New subject: Regarding smatrix and spin

Feb. 14, 2017

      Dear Anton,

sorry for troubling you again.

so I finally got hold of a linux-computer, and managed to run the commands and install all the things in the notebook. When I try to run the code you have in your notebook however, I get the following error message:

  File "/home/camilla/.local/lib/python3.5/site-packages/kwant/builder.py", line 1371, in attach_lead
    self.leads.append(BuilderLead(lead_builder, tuple(interface)))
  File "/home/camilla/.local/lib/python3.5/site-packages/kwant/builder.py", line 565, in __init__
    self.interface = tuple(sorted(interface))
TypeError: unorderable types: tinyarray.ndarray_int() < tinyarray.ndarray_int()

Do you know what the problem could be?

Best,

Camilla

The code if needed:

import kwant
import tinyarray as ta
import numpy as np
from scipy import sparse
from matplotlib import pyplot
import matplotlib

s0 = ta.array([[1, 0], [0, 1]])
sx = ta.array([[0, 1], [1, 0]])
sy = ta.array([[0, 1j], [-1j, 0]])
sz = ta.array([[1, 0], [0, -1]])

# Adapted from https://kwant-project.org/doc/1/tutorial/tutorial2

def make_system(t=1.0, W=10, L=10):
    # Now we must specify the number of orbitals per site.
    lat = kwant.lattice.square(norbs=2)
    syst = kwant.Builder()

    syst[(lat(x, y) for x in range(L) for y in range(W))] = \
            lambda s, alpha, E_z: 4 * t * s0 + E_z * sz
    syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
            lambda s1, s2, alpha, E_z: -t * s0 - 1j * alpha * sy
    syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
            lambda s1, s2, alpha, E_z: -t * s0 + 1j * alpha * sx

    # The new bit: specifying the conservation law.
    lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)),
                         conservation_law=-sz, time_reversal=s0)
    lead[(lat(0, j) for j in range(W))] = 4 * t * s0
    lead[lat.neighbors()] = -t * s0 # Note: no spin-orbit in the lead.

    syst.attach_lead(lead)
    syst.attach_lead(lead.reversed())

    syst = syst.finalized()

    return syst

syst = make_system(t=1.0, W=10, L=10)
energies = np.linspace(0, 1, 200)
smatrices = [kwant.smatrix(syst, energy, args=(0.2, 0.05)) for energy in energies]

fig = pyplot.figure(figsize=(13, 8))
ax = fig.add_subplot(1, 1, 1)

# Like previously smatrix.transmission(lead1, lead0) is transmission from lead0 to lead1
ax.plot(energies, [smatrix.transmission(1, 0) for smatrix in smatrices], label='total')

# The new bit: smatrix.transmission((lead1, q1), (lead0, q0)) is the transmission from the
# q0 block of the lead0 into the q1 block of lead1. The subblock ordering is same as we used
# in set_symmetry.
ax.plot(energies, [smatrix.transmission((1, 0), (0, 0)) for smatrix in smatrices], label='$G_{↑↑}$')
ax.plot(energies, [smatrix.transmission((1, 1), (0, 0)) for smatrix in smatrices], label='$G_{↑↓}$')
ax.plot(energies, [smatrix.transmission((1, 0), (0, 1)) for smatrix in smatrices], label='$G_{↓↑}$')
ax.plot(energies, [smatrix.transmission((1, 1), (0, 1)) for smatrix in smatrices], label='$G_{↓↓}$')
ax.set_ylabel('$G [e^2/h]$', fontsize='xx-large')
ax.set_xlabel('$E/t$', fontsize='xx-large')
ax.legend(fontsize='x-large');

On 24. jan. 2017 13:18, Anton Akhmerov wrote:

OK, please double-check the remaining simulation parameters.

Anton

On Tue, Jan 24, 2017 at 1:00 PM, Camilla Espedal
<camilla.espedal@ntnu.no> wrote:

Dear Anton,

I triend changing it, but that does not solve the problem.

Best,

Camilla

From: Anton Akhmerov [mailto:anton.akhmerov+kd@gmail.com]
Sent: 24. januar 2017 12:36

To: Camilla Espedal <camilla.espedal@ntnu.no>
Cc: kwant-discuss@kwant-project.org
Subject: Re: [Kwant] Regarding smatrix and spin

Dear Camilla,

Could the difference originate from you using a lattice constant of 2
instead of 1?

Anton

On Tue, Jan 24, 2017, 10:41 Camilla Espedal <camilla.espedal@ntnu.no> wrote:

Dear Anton,

Thanks again for all your help. I will try to do it the linux way. Just one
more thing regarding this. I wrote a script in the old Kwant to find the
total conductance and compare it to the one in your notebook. While the two
plots are qualitatively similar, they are not the same. Am I missing
something, or am I calculating different things?

Best, Camilla

(my code):

# Tutorial 2.3.1. Matrix structure of on-site and hopping elements
# ================================================================
#
# Physics background
# ------------------
#  Gaps in quantum wires with spin-orbit coupling and Zeeman splititng,
#  as theoretically predicted in
#   http://prl.aps.org/abstract/PRL/v90/i25/e256601
#  and (supposedly) experimentally oberved in
#   http://www.nature.com/nphys/journal/v6/n5/abs/nphys1626.html
#
# Kwant features highlighted
# --------------------------
#  - Numpy matrices as values in Builder

import kwant

# For plotting
import matplotlib.pyplot as plt

# For matrix support
import tinyarray
import numpy as np

# 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(a=2, t=1.0, alpha=0.1, e_z=0.05, W=10, L=10):
    # 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(a)

    sys = kwant.Builder()

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

    #### Define the left lead. ####
    lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0)))

    lead[(lat(0, j) for j in range(W))] = 4 * t * sigma_0
    # hoppings in x-direction
    lead[lat.neighbors()] = \
        -t * sigma_0

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

    return sys

def plot_conductance(sys, energies):
    # Compute conductance
    data = []
    for energy in energies:
        smatrix = kwant.smatrix(sys, energy)
        data.append(smatrix.transmission(1, 0))

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

def main():
    sys = make_system()

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

    # Finalize the system.
    sys = sys.finalized()
    energies = np.linspace(0, 1, 200)
    smatrices = [kwant.smatrix(sys, energy) for energy in energies]

    fig = plt.figure(figsize=(13, 8))
    ax = fig.add_subplot(1, 1, 1)

    ax.plot(energies, [smatrix.transmission(1,0) for smatrix in smatrices],
label='total')

    ax.set_ylabel('$G [e^2/h]$', fontsize='xx-large')
    ax.set_xlabel('$E/t$', fontsize='xx-large')
    ax.legend(fontsize='x-large')

    plt.show()

# 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()

-----Original Message-----
From: anton.akhmerov@gmail.com [mailto:anton.akhmerov@gmail.com] On Behalf
Of Anton Akhmerov
Sent: 17. januar 2017 10:48
To: Camilla Espedal <camilla.espedal@ntnu.no>
Cc: kwant-discuss@kwant-project.org
Subject: Re: [Kwant] Regarding smatrix and spin

Dear Camilla,

It seems that you are trying to install Kwant on windows. This is a very
hard task, and I fear none of the Kwant developers has enough knowledge of
it right now (our Windows packages are built by Christoph Gohlke, see [1]
for the build environment description). However if you are using windows 10,
I suggest to try to install Kwant using the windows subsystem for linux.
That way the standard Ubuntu build procedure should work for you.

Best,
Anton

[1]: http://www.lfd.uci.edu/~gohlke/pythonlibs/

On Mon, Jan 16, 2017 at 9:45 AM, Camilla Espedal <camilla.espedal@ntnu.no>
wrote:

Thanks a lot. I tried to install the cons_laws_combined, but I get the
following error message:

"LINK: fatal error LNK1181: cannot open input file 'lapack.lib'"

Is there some package or installation I am missing?

Best regards,
Camilla

-----Original Message-----
From: anton.akhmerov@gmail.com [mailto:anton.akhmerov@gmail.com] On
Behalf Of Anton Akhmerov
Sent: 8. januar 2017 16:35
To: Tómas Örn Rosdahl <torosdahl@gmail.com>
Cc: Camilla Espedal <camilla.espedal@ntnu.no>;
kwant-discuss@kwant-project.org
Subject: Re: [Kwant] Regarding smatrix and spin

Hi Camilla, everyone,

I've slightly modified Tómas's example to a case where the spins do get
coupled, check it out:
http://nbviewer.jupyter.org/url/antonakhmerov.org/misc/spin_conductanc
e.ipynb

I've also provided more detailed installation instructions in the
notebook.

Cheers,
Anton

On Sun, Jan 8, 2017 at 2:45 PM, Tómas Örn Rosdahl <torosdahl@gmail.com>
wrote:

Dear Camilla,

For a Hamiltonian with degeneracies due to a conservation law, the
scattering states will in general not have a definite value of the
conservation law. In your case, Kwant returns scattering states that
are arbitrary linear combinations of spin up and down, so it is not
possible to label the amplitudes in the scattering matrix by spin.

However, in Kwant 1.3 a feature will be added that allows for the
construction of scattering states with definite values of a
conservation law. See here for an explanation of the basic idea behind
the algorithm.

We're currently working on implementing this feature in Kwant itself.
The good news is that we're practically done - here is a link to a
git repo with a functioning implementation. After you clone the repo,
check out the branch cons_laws_combined, which contains a version of
Kwant with conservation laws implemented. This notebook contains a
simple example to illustrate how to work with conservation laws and the
scattering matrix.

I invite you and anyone else who is interested to give it a try. We'd
appreciate any feedback!

In your case specifically, there would be two projectors in the new
implementation - P0 which projects out the spin up block, and P1 that
projects out the spin down block. If they are specified in this
order, then the spin up and down blocks in the Hamiltonian have block
indices
0 and 1, respectively. In the new implementation, it is possible to
ask for subblocks of the scattering matrix relating not only any two
leads, but also any two conservation law blocks in any leads. To get
the reflection amplitude of an incident spin up electron from lead 0
into an outgoing spin down electron in lead 0, you could simply do
smat.submatrix((0, 1), (0, 0)). Here, the arguments are tuples of indices
(lead index, block index).

Best regards,
Tómas

On Fri, Jan 6, 2017 at 3:46 PM, Camilla Espedal
<camilla.espedal@ntnu.no>
wrote:

Hi again,

This question is basically the same as this:
https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg0007
6
.html

I want to calculate some things using the scattering matrix. I
started out with a very simple system, most basic two-terminal
system. For some energy there is one propagating mode. I now add
matrix structure to the mix (just multiply by s_0 everywhere) and
there are now 2 propagating modes (which makes sense).

Now, if I look at the reflection coefficients for lead 0 by using
submatrix(0,0), it is now a 2x2 matrix after I introduced the
matrices. How are the elements ordered? Is it

[[r_upup, r_updown],[r_downup, r_downdown]]

I know that I could make two lattices, but since I do not plan to
use the other functions such as transmission. I  just want the smatrix.

Hope you can help me, and thanks in advance.

Best regards,

Camilla

Re: [Kwant] Regarding smatrix and spin

Camilla Espedal

Anton Akhmerov

Camilla Espedal

Anton Akhmerov

Joseph Weston

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