[Kwant] Extract wavefunction for one lattice

[Yuhao Kang]

Hello,

In the system I use two lattices to represent spin up and down.

lat_u = kwant.lattice.honeycomb(a=1, name='up')
lat_d = kwant.lattice.honeycomb(a=1, name='down')

When I obtain the scattering wave functions from wf=kwant.wave_function(sys, en),
it contains the value for both lat_u and lat_d.

Is that possible to extract the wave function only for lat_u or lat_d?

I tried to separate the wave function based on the odd/even index, but it is not the right sequence.

Thank you in advance.

Regards,
Yuhao

[Abbout Adel]

Dear Yuhao,

The order of the wavefunction elements is the same as that of the sites.
Since you know the order of sites from sys.sites the task becomes very
simple.

In the following example, you can check the family of the sites in
sys.sites and notice that you have the electrons first and then the holes.
The wavefunction order becomes straightforward.

I hope this helps
Adel



import kwant
from matplotlib import pyplot
import numpy as np

def make_system(a=1, W=30, L=20, barrier=1., barrierpos=(3, 8),
mu=0.4, Delta=0.1, Deltapos=4, t=1.0):
# Start with an empty tight-binding system and two square lattices,
# corresponding to electron and hole degree of freedom
lat_e = kwant.lattice.square(a, name='e')
lat_h = kwant.lattice.square(a, name='h')

sys = kwant.Builder()
sys1 = kwant.Builder()
sys2 = kwant.Builder()
#### Define the scattering region. ####
sys1[(lat_e(x, y) for x in range(L) for y in range(W))] = 4 * t - mu
sys2[(lat_h(x, y) for x in range(L) for y in range(W))] = mu - 4 * t

# the tunnel barrier
sys1[(lat_e(x, y) for x in range(barrierpos[0], barrierpos[1])
for y in range(W))] = 4 * t + barrier - mu
sys2[(lat_h(x, y) for x in range(barrierpos[0], barrierpos[1])
for y in range(W))] = mu - 4 * t - barrier

# hoppings for both electrons and holes
sys1[lat_e.neighbors()] = -t
sys2[lat_h.neighbors()] = t
sys+=sys1
sys+=sys2
# Superconducting order parameter enters as hopping between
# electrons and holes
sys[((lat_e(x, y), lat_h(x, y)) for x in range(Deltapos, L)
for y in range(W))] = Delta

#### Define the leads. ####
# Symmetry for the left leads.
sym_left = kwant.TranslationalSymmetry((-a, 0))

# left electron lead
lead0 = kwant.Builder(sym_left)
lead0[(lat_e(0, j) for j in range(W))] = 4 * t - mu
lead0[lat_e.neighbors()] = -t

# left hole lead
lead1 = kwant.Builder(sym_left)
lead1[(lat_h(0, j) for j in range(W))] = mu - 4 * t
lead1[lat_h.neighbors()] = t

# Then the lead to the right
# this one is superconducting and thus is comprised of electrons
# AND holes
sym_right = kwant.TranslationalSymmetry((a, 0))
lead2 = kwant.Builder(sym_right)
lead2 += lead0
lead2 += lead1
lead2[((lat_e(0, j), lat_h(0, j)) for j in range(W))] = Delta

#### Attach the leads and return the system. ####
sys.attach_lead(lead0)
sys.attach_lead(lead1)
sys.attach_lead(lead2)

return sys,sys1,sys2



#I used sys1 and sys2 to help in the 2D-plot
sys,sys1,sys2 = make_system()

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

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


#sys and sys1 have the same positions of sites

wf=kwant.wave_function(sys,energy=1)

wavefunction=wf(0)[1].reshape(2,-1)# lead number 0, mode number 2

#plot of wf for electrons:
kwant.plotter.map(sys1.finalized(),abs(wavefunction[0])**2 )


#plot of wf for holes:
kwant.plotter.map(sys1.finalized(),abs(wavefunction[1])**2 )

On Thu, Nov 1, 2018 at 11:15 AM Yuhao Kang <***@gradcenter.cuny.edu>
wrote:

> Hello,
>
> In the system I use two lattices to represent spin up and down.
>
> lat_u = kwant.lattice.honeycomb(a=1, name='up')
> lat_d = kwant.lattice.honeycomb(a=1, name='down')
>
> When I obtain the scattering wave functions from
> wf=kwant.wave_function(sys, en),
> it contains the value for both lat_u and lat_d.
>
> Is that possible to extract the wave function only for lat_u or lat_d?
>
> I tried to separate the wave function based on the odd/even index, but it
> is not the right sequence.
>
> Thank you in advance.
>
> Regards,
> Yuhao
>


--
Abbout Adel

[Joseph Weston]

Hi Yuhao


> In the system I use two lattices to represent spin up and down.
>
> lat_u = kwant.lattice.honeycomb(a=1, name='up') 
> lat_d = kwant.lattice.honeycomb(a=1, name='down')
>
> When I obtain the scattering wave functions from
> wf=kwant.wave_function(sys, en), 
> it contains the value for both lat_u and lat_d.


On 11/01/2018 01:33 PM, Abbout Adel wrote:
> Dear Yuhao,
>
> The order of the wavefunction elements is the same as that of the
> sites. Since you know the order of sites from sys.sites the task
> becomes very simple.
>
> In the following example, you can check the family of the sites in
> sys.sites and notice that you have the electrons first and then the
> holes. The wavefunction order becomes straightforward.
>

Adel's answer is correct, however I wanted to point out that there's
little need to separate internal degrees of freedom onto separate
lattices in modern Kwant.

You can define your system much more concisely using matrix-valued
onsites, not to mention you can use more advanced functionality such as
the kwant.operator module to get spatially resolved spin currents etc.

Historically the only reason to separate internal degrees of freedom
(e.g. spin) onto separate lattices was to force Kwant to choose a
certain mode basis  in the leads, e.g. modes with spin up and spin down,
which would mean that the scattering matrix blocks could be more easily
interpreted (e.g. reflection from spin up channel into spin down
channel). In modern Kwant you can declare that your lead satisfies a
certain conservation law (e.g. spin conservation), as illustrated in the
documentation [1], which forces Kwant to use a mode basis that is
compatible with the conservation law.

For example, if your on-site degrees of freedom are (spin up, spin down)
and your lead Hamiltonian conserves spin, then you could declare your
lead like so:

    sigma_z = np.array([[1, 0], [0, -1]])
    lead = kwant.Builder(lead_symmetry, conservation_law=-sigma_z)
    ...

This will guarantee that each of the lead modes has a well-defined spin,
i.e. that a mode wavefunctions are non-zero only on the spin up or spin
down degrees of freedom.

Happy Kwanting,

Joe


[1]: https://kwant-project.org/doc/1/tutorial/superconductors

[Yuhao Kang]

Thank you! That's very helpful.


Best,

yuhao

________________________________
From: Joseph Weston <***@gmail.com>
Sent: Thursday, November 1, 2018 10:22:52 AM
To: Abbout Adel; Yuhao Kang; kwant-discuss
Subject: Re: [Kwant] Extract wavefunction for one lattice

Hi Yuhao


> In the system I use two lattices to represent spin up and down.
>
> lat_u = kwant.lattice.honeycomb(a=1, name='up')
> lat_d = kwant.lattice.honeycomb(a=1, name='down')
>
> When I obtain the scattering wave functions from
> wf=kwant.wave_function(sys, en),
> it contains the value for both lat_u and lat_d.


On 11/01/2018 01:33 PM, Abbout Adel wrote:
> Dear Yuhao,
>
> The order of the wavefunction elements is the same as that of the
> sites. Since you know the order of sites from sys.sites the task
> becomes very simple.
>
> In the following example, you can check the family of the sites in
> sys.sites and notice that you have the electrons first and then the
> holes. The wavefunction order becomes straightforward.
>

Adel's answer is correct, however I wanted to point out that there's
little need to separate internal degrees of freedom onto separate
lattices in modern Kwant.

You can define your system much more concisely using matrix-valued
onsites, not to mention you can use more advanced functionality such as
the kwant.operator module to get spatially resolved spin currents etc.

Historically the only reason to separate internal degrees of freedom
(e.g. spin) onto separate lattices was to force Kwant to choose a
certain mode basis in the leads, e.g. modes with spin up and spin down,
which would mean that the scattering matrix blocks could be more easily
interpreted (e.g. reflection from spin up channel into spin down
channel). In modern Kwant you can declare that your lead satisfies a
certain conservation law (e.g. spin conservation), as illustrated in the
documentation [1], which forces Kwant to use a mode basis that is
compatible with the conservation law.

For example, if your on-site degrees of freedom are (spin up, spin down)
and your lead Hamiltonian conserves spin, then you could declare your
lead like so:

sigma_z = np.array([[1, 0], [0, -1]])
lead = kwant.Builder(lead_symmetry, conservation_law=-sigma_z)
...

This will guarantee that each of the lead modes has a well-defined spin,
i.e. that a mode wavefunctions are non-zero only on the spin up or spin
down degrees of freedom.

Happy Kwanting,

Joe


[1]: https://kwant-project.org/doc/1/tutorial/superconductors