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Commit f1929ee7 authored by Joseph Weston's avatar Joseph Weston
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convert faq to jupyter-sphinx

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doc/source/tutorial/faq.rst
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...@@ -5,6 +5,14 @@ questions that are discussed there. The `Kwant paper ...@@ -5,6 +5,14 @@ questions that are discussed there. The `Kwant paper
<https://downloads.kwant-project.org/doc/kwant-paper.pdf>`_ also digs deeper <https://downloads.kwant-project.org/doc/kwant-paper.pdf>`_ also digs deeper
into Kwant's structure. into Kwant's structure.
.. jupyter-execute::
:hide-code:
import kwant
import numpy as np
import tinyarray
import matplotlib
from matplotlib import pyplot as plt
What is a system, and what is a builder? What is a system, and what is a builder?
======================================== ========================================
...@@ -48,11 +56,16 @@ combination of family and tag uniquely defines a site. ...@@ -48,11 +56,16 @@ combination of family and tag uniquely defines a site.
For example let us create an empty tight binding system and add two sites: For example let us create an empty tight binding system and add two sites:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_site
:end-before: #HIDDEN_END_site a = 1
lat = kwant.lattice.square(a)
syst = kwant.Builder()
syst[lat(1, 0)] = 4
syst[lat(1, 1)] = 4
.. image:: /code/figure/faq_site.* kwant.plot(syst);
In the above snippet we added 2 sites: ``lat(1, 0)`` and ``lat(0, 1)``. Both In the above snippet we added 2 sites: ``lat(1, 0)`` and ``lat(0, 1)``. Both
of these sites belong to the same family, ``lat``, but have different tags: of these sites belong to the same family, ``lat``, but have different tags:
...@@ -79,11 +92,11 @@ tuples, for example lists, are not treated as hoppings. ...@@ -79,11 +92,11 @@ tuples, for example lists, are not treated as hoppings.
Starting from the example code from `What is a site?`_, we can add a hopping Starting from the example code from `What is a site?`_, we can add a hopping
to our system in the following way: to our system in the following way:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_hopping
:end-before: #HIDDEN_END_hopping
.. image:: /code/figure/faq_hopping.* syst[(lat(1, 0), lat(1, 1))] = 1j
kwant.plot(syst);
Visually, a hopping is represented as a line that joins two sites. Visually, a hopping is represented as a line that joins two sites.
...@@ -139,18 +152,29 @@ Let's create two monatomic lattices (``lat_a`` and ``lat_b``). ``(1, 0)`` ...@@ -139,18 +152,29 @@ Let's create two monatomic lattices (``lat_a`` and ``lat_b``). ``(1, 0)``
and ``(0, 1)`` will be the primitive vectors and ``(0, 0)`` and ``(0.5, 0.5)`` and ``(0, 1)`` will be the primitive vectors and ``(0, 0)`` and ``(0.5, 0.5)``
the origins of the two lattices: the origins of the two lattices:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_lattice_monatomic
:end-before: #HIDDEN_END_lattice_monatomic # Two monatomic lattices
primitive_vectors = [(1, 0), (0, 1)]
lat_a = kwant.lattice.Monatomic(primitive_vectors, offset=(0, 0))
lat_b = kwant.lattice.Monatomic(primitive_vectors, offset=(0.5, 0.5))
# lat1 is equivalent to kwant.lattice.square()
syst = kwant.Builder()
.. image:: /code/figure/faq_lattice.* syst[lat_a(0, 0)] = 4
syst[lat_b(0, 0)] = 4
kwant.plot(syst);
We can also create a ``Polyatomic`` lattice with the same primitive vectors and We can also create a ``Polyatomic`` lattice with the same primitive vectors and
two sites in the basis: two sites in the basis:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_lattice_polyatomic
:end-before: #HIDDEN_END_lattice_polyatomic # One polyatomic lattice containing two sublattices
lat = kwant.lattice.Polyatomic([(1, 0), (0, 1)], [(0, 0), (0.5, 0.5)])
sub_a, sub_b = lat.sublattices
The two sublattices ``sub_a`` and ``sub_b`` are nothing else than ``Monatomic`` The two sublattices ``sub_a`` and ``sub_b`` are nothing else than ``Monatomic``
instances, and are equivalent to ``lat_a`` and ``lat_b`` that we created instances, and are equivalent to ``lat_a`` and ``lat_b`` that we created
...@@ -169,17 +193,40 @@ When plotting, how to color the different sublattices differently? ...@@ -169,17 +193,40 @@ When plotting, how to color the different sublattices differently?
================================================================== ==================================================================
In the following example we shall use a kagome lattice, which has three sublattices. In the following example we shall use a kagome lattice, which has three sublattices.
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_colors1
:end-before: #HIDDEN_END_colors1 lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
As we can see below, we create a new plotting function that assigns a color for each family, and a different size for the hoppings depending on the family of the two sites. Finally we add sites and hoppings to our system and plot it with the new function. As we can see below, we create a new plotting function that assigns a color for each family, and a different size for the hoppings depending on the family of the two sites. Finally we add sites and hoppings to our system and plot it with the new function.
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_colors2
:end-before: #HIDDEN_END_colors2 # Plot sites from different families in different colors
def family_color(site):
if site.family == a:
return 'red'
if site.family == b:
return 'green'
else:
return 'blue'
.. image:: /code/figure/faq_colors.* def plot_system(syst):
kwant.plot(syst, site_lw=0.1, site_color=family_color)
## Add sites and hoppings.
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4
syst[b(i, j)] = 4
syst[c(i, j)] = 4
syst[lat.neighbors()] = -1
## Plot the system.
plot_system(syst)
How to create many similar hoppings in one go? How to create many similar hoppings in one go?
...@@ -197,31 +244,64 @@ integers). ...@@ -197,31 +244,64 @@ integers).
The following example shows how this can be used: The following example shows how this can be used:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_direction1
:end-before: #HIDDEN_END_direction1
.. image:: /code/figure/faq_direction1.* # Create hopping between neighbors with HoppingKind
a = 1
syst = kwant.Builder()
lat = kwant.lattice.square(a)
syst[ (lat(i, j) for i in range(5) for j in range(5)) ] = 4
syst[kwant.builder.HoppingKind((1, 0), lat)] = -1
kwant.plot(syst);
Note that ``HoppingKind`` only works with site families so you cannot use Note that ``HoppingKind`` only works with site families so you cannot use
them directly with ``Polyatomic`` lattices; you have to explicitly specify them directly with ``Polyatomic`` lattices; you have to explicitly specify
the sublattices when creating a ``HoppingKind``: the sublattices when creating a ``HoppingKind``:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_direction2 :hide-code:
:end-before: #HIDDEN_END_direction2
lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4
syst[b(i, j)] = 4
syst[c(i, j)] = 4
.. jupyter-execute::
# equivalent to syst[kwant.builder.HoppingKind((0, 1), b)] = -1
syst[kwant.builder.HoppingKind((0, 1), b, b)] = -1
Here, we want the hoppings between the sites from sublattice b with a direction of (0,1) in the lattice coordinates. Here, we want the hoppings between the sites from sublattice b with a direction of (0,1) in the lattice coordinates.
.. image:: /code/figure/faq_direction2.* .. jupyter-execute::
:hide-code:
plot_system(syst)
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_direction3 :hide-code:
:end-before: #HIDDEN_END_direction3
# Delete the hoppings previously created
del syst[kwant.builder.HoppingKind((0, 1), b, b)]
.. jupyter-execute::
syst[kwant.builder.HoppingKind((0, 0), a, b)] = -1
syst[kwant.builder.HoppingKind((0, 0), a, c)] = -1
syst[kwant.builder.HoppingKind((0, 0), c, b)] = -1
Here, we create hoppings between the sites of the same lattice coordinates but from different families. Here, we create hoppings between the sites of the same lattice coordinates but from different families.
.. image:: /code/figure/faq_direction3.* .. jupyter-execute::
plot_system(syst)
How to set the hoppings between adjacent sites? How to set the hoppings between adjacent sites?
...@@ -230,32 +310,53 @@ How to set the hoppings between adjacent sites? ...@@ -230,32 +310,53 @@ How to set the hoppings between adjacent sites?
that returns a list of ``HoppingKind`` instances that connect sites with their that returns a list of ``HoppingKind`` instances that connect sites with their
(n-nearest) neighors: (n-nearest) neighors:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_adjacent1
:end-before: #HIDDEN_END_adjacent1 # Create hoppings with lat.neighbors()
syst = kwant.Builder()
lat = kwant.lattice.square()
syst[(lat(i, j) for i in range(3) for j in range(3))] = 4
.. image:: /code/figure/faq_adjacent1.* syst[lat.neighbors()] = -1 # Equivalent to lat.neighbors(1)
.. image:: /code/figure/faq_adjacent2.* kwant.plot(syst);
As we can see in the figure above, ``lat.neighbors()`` (on the left) returns the hoppings between the first nearest neighbors and ``lat.neighbors(2)`` (on the right) returns the hoppings between the second nearest neighbors. .. jupyter-execute::
del syst[lat.neighbors()] # Delete all nearest-neighbor hoppings
syst[lat.neighbors(2)] = -1
kwant.plot(syst);
As we can see in the figures above, ``lat.neighbors()`` returns the hoppings between the first nearest neighbors and ``lat.neighbors(2)`` returns the hoppings between the second nearest neighbors.
When using a ``Polyatomic`` lattice ``neighbors()`` knows about the different When using a ``Polyatomic`` lattice ``neighbors()`` knows about the different
sublattices: sublattices:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_adjacent2
:end-before: #HIDDEN_END_adjacent2 # Create the system
lat = kwant.lattice.kagome()
syst = kwant.Builder()
a, b, c = lat.sublattices # The kagome lattice has 3 sublattices
for i in range(4):
for j in range (4):
syst[a(i, j)] = 4 # red
syst[b(i, j)] = 4 # green
syst[c(i, j)] = 4 # blue
.. image:: /code/figure/faq_adjacent3.* syst[lat.neighbors()] = -1
plot_system(syst)
However, if we use the ``neighbors()`` method of a single sublattice, we will However, if we use the ``neighbors()`` method of a single sublattice, we will
only get the neighbors *on that sublattice*: only get the neighbors *on that sublattice*:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_adjacent3
:end-before: #HIDDEN_END_adjacent3
.. image:: /code/figure/faq_adjacent4.* del syst[lat.neighbors()] # Delete the hoppings previously created
syst[a.neighbors()] = -1
plot_system(syst)
Note in the above that there are *only* hoppings between the red sites. This Note in the above that there are *only* hoppings between the red sites. This
is an artifact of the visualisation: the blue and green sites just happen to lie is an artifact of the visualisation: the blue and green sites just happen to lie
...@@ -268,12 +369,35 @@ To make a hole in the system, use ``del syst[site]``, just like with any other ...@@ -268,12 +369,35 @@ To make a hole in the system, use ``del syst[site]``, just like with any other
mapping. In the following example we remove all sites inside some "hole" mapping. In the following example we remove all sites inside some "hole"
region: region:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_hole
:end-before: #HIDDEN_END_hole # Define the lattice and the (empty) system
a = 2
lat = kwant.lattice.cubic(a)
syst = kwant.Builder()
L = 10
W = 10
H = 2
# Add sites to the system in a cuboid
syst[(lat(i, j, k) for i in range(L) for j in range(W) for k in range(H))] = 4
kwant.plot(syst);
.. jupyter-execute::
# Delete sites to create a hole
def in_hole(site):
x, y, z = site.pos / a - (L/2, W/2, H/2) # position relative to centre
return abs(x) < L / 4 and abs(y) < W / 4
for site in filter(in_hole, list(syst.sites())):
del syst[site]
kwant.plot(syst);
.. image:: /code/figure/faq_hole1.*
.. image:: /code/figure/faq_hole2.*
``del syst[site]`` also works after hoppings have been added to the system. ``del syst[site]`` also works after hoppings have been added to the system.
If a site is deleted, then all the hoppings to/from that site are also deleted. If a site is deleted, then all the hoppings to/from that site are also deleted.
...@@ -287,9 +411,18 @@ what is a builder?`_ if you do not know the difference). ...@@ -287,9 +411,18 @@ what is a builder?`_ if you do not know the difference).
We can access the sites of a ``Builder`` by using its `~kwant.builder.Builder.sites` method: We can access the sites of a ``Builder`` by using its `~kwant.builder.Builder.sites` method:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_sites1 :hide-code:
:end-before: #HIDDEN_END_sites1
builder = kwant.Builder()
lat = kwant.lattice.square()
builder[(lat(i, j) for i in range(3) for j in range(3))] = 4
.. jupyter-execute::
# Before finalizing the system
sites = list(builder.sites()) # sites() doe *not* return a list
The ``sites()`` method returns an *iterator* over the system sites, and in the The ``sites()`` method returns an *iterator* over the system sites, and in the
above example we create a list from the contents of this iterator, which above example we create a list from the contents of this iterator, which
...@@ -300,9 +433,11 @@ well be returned in a different order. ...@@ -300,9 +433,11 @@ well be returned in a different order.
After finalization, when we are dealing with a ``System``, the sites themselves After finalization, when we are dealing with a ``System``, the sites themselves
are stored in a list, which can be accessed via the ``sites`` attribute: are stored in a list, which can be accessed via the ``sites`` attribute:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_sites2
:end-before: #HIDDEN_END_sites2 # After finalizing the system
syst = builder.finalized()
sites = syst.sites # syst.sites is an actual list
The order of sites in a ``System`` is fixed, and also defines the ordering of The order of sites in a ``System`` is fixed, and also defines the ordering of
the system Hamiltonian, system wavefunctions etc. (see `How does Kwant order components of an individual wavefunction?`_ for details). the system Hamiltonian, system wavefunctions etc. (see `How does Kwant order components of an individual wavefunction?`_ for details).
...@@ -310,9 +445,9 @@ the system Hamiltonian, system wavefunctions etc. (see `How does Kwant order com ...@@ -310,9 +445,9 @@ the system Hamiltonian, system wavefunctions etc. (see `How does Kwant order com
``System`` also contains the inverse mapping, ``id_by_site`` which gives us ``System`` also contains the inverse mapping, ``id_by_site`` which gives us
the index of a given site within the system: the index of a given site within the system:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_sites3
:end-before: #HIDDEN_END_sites3 i = syst.id_by_site[lat(0, 2)] # we want the id of the site lat(0, 2)
How to use different lattices for the scattering region and a lead? How to use different lattices for the scattering region and a lead?
...@@ -322,19 +457,35 @@ which we want to connect to leads that contain sites from a square lattice. ...@@ -322,19 +457,35 @@ which we want to connect to leads that contain sites from a square lattice.
First we construct the central system: First we construct the central system:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_different_lattice1
:end-before: #HIDDEN_END_different_lattice1
.. image:: /code/figure/faq_different_lattice1.* # Define the scattering Region
L = 5
W = 5
lat = kwant.lattice.honeycomb()
subA, subB = lat.sublattices
syst = kwant.Builder()
syst[(subA(i, j) for i in range(L) for j in range(W))] = 4
syst[(subB(i, j) for i in range(L) for j in range(W))] = 4
syst[lat.neighbors()] = -1
kwant.plot(syst);
and the lead: and the lead:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_different_lattice2
:end-before: #HIDDEN_END_different_lattice2 # Create a lead
lat_lead = kwant.lattice.square()
sym_lead1 = kwant.TranslationalSymmetry((0, 1))
lead1 = kwant.Builder(sym_lead1)
lead1[(lat_lead(i, 0) for i in range(2, 7))] = 4
lead1[lat_lead.neighbors()] = -1
.. image:: /code/figure/faq_different_lattice2.* kwant.plot(syst);
We cannot simply use `~kwant.builder.Builder.attach_lead` to attach this lead to the We cannot simply use `~kwant.builder.Builder.attach_lead` to attach this lead to the
system with the honeycomb lattice because Kwant does not know how sites from system with the honeycomb lattice because Kwant does not know how sites from
...@@ -343,19 +494,22 @@ these two lattices should be connected. ...@@ -343,19 +494,22 @@ these two lattices should be connected.
We must first add a layer of sites from the square lattice to the system and manually We must first add a layer of sites from the square lattice to the system and manually
add the hoppings from these sites to the sites from the honeycomb lattice: add the hoppings from these sites to the sites from the honeycomb lattice:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_different_lattice3
:end-before: #HIDDEN_END_different_lattice3
.. image:: /code/figure/faq_different_lattice3.* syst[(lat_lead(i, 5) for i in range(2, 7))] = 4
syst[lat_lead.neighbors()] = -1
# Manually attach sites from graphene to square lattice
syst[((lat_lead(i+2, 5), subB(i, 4)) for i in range(5))] = -1
kwant.plot(syst);
``attach_lead()`` will now be able to attach the lead: ``attach_lead()`` will now be able to attach the lead:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_different_lattice4
:end-before: #HIDDEN_END_different_lattice4
.. image:: /code/figure/faq_different_lattice4.* syst.attach_lead(lead1)
kwant.plot(syst);
How to cut a finite system out of a system with translational symmetries? How to cut a finite system out of a system with translational symmetries?
...@@ -369,30 +523,50 @@ will be repeated in the desired shape to create the final system. ...@@ -369,30 +523,50 @@ will be repeated in the desired shape to create the final system.
For example, say we want to create a simple model on a cubic lattice: For example, say we want to create a simple model on a cubic lattice:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_fill1
:end-before: #HIDDEN_END_fill1 # Create 3d model.
cubic = kwant.lattice.cubic()
sym_3d = kwant.TranslationalSymmetry([1, 0, 0], [0, 1, 0], [0, 0, 1])
model = kwant.Builder(sym_3d)
model[cubic(0, 0, 0)] = 4
model[cubic.neighbors()] = -1
We have now created our "template" ``Builder`` which has 3 translational We have now created our "template" ``Builder`` which has 3 translational
symmetries. Next we will fill a system with no translational symmetries with symmetries. Next we will fill a system with no translational symmetries with
sites and hoppings from the template inside a cuboid: sites and hoppings from the template inside a cuboid:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_fill2
:end-before: #HIDDEN_END_fill2
.. image:: /code/figure/faq_fill2.* # Build scattering region (white).
def cuboid_shape(site):
x, y, z = abs(site.pos)
return x < 4 and y < 10 and z < 3
cuboid = kwant.Builder()
cuboid.fill(model, cuboid_shape, (0, 0, 0));
kwant.plot(cuboid);
We can then use the original template to create a lead, which has 1 translational We can then use the original template to create a lead, which has 1 translational
symmetry. We can then use this lead as a template to fill another section of symmetry. We can then use this lead as a template to fill another section of
the system with a cylinder of sites and hoppings: the system with a cylinder of sites and hoppings:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_fill3
:end-before: #HIDDEN_END_fill3 # Build electrode (black).
def electrode_shape(site):
x, y, z = site.pos - (0, 5, 2)
return y**2 + z**2 < 2.3**2
.. image:: /code/figure/faq_fill3.* electrode = kwant.Builder(kwant.TranslationalSymmetry([1, 0, 0]))
electrode.fill(model, electrode_shape, (0, 5, 2)) # lead
# Scattering region
cuboid.fill(electrode, lambda s: abs(s.pos[0]) < 7, (0, 5, 4))
cuboid.attach_lead(electrode)
kwant.plot(cuboid);
How does Kwant order the propagating modes of a lead? How does Kwant order the propagating modes of a lead?
===================================================== =====================================================
...@@ -402,16 +576,44 @@ achieved with the `~kwant.system.InfiniteSystem.modes` method, which returns a ...@@ -402,16 +576,44 @@ achieved with the `~kwant.system.InfiniteSystem.modes` method, which returns a
pair of objects, the first of which contains the propagating modes of the pair of objects, the first of which contains the propagating modes of the
system in a `~kwant.physics.PropagatingModes` object: system in a `~kwant.physics.PropagatingModes` object:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_pm
:end-before: #HIDDEN_END_pm lat = kwant.lattice.square()
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead[(lat(0, i) for i in range(3))] = 4
lead[lat.neighbors()] = -1
flead = lead.finalized()
E = 2.5
prop_modes, _ = flead.modes(energy=E)
``PropagatingModes`` contains the wavefunctions, velocities and momenta of the ``PropagatingModes`` contains the wavefunctions, velocities and momenta of the
modes at the requested energy (2.5 in this example). In order to understand modes at the requested energy (2.5 in this example). In order to understand
the order in which these quantities are returned it is often useful to look at the order in which these quantities are returned it is often useful to look at
the a section of the band structure for the system in question: the a section of the band structure for the system in question:
.. image:: /code/figure/faq_pm1.* .. jupyter-execute::
:hide-code:
def plot_and_label_modes(lead, E):
# Plot the different modes
pmodes, _ = lead.modes(energy=E)
kwant.plotter.bands(lead, show=False)
for i, k in enumerate(pmodes.momenta):
plt.plot(k, E, 'ko')
plt.annotate(str(i), xy=(k, E), xytext=(-5, 8),
textcoords='offset points',
bbox=dict(boxstyle='round,pad=0.1',fc='white', alpha=0.7))
plt.plot([-3, 3], [E, E], 'r--')
plt.ylim(E-1, E+1)
plt.xlim(-2, 2)
plt.xlabel("momentum")
plt.ylabel("energy")
plt.show()
plot_and_label_modes(flead, E)
On the above band structure we have labelled the 4 modes in the order On the above band structure we have labelled the 4 modes in the order
that they appear in the output of ``modes()`` at energy 2.5. Note that that they appear in the output of ``modes()`` at energy 2.5. Note that
...@@ -425,7 +627,20 @@ the modes are sorted in the following way: ...@@ -425,7 +627,20 @@ the modes are sorted in the following way:
For more complicated systems and band structures this can lead to some For more complicated systems and band structures this can lead to some
unintuitive orderings: unintuitive orderings:
.. image:: /code/figure/faq_pm2.* .. jupyter-execute::
:hide-code:
s0 = np.eye(2)
sz = np.array([[1, 0], [0, -1]])
lead2 = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead2[(lat(0, i) for i in range(2))] = np.diag([1.8, -1])
lead2[lat.neighbors()] = -1 * sz
flead2 = lead2.finalized()
plot_and_label_modes(flead2, 1)
How does Kwant order scattering states? How does Kwant order scattering states?
...@@ -452,10 +667,38 @@ When all the site families present in a system have only 1 degree of freedom ...@@ -452,10 +667,38 @@ When all the site families present in a system have only 1 degree of freedom
per site (i.e. the all the onsites are scalars) then the index into a per site (i.e. the all the onsites are scalars) then the index into a
wavefunction defined over the system is exactly the site index: wavefunction defined over the system is exactly the site index:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_ord1 :hide-code:
:end-before: #HIDDEN_END_ord1
.. literalinclude:: /code/figure/faq_ord1.txt def circle(R):
return lambda r: np.linalg.norm(r) < R
def make_system(lat):
norbs = lat.norbs
syst = kwant.Builder()
syst[lat.shape(circle(3), (0, 0))] = 4 * np.eye(norbs)
syst[lat.neighbors()] = -1 * np.eye(norbs)
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
lead[(lat(0, i) for i in range(-1, 2))] = 4 * np.eye(norbs)
lead[lat.neighbors()] = -1 * np.eye(norbs)
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst.finalized()
.. jupyter-execute::
lat = kwant.lattice.square(norbs=1)
syst = make_system(lat)
scattering_states = kwant.wave_function(syst, energy=1)
wf = scattering_states(0)[0] # scattering state from lead 0 incoming in mode 0
idx = syst.id_by_site[lat(0, 0)] # look up index of site
print('wavefunction on lat(0, 0): ', wf[idx])
We see that the wavefunction on a single site is a single complex number, as We see that the wavefunction on a single site is a single complex number, as
expected. expected.
...@@ -467,10 +710,20 @@ to one another. In the case where all site families in the system have the ...@@ -467,10 +710,20 @@ to one another. In the case where all site families in the system have the
wavefunction into a matrix, where the row number indexes the site, and the wavefunction into a matrix, where the row number indexes the site, and the
column number indexes the degree of freedom on that site: column number indexes the degree of freedom on that site:
.. literalinclude:: /code/include/faq.py .. jupyter-execute::
:start-after: #HIDDEN_BEGIN_ord2
:end-before: #HIDDEN_END_ord2 lat = kwant.lattice.square(norbs=2)
.. literalinclude:: /code/figure/faq_ord2.txt syst = make_system(lat)
scattering_states = kwant.wave_function(syst, energy=1)
wf = scattering_states(0)[0] # scattering state from lead 0 incoming in mode 0
idx = syst.id_by_site[lat(0, 0)] # look up index of site
# Group consecutive degrees of freedom from 'wf' together; these correspond
# to degrees of freedom on the same site.
wf = wf.reshape(-1, 2)
print('wavefunction on lat(0, 0): ', wf[idx])
We see that the wavefunction on a single site is a *vector* of 2 complex numbers, We see that the wavefunction on a single site is a *vector* of 2 complex numbers,
as we expect. as we expect.
......
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