight-binding system and a single square lattice. # `a` is the lattice constant (by default set to 1 for simplicity). lat = kwant.lattice.square(norbs = 2) # these orbitals will be one to spin up and one to spin down

syst = kwant.Builder()

#### Define the scattering region. #### 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 syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_y / 2 # hoppings in y-directions syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2

#### Define the left lead. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)), conservation_law=-sigma_z) # this sorts the blocks: "up" modes come first, and then the "down"

lead[(lat(0, j) for j in range(W))] = 4 * t * sigma_0 + e_z * sigma_z # hoppings in x-direction lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_y / 2 # hoppings in y-directions lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2

#### Attach the leads and return the finalized system. #### syst.attach_lead(lead) syst.attach_lead(lead.reversed())

return syst

def plot_conductance(syst, energies): # Compute conductance data = [] #total transmission data_uu = [] #up-up transmission data_ud = [] #up-down transmission data_du = [] #down-up transmission data_du = [] #down-up transmission

for energy in energies: smatrix = kwant.smatrix(syst, energy) t = np.matrix(smat.submatrix(1, 0)) #transmission block tt = np.matmul(t, t.getH()) #t*t_dagger matrix in case of especific cases, i prefer work with this form data.append(smatrix.transmission(1, 0)) # transmission lead 0 to lead 1 data_uu.append(smatrix.transmission((1,0), (0,0))) # transmission lead 0 up to lead 1 up data_ud.append(smatrix.transmission((1,1), (0,0))) # transmission lead 0 up to lead 1 down data_du.append(smatrix.transmission((1,0), (0,1))) # transmission lead 0 down to lead 1 up data_dd.append(smatrix.transmission((1,0), (0,0))) # transmission lead 0 down to lead 1 down

pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show()

pyplot.figure() pyplot.plot(energies, data_uu) pyplot.xlabel("energy [t]") pyplot.ylabel("up-up conductance [e^2/h]") pyplot.show()

pyplot.figure() pyplot.plot(energies, data_ud) pyplot.xlabel("energy [t]") pyplot.ylabel("up-down conductance [e^2/h]") pyplot.show()

pyplot.figure() pyplot.plot(energies, data_du) pyplot.xlabel("energy [t]") pyplot.ylabel("down-up conductance [e^2/h]") pyplot.show()

pyplot.figure() pyplot.plot(energies, data_dd) pyplot.xlabel("energy [t]") pyplot.ylabel("down-down conductance [e^2/h]") pyplot.show()

def main(): syst = make_system()

# Check that the system looks as intended. kwant.plot(syst)

# Finalize the system. syst = syst.finalized()

# We should see non-monotonic conductance steps. plot_conductance(syst, energies=[0.01 * i - 0.3 for i in range(100)])

# Call the main function if the script gets executed (as opposed to imported). # See http://docs.python.org/library/__main__.html. if __name__ == '__main__': main()

Hope it helps!