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doc/source/tutorial/kpm.rst
View file @ bc9b2edf
......@@ -15,10 +15,38 @@ expansion of the density of states. It can also be used to calculate the
spectral density of arbitrary operators. Kwant has an implementation of the
KPM method `kwant.kpm`, that is based on the algorithms presented in Ref. [1]_.
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:download:`kernel_polynomial_method.py </code/download/kernel_polynomial_method.py>`
:jupyter-download:script:`kernel_polynomial_method`
.. jupyter-kernel::
:id: kernel_polynomial_method
.. jupyter-execute::
:hide-code:
# Tutorial 2.8. Calculating spectral density with the Kernel Polynomial Method
# ============================================================================
#
# Physics background
# ------------------
# - Chebyshev polynomials, random trace approximation, spectral densities.
#
# Kwant features highlighted
# --------------------------
# - kpm module,kwant operators.
import scipy
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
Introduction
************
......@@ -103,35 +131,104 @@ to obtain the (global) density of states of a graphene disk.
We start by importing kwant and defining our system.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_sys1
:end-before: #HIDDEN_END_sys1
.. jupyter-execute::
# necessary imports
import kwant
import numpy as np
# define the system
def make_syst(r=30, t=-1, a=1):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1)
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.shape(circle, (0, 0))] = 0.
syst[lat.neighbors()] = t
syst.eradicate_dangling()
return syst
.. jupyter-execute::
:hide-code:
## common plotting routines ##
# Plot several density of states curves on the same axes.
def plot_dos(labels_to_data):
pyplot.figure()
for label, (x, y) in labels_to_data:
pyplot.plot(x, y.real, label=label, linewidth=2)
pyplot.legend(loc=2, framealpha=0.5)
pyplot.xlabel("energy [t]")
pyplot.ylabel("DoS [a.u.]")
pyplot.show()
# Plot fill density of states plus curves on the same axes.
def plot_dos_and_curves(dos, labels_to_data):
pyplot.figure()
pyplot.fill_between(dos[0], dos[1], label="DoS [a.u.]",
alpha=0.5, color='gray')
for label, (x, y) in labels_to_data:
pyplot.plot(x, y, label=label, linewidth=2)
pyplot.legend(loc=2, framealpha=0.5)
pyplot.xlabel("energy [t]")
pyplot.ylabel("$σ [e^2/h]$")
pyplot.show()
def site_size_conversion(densities):
return 3 * np.abs(densities) / max(densities)
# Plot several local density of states maps in different subplots
def plot_ldos(syst, densities):
fig, axes = pyplot.subplots(1, len(densities), figsize=(7*len(densities), 7))
for ax, (title, rho) in zip(axes, densities):
kwant.plotter.density(syst, rho.real, ax=ax)
ax.set_title(title)
ax.set(adjustable='box', aspect='equal')
pyplot.show()
After making a system we can then create a `~kwant.kpm.SpectralDensity`
object that represents the density of states for this system.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_kpm1
:end-before: #HIDDEN_END_kpm1
.. jupyter-execute::
fsyst = make_syst().finalized()
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
The `~kwant.kpm.SpectralDensity` can then be called like a function to obtain a
sequence of energies in the spectrum of the Hamiltonian, and the corresponding
density of states at these energies.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_kpm2
:end-before: #HIDDEN_END_kpm2
.. jupyter-execute::
energies, densities = spectrum()
When called with no arguments, an optimal set of energies is chosen (these are
not evenly distributed over the spectrum, see Ref. [1]_ for details), however
it is also possible to provide an explicit sequence of energies at which to
evaluate the density of states.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_kpm3
:end-before: #HIDDEN_END_kpm3
.. jupyter-execute::
energy_subset = np.linspace(0, 2)
density_subset = spectrum(energy_subset)
.. image:: /code/figure/kpm_dos.*
.. jupyter-execute::
:hide-code:
plot_dos([
('densities', (energies, densities)),
('density subset', (energy_subset, density_subset)),
])
In addition to being called like functions, `~kwant.kpm.SpectralDensity`
objects also have a method `~kwant.kpm.SpectralDensity.integrate` which can be
......@@ -139,22 +236,21 @@ used to integrate the density of states against some distribution function over
the whole spectrum. If no distribution function is specified, then the uniform
distribution is used:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_int1
:end-before: #HIDDEN_END_int1
.. jupyter-execute::
.. literalinclude:: /code/figure/kpm_normalization.txt
print('identity resolution:', spectrum.integrate())
We see that the integral of the density of states is normalized to the total
number of available states in the system. If we wish to calculate, say, the
number of states populated in equilibrium, then we should integrate with
respect to a Fermi-Dirac distribution:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_int2
:end-before: #HIDDEN_END_int2
.. jupyter-execute::
# Fermi energy 0.1 and temperature 0.2
fermi = lambda E: 1 / (np.exp((E - 0.1) / 0.2) + 1)
.. literalinclude:: /code/figure/kpm_total_states.txt
print('number of filled states:', spectrum.integrate(fermi))
.. specialnote:: Stability and performance: spectral bounds
......@@ -194,23 +290,52 @@ In the following example, we compute the local density of states at the center
of the graphene disk, and we add a staggering potential between the two
sublattices.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_sys3
:end-before: #HIDDEN_END_sys3
.. jupyter-execute::
def make_syst_staggered(r=30, t=-1, a=1, m=0.1):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1)
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.a.shape(circle, (0, 0))] = m
syst[lat.b.shape(circle, (0, 0))] = -m
syst[lat.neighbors()] = t
syst.eradicate_dangling()
return syst
Next, we choose one site of each sublattice "A" and "B",
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op5
:end-before: #HIDDEN_END_op5
.. jupyter-execute::
fsyst_staggered = make_syst_staggered().finalized()
# find 'A' and 'B' sites in the unit cell at the center of the disk
center_tag = np.array([0, 0])
where = lambda s: s.tag == center_tag
# make local vectors
vector_factory = kwant.kpm.LocalVectors(fsyst_staggered, where)
and plot their respective local density of states.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op6
:end-before: #HIDDEN_END_op6
.. jupyter-execute::
.. image:: /code/figure/kpm_ldos_sites.*
# 'num_vectors' can be unspecified when using 'LocalVectors'
local_dos = kwant.kpm.SpectralDensity(fsyst_staggered, num_vectors=None,
vector_factory=vector_factory,
mean=False,
rng=0)
energies, densities = local_dos()
.. jupyter-execute::
:hide-code:
plot_dos([
('A sublattice', (energies, densities[:, 0])),
('B sublattice', (energies, densities[:, 1])),
])
Note that there is no noise comming from the random vectors.
......@@ -225,9 +350,15 @@ exactly is changed.
The simplest way to obtain a more accurate solution is to use the
``add_moments`` method:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_acc1
:end-before: #HIDDEN_END_acc1
.. jupyter-execute::
:hide-code:
spectrum = kwant.kpm.SpectralDensity(fsyst, rng=0)
original_dos = spectrum()
.. jupyter-execute::
spectrum.add_moments(energy_resolution=0.03)
This will update the number of calculated moments and also the default
number of sampling points such that the maximum distance between successive
......@@ -237,12 +368,19 @@ Alternatively, you can directly increase the number of moments
with ``add_moments``, or the number of random vectors with ``add_vectors``.
The added vectors will be generated with the ``vector_factory``.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_acc2
:end-before: #HIDDEN_END_acc2
.. jupyter-execute::
spectrum.add_moments(100)
spectrum.add_vectors(5)
.. image:: /code/figure/kpm_dos_r.*
.. jupyter-execute::
:hide-code:
increased_moments_dos = spectrum()
plot_dos([
('density', original_dos),
('higher number of moments', increased_moments_dos),
])
.. _operator_spectral_density:
......@@ -264,17 +402,19 @@ the spectral density of the given operator that is calculated.
If an explicit matrix is provided then it must have the same
shape as the system Hamiltonian.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op1
:end-before: #HIDDEN_END_op1
.. jupyter-execute::
# identity matrix
matrix_op = scipy.sparse.eye(len(fsyst.sites))
matrix_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=matrix_op, rng=0)
Or, to do the same calculation using `kwant.operator.Density`:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op2
:end-before: #HIDDEN_END_op2
.. jupyter-execute::
# 'sum=True' means we sum over all the sites
kwant_op = kwant.operator.Density(fsyst, sum=True)
operator_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=kwant_op, rng=0)
Spectral density with random vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
......@@ -283,9 +423,11 @@ Using operators from `kwant.operator` allows us to calculate quantities
such as the *local* density of states by telling the operator not to
sum over all the sites of the system:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op3
:end-before: #HIDDEN_END_op3
.. jupyter-execute::
# 'sum=False' is the default, but we include it explicitly here for clarity.
kwant_op = kwant.operator.Density(fsyst, sum=False)
local_dos = kwant.kpm.SpectralDensity(fsyst, operator=kwant_op, rng=0)
`~kwant.kpm.SpectralDensity` will properly handle this vector output,
and will average the local density obtained with random vectors.
......@@ -294,12 +436,18 @@ The accuracy of this approximation depends on the number of random vectors used.
This allows us to plot an approximate local density of states at different
points in the spectrum:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_op4
:end-before: #HIDDEN_END_op4
.. jupyter-execute::
zero_energy_ldos = local_dos(energy=0)
finite_energy_ldos = local_dos(energy=1)
.. image:: /code/figure/kpm_ldos.*
.. jupyter-execute::
:hide-code:
plot_ldos(fsyst, [
('energy = 0', zero_energy_ldos),
('energy = 1', finite_energy_ldos)
])
Calculating Kubo conductivity
*****************************
......@@ -325,17 +473,55 @@ with nearest neigbours hoppings. To turn it into a topological
insulator we add imaginary second nearest neigbours hoppings, where
the sign changes for each sublattice.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_sys2
:end-before: #HIDDEN_END_sys2
.. jupyter-execute::
def make_syst_topo(r=30, a=1, t=1, t2=0.5):
syst = kwant.Builder()
lat = kwant.lattice.honeycomb(a, norbs=1, name=['a', 'b'])
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst[lat.shape(circle, (0, 0))] = 0.
syst[lat.neighbors()] = t
# add second neighbours hoppings
syst[lat.a.neighbors()] = 1j * t2
syst[lat.b.neighbors()] = -1j * t2
syst.eradicate_dangling()
return lat, syst.finalized()
To calculate the bulk conductivity, we will select sites in the unit cell
in the middle of the sample, and create a vector factory that outputs local
vectors
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_cond
:end-before: #HIDDEN_END_cond
.. jupyter-execute::
# construct the Haldane model
lat, fsyst_topo = make_syst_topo()
# find 'A' and 'B' sites in the unit cell at the center of the disk
where = lambda s: np.linalg.norm(s.pos) < 1
# component 'xx'
s_factory = kwant.kpm.LocalVectors(fsyst_topo, where)
cond_xx = kwant.kpm.conductivity(fsyst_topo, alpha='x', beta='x', mean=True,
num_vectors=None, vector_factory=s_factory,
rng=0)
# component 'xy'
s_factory = kwant.kpm.LocalVectors(fsyst_topo, where)
cond_xy = kwant.kpm.conductivity(fsyst_topo, alpha='x', beta='y', mean=True,
num_vectors=None, vector_factory=s_factory,
rng=0)
energies = cond_xx.energies
cond_array_xx = np.array([cond_xx(e, temperature=0.01) for e in energies])
cond_array_xy = np.array([cond_xy(e, temperature=0.01) for e in energies])
# area of the unit cell per site
area_per_site = np.abs(np.cross(*lat.prim_vecs)) / len(lat.sublattices)
cond_array_xx /= area_per_site
cond_array_xy /= area_per_site
Note that the Kubo conductivity must be normalized with the area covered
by the vectors. In this case, each local vector represents a site, and
......@@ -345,8 +531,23 @@ value of the conductivity over large parts of the system. In this
case, the area that normalizes the result, is the area covered by
the random vectors.
.. image:: /code/figure/kpm_cond.*
.. jupyter-execute::
:hide-code:
s_factory = kwant.kpm.LocalVectors(fsyst_topo, where)
spectrum = kwant.kpm.SpectralDensity(fsyst_topo, num_vectors=None,
vector_factory=s_factory,
rng=0)
plot_dos_and_curves(
(spectrum.energies, spectrum.densities * 8),
[
(r'Longitudinal conductivity $\sigma_{xx} / 4$',
(energies, cond_array_xx.real / 4)),
(r'Hall conductivity $\sigma_{xy}$',
(energies, cond_array_xy.real))],
)
.. _advanced_topics:
......@@ -370,9 +571,17 @@ To change how the random vectors are generated, you need only specify a
function that takes the dimension of the Hilbert space as a single parameter,
and which returns a vector in that Hilbert space:
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_fact1
:end-before: #HIDDEN_END_fact1
.. jupyter-execute::
# construct a generator of vectors with n random elements -1 or +1.
n = fsyst.hamiltonian_submatrix(sparse=True).shape[0]
def binary_vectors():
while True:
yield np.rint(np.random.random_sample(n)) * 2 - 1
custom_factory = kwant.kpm.SpectralDensity(fsyst,
vector_factory=binary_vectors(),
rng=0)
Aditionally, a `~kwant.kpm.LocalVectors` generator is also available, that
returns local vectors that correspond to the sites passed. Note that
......@@ -407,9 +616,16 @@ first argument, and linear in its second argument. Below, we compare two
methods for computing the local density of states, one using
`kwant.operator.Density`, and the other using a custom function.
.. literalinclude:: /code/include/kernel_polynomial_method.py
:start-after: #HIDDEN_BEGIN_blm
:end-before: #HIDDEN_END_blm
.. jupyter-execute::
rho = kwant.operator.Density(fsyst, sum=True)
# sesquilinear map that does the same thing as `rho`
def rho_alt(bra, ket):
return np.vdot(bra, ket)
rho_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=rho, rng=0)
rho_alt_spectrum = kwant.kpm.SpectralDensity(fsyst, operator=rho_alt, rng=0)
__ operator_spectral_density_
......
doc/source/tutorial/magnetic_field.rst 0 → 100644
View file @ bc9b2edf
Adding magnetic field
---------------------
Computing Landau levels in a harmonic oscillator basis
......................................................
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:jupyter-download:script:`landau-levels`
.. jupyter-kernel::
:id: landau-levels
When electrons move in an external magnetic field, their motion perpendicular
to the field direction is quantized into discrete Landau levels. Kwant implements
an efficient scheme for computing the Landau levels of arbitrary continuum
Hamiltonians. The general scheme revolves around rewriting the Hamiltonian in terms
of a basis of harmonic oscillator states [#]_, and is commonly illustrated in textbooks
for quadratic Hamiltonians.
To demonstrate the general scheme, let us consider a magnetic field oriented along
the :math:`z` direction :math:`\vec{B} = (0, 0, B)`, such that electron motion
in the :math:`xy` plane is Landau quantized. The magnetic field enters the Hamiltonian
through the kinetic momentum
.. math:: \hbar \vec{k} = - i \hbar \nabla + e\vec{A}(x, y).
In the symmetric gauge :math:`\vec{A}(x, y) = (B/2)[-y, x, 0]`, we introduce ladder
operators with the substitution
.. math::
k_x = \frac{1}{\sqrt{2} l_B} (a + a^\dagger), \quad \quad
k_y = \frac{i}{\sqrt{2} l_B} (a - a^\dagger),
with the magnetic length :math:`l_B = \sqrt{\hbar/eB}`. The ladder operators obey the
commutation relation
.. math:: [a, a^\dagger] = 1,
and define a quantum harmonic oscillator. We can thus write any electron continuum
Hamiltonian in terms of :math:`a` and :math:`a^\dagger`. Such a Hamiltonian has a
simple matrix representation in the eigenbasis of the number operator :math:`a^\dagger a`.
The eigenstates satisfy :math:`a^\dagger a | n \rangle = n | n \rangle` with the integer
Landau level index :math:`n \geq 0`, and in coordinate representation are proportional to
.. math::
\psi_n (x, y) = \left( \frac{\partial}{ \partial w} - \frac{w^*}{4 l_B^2} \right)
w^n e^{-|w|^2/4l_B^2},
with :math:`w = x + i y`. The matrix elements of the ladder operators are
.. math::
\langle n | a | m \rangle = \sqrt{m}~\delta_{n, m-1}, \quad \quad
\langle n | a^\dagger | m \rangle = \sqrt{m + 1}~\delta_{n, m+1}.
Truncating the basis to the first :math:`N` Landau levels allows us to approximate
the Hamiltonian as a discrete, finite matrix.
We can now formulate the algorithm that Kwant uses to compute Landau levels.
1. We take a generic continuum Hamiltonian, written in terms of the kinetic
momentum :math:`\vec{k}`. The Hamiltonian must be translationally
invariant along the directions perpendicular to the field direction.
2. We substitute the momenta perpendicular to the magnetic field with the ladder
operators :math:`a` and :math:`a^\dagger`.
3. We construct a `kwant.builder.Builder` using a special lattice which includes
the Landau level index :math:`n` as a degree of freedom on each site. The directions
normal to the field direction are not included in the builder, because they are
encoded in the Landau level index.
This procedure is automated with `kwant.continuum.discretize_landau`.
As an example, let us take the Bernevig-Hughes-Zhang model that we first considered in the
discretizer tutorial ":ref:`discretize-bhz-model`":
.. math::
C + M σ_0 \otimes σ_z + F(k_x^2 + k_y^2) σ_0 \otimes σ_z + D(k_x^2 + k_y^2) σ_0 \otimes σ_0
+ A k_x σ_z \otimes σ_x + A k_y σ_0 \otimes σ_y.
We can discretize this Hamiltonian in a basis of Landau levels as follows
.. jupyter-execute::
import numpy as np
import scipy.linalg
from matplotlib import pyplot
import kwant
import kwant.continuum
.. jupyter-execute:: boilerplate.py
:hide-code:
.. jupyter-execute::
hamiltonian = """
+ C * identity(4) + M * kron(sigma_0, sigma_z)
- F * (k_x**2 + k_y**2) * kron(sigma_0, sigma_z)
- D * (k_x**2 + k_y**2) * kron(sigma_0, sigma_0)
+ A * k_x * kron(sigma_z, sigma_x)
- A * k_y * kron(sigma_0, sigma_y)
"""
syst = kwant.continuum.discretize_landau(hamiltonian, N=10)
syst = syst.finalized()
We can then plot the spectrum of the system as a function of magnetic field, and
observe a crossing of Landau levels at finite magnetic field near zero energy,
characteristic of a quantum spin Hall insulator with band inversion.
.. jupyter-execute::
params = dict(A=3.645, F =-68.6, D=-51.2, M=-0.01, C=0)
b_values = np.linspace(0.0001, 0.0004, 200)
fig = kwant.plotter.spectrum(syst, ('B', b_values), params=params, show=False)
pyplot.ylim(-0.1, 0.2);
Comparing with tight-binding
============================
In the limit where fewer than one quantum of flux is threaded through a plaquette of
the discretization lattice we can compare the discretization in Landau levels with
a discretization in realspace.
.. jupyter-execute::
lat = kwant.lattice.square(norbs=1)
syst = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
def peierls(to_site, from_site, B):
y = from_site.tag[1]
return -1 * np.exp(-1j * B * y)
syst[(lat(0, j) for j in range(-19, 20))] = 4
syst[lat.neighbors()] = -1
syst[kwant.HoppingKind((1, 0), lat)] = peierls
syst = syst.finalized()
landau_syst = kwant.continuum.discretize_landau("k_x**2 + k_y**2", N=5)
landau_syst = landau_syst.finalized()
Here we plot the dispersion relation for the discretized ribbon and compare it
with the Landau levels shown as dashed lines.
.. jupyter-execute::
fig, ax = pyplot.subplots(1, 1)
ax.set_xlabel("momentum")
ax.set_ylabel("energy")
ax.set_ylim(0, 1)
params = dict(B=0.1)
kwant.plotter.bands(syst, ax=ax, params=params)
h = landau_syst.hamiltonian_submatrix(params=params)
for ev in scipy.linalg.eigvalsh(h):
ax.axhline(ev, linestyle='--')
The dispersion and the Landau levels diverge with increasing energy, because the real space
discretization of the ribbon gives a worse approximation to the dispersion at higher energies.
Discretizing 3D models
======================
Although the preceding examples have only included the plane perpendicular to the
magnetic field, the Landau level quantization also works if the direction
parallel to the field is included. In fact, although the system must be
translationally invariant in the plane perpendicular to the field, the system may
be arbitrary along the parallel direction. For example, it is therefore possible to
model a heterostructure and/or set up a scattering problem along the field direction.
Let's say that we wish to to model a heterostructure with a varying potential
:math:`V` along the direction of a magnetic field, :math:`z`, that includes
Zeeman splitting and Rashba spin-orbit coupling:
.. math::
\frac{\hbar^2}{2m}\sigma_0(k_x^2 + k_y^2 + k_z^2)
+ V(z)\sigma_0
+ \frac{\mu_B B}{2}\sigma_z
+ \hbar\alpha(\sigma_x k_y - \sigma_y k_x).
We can discretize this Hamiltonian in a basis of Landau levels as before:
.. jupyter-execute::
continuum_hamiltonian = """
(k_x**2 + k_y**2 + k_z**2) * sigma_0
+ V(z) * sigma_0
+ mu * B * sigma_z / 2
+ alpha * (sigma_x * k_y - sigma_y * k_x)
"""
template = kwant.continuum.discretize_landau(continuum_hamiltonian, N=10)
This creates a system with a single translational symmetry, along
the :math:`z` direction, which we can use as a template
to construct our heterostructure:
.. jupyter-execute::
def hetero_structure(site):
z, = site.pos
return 0 <= z < 10
def hetero_potential(z):
if z < 2:
return 0
elif z < 7:
return 0.5
else:
return 0.7
syst = kwant.Builder()
syst.fill(template, hetero_structure, (0,))
syst = syst.finalized()
params = dict(
B=0.5,
mu=0.2,
alpha=0.4,
V=hetero_potential,
)
syst.hamiltonian_submatrix(params=params);
.. rubric:: Footnotes
.. [#] `Wikipedia <https://en.wikipedia.org/wiki/Landau_quantization>`_ has
a nice introduction to Landau quantization.
doc/source/tutorial/operators.rst
View file @ bc9b2edf
......@@ -12,9 +12,53 @@ simple densities by studying spin transport in a system with a magnetic
texture.
.. seealso::
The complete source code of this example can be found in
:download:`magnetic_texture.py </code/download/magnetic_texture.py>`
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:jupyter-download:script:`magnetic_texture`
.. jupyter-kernel::
:id: magnetic_texture
.. jupyter-execute::
:hide-code:
# Tutorial 2.7. Spin textures
# ===========================
#
# Physics background
# ------------------
# - Spin textures
# - Skyrmions
#
# Kwant features highlighted
# --------------------------
# - operators
# - plotting vector fields
from math import sin, cos, tanh, pi
import itertools
import numpy as np
import tinyarray as ta
import matplotlib.pyplot as plt
import kwant
sigma_0 = ta.array([[1, 0], [0, 1]])
sigma_x = ta.array([[0, 1], [1, 0]])
sigma_y = ta.array([[0, -1j], [1j, 0]])
sigma_z = ta.array([[1, 0], [0, -1]])
# vector of Pauli matrices σ_αiβ where greek
# letters denote spinor indices
sigma = np.rollaxis(np.array([sigma_x, sigma_y, sigma_z]), 1)
.. jupyter-execute:: boilerplate.py
:hide-code:
Introduction
------------
......@@ -47,28 +91,119 @@ of site :math:`i`, and :math:`r_i = \sqrt{x_i^2 + y_i^2}`.
To define this model in Kwant we start as usual by defining functions that
depend on the model parameters:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_model
:end-before: #HIDDEN_END_model
.. jupyter-execute::
def field_direction(pos, r0, delta):
x, y = pos
r = np.linalg.norm(pos)
r_tilde = (r - r0) / delta
theta = (tanh(r_tilde) - 1) * (pi / 2)
if r == 0:
m_i = [0, 0, -1]
else:
m_i = [
(x / r) * sin(theta),
(y / r) * sin(theta),
cos(theta),
]
return np.array(m_i)
def scattering_onsite(site, r0, delta, J):
m_i = field_direction(site.pos, r0, delta)
return J * np.dot(m_i, sigma)
def lead_onsite(site, J):
return J * sigma_z
and define our system as a square shape on a square lattice with two orbitals
per site, with leads attached on the left and right:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_syst
:end-before: #HIDDEN_END_syst
.. jupyter-execute::
lat = kwant.lattice.square(norbs=2)
def make_system(L=80):
syst = kwant.Builder()
def square(pos):
return all(-L/2 < p < L/2 for p in pos)
syst[lat.shape(square, (0, 0))] = scattering_onsite
syst[lat.neighbors()] = -sigma_0
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)),
conservation_law=-sigma_z)
lead[lat.shape(square, (0, 0))] = lead_onsite
lead[lat.neighbors()] = -sigma_0
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst
Below is a plot of a projection of :math:`\mathbf{m}_i` onto the x-y plane
inside the scattering region. The z component is shown by the color scale:
.. image:: /code/figure/mag_field_direction.*
.. jupyter-execute::
:hide-code:
def plot_vector_field(syst, params):
xmin, ymin = min(s.tag for s in syst.sites)
xmax, ymax = max(s.tag for s in syst.sites)
x, y = np.meshgrid(np.arange(xmin, xmax+1), np.arange(ymin, ymax+1))
m_i = [field_direction(p, **params) for p in zip(x.flat, y.flat)]
m_i = np.reshape(m_i, x.shape + (3,))
m_i = np.rollaxis(m_i, 2, 0)
fig, ax = plt.subplots(1, 1)
im = ax.quiver(x, y, *m_i, pivot='mid', scale=75)
fig.colorbar(im)
plt.show()
def plot_densities(syst, densities):
fig, axes = plt.subplots(1, len(densities), figsize=(13, 10))
for ax, (title, rho) in zip(axes, densities):
kwant.plotter.density(syst, rho, ax=ax)
ax.set_title(title)
plt.show()
def plot_currents(syst, currents):
fig, axes = plt.subplots(1, len(currents), figsize=(13, 10))
if not hasattr(axes, '__len__'):
axes = (axes,)
for ax, (title, current) in zip(axes, currents):
kwant.plotter.current(syst, current, ax=ax, colorbar=False,
fig_size=(13, 10))
ax.set_title(title)
plt.show()
.. jupyter-execute::
:hide-code:
syst = make_system().finalized()
.. jupyter-execute::
:hide-code:
plot_vector_field(syst, dict(r0=20, delta=10))
We will now be interested in analyzing the form of the scattering states
that originate from the left lead:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_wavefunction
:end-before: #HIDDEN_END_wavefunction
.. jupyter-execute::
params = dict(r0=20, delta=10, J=1)
wf = kwant.wave_function(syst, energy=-1, params=params)
psi = wf(0)[0]
Local densities
---------------
......@@ -82,34 +217,45 @@ When there are multiple degrees of freedom per site, however, one has to be
more careful. In the present case with two (spin) degrees of freedom per site
one could calculate the per-site density like:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_ldos
:end-before: #HIDDEN_END_ldos
.. jupyter-execute::
# even (odd) indices correspond to spin up (down)
up, down = psi[::2], psi[1::2]
density = np.abs(up)**2 + np.abs(down)**2
With more than one degree of freedom per site we have more freedom as to what
local quantities we can meaningfully compute. For example, we may wish to
calculate the local z-projected spin density. We could calculate
this in the following way:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_lsdz
:end-before: #HIDDEN_END_lsdz
.. jupyter-execute::
# spin down components have a minus sign
spin_z = np.abs(up)**2 - np.abs(down)**2
If we wanted instead to calculate the local y-projected spin density, we would
need to use an even more complicated expression:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_lsdy
:end-before: #HIDDEN_END_lsdy
.. jupyter-execute::
# spin down components have a minus sign
spin_y = 1j * (down.conjugate() * up - up.conjugate() * down)
The `kwant.operator` module aims to alleviate somewhat this tedious
book-keeping by providing a simple interface for defining operators that act on
wavefunctions. To calculate the above quantities we would use the
`~kwant.operator.Density` operator like so:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_lden
:end-before: #HIDDEN_END_lden
.. jupyter-execute::
rho = kwant.operator.Density(syst)
rho_sz = kwant.operator.Density(syst, sigma_z)
rho_sy = kwant.operator.Density(syst, sigma_y)
# calculate the expectation values of the operators with 'psi'
density = rho(psi)
spin_z = rho_sz(psi)
spin_y = rho_sy(psi)
`~kwant.operator.Density` takes a `~kwant.system.System` as its first parameter
as well as (optionally) a square matrix that defines the quantity that you wish
......@@ -126,8 +272,14 @@ Below we can see colorplots of the above-calculated quantities. The array that
is returned by evaluating a `~kwant.operator.Density` can be used directly with
`kwant.plotter.density`:
.. image:: /code/figure/spin_densities.*
.. jupyter-execute::
:hide-code:
plot_densities(syst, [
('$σ_0$', density),
('$σ_z$', spin_z),
('$σ_y$', spin_y),
])
.. specialnote:: Technical Details
......@@ -180,14 +332,27 @@ where :math:`\mathbf{H}_{ab}` is the hopping matrix from site :math:`b` to site
:math:`a`. For example, to calculate the local current and
spin current:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_current
:end-before: #HIDDEN_END_current
.. jupyter-execute::
J_0 = kwant.operator.Current(syst)
J_z = kwant.operator.Current(syst, sigma_z)
J_y = kwant.operator.Current(syst, sigma_y)
# calculate the expectation values of the operators with 'psi'
current = J_0(psi)
spin_z_current = J_z(psi)
spin_y_current = J_y(psi)
Evaluating a `~kwant.operator.Current` operator on a wavefunction returns a
1D array of values that can be directly used with `kwant.plotter.current`:
.. image:: /code/figure/spin_currents.*
.. jupyter-execute::
plot_currents(syst, [
('$J_{σ_0}$', current),
('$J_{σ_z}$', spin_z_current),
('$J_{σ_y}$', spin_y_current),
])
.. note::
......@@ -262,9 +427,16 @@ scattering region.
Doing this is as simple as passing a *function* when instantiating
the `~kwant.operator.Current`, instead of a constant matrix:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_following
:end-before: #HIDDEN_END_following
.. jupyter-execute::
def following_m_i(site, r0, delta):
m_i = field_direction(site.pos, r0, delta)
return np.dot(m_i, sigma)
J_m = kwant.operator.Current(syst, following_m_i)
# evaluate the operator
m_current = J_m(psi, params=dict(r0=25, delta=10))
The function must take a `~kwant.builder.Site` as its first parameter,
and may optionally take other parameters (i.e. it must have the same
......@@ -274,7 +446,7 @@ matrix that defines the operator we wish to calculate.
.. note::
In the above example we had to pass the extra parameters needed by the
``following_operator`` function via the ``param`` keyword argument. In
``following_operator`` function via the ``params`` keyword argument. In
general you must pass all the parameters needed by the Hamiltonian via
``params`` (as you would when calling `~kwant.solvers.default.smatrix` or
`~kwant.solvers.default.wave_function`). In the previous examples,
......@@ -286,7 +458,13 @@ Using this we can see that the spin current is essentially oriented along
the direction of :math:`m_i` in the present regime where the onsite term
in the Hamiltonian is dominant:
.. image:: /code/figure/spin_current_comparison.*
.. jupyter-execute::
:hide-code:
plot_currents(syst, [
(r'$J_{\mathbf{m}_i}$', m_current),
('$J_{σ_z}$', spin_z_current),
])
.. note:: Although this example used exclusively `~kwant.operator.Current`,
you can do the same with `~kwant.operator.Density`.
......@@ -308,11 +486,16 @@ Density of states in a circle
To calculate the density of states inside a circle of radius
20 we can simply do:
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_density_cut
:end-before: #HIDDEN_END_density_cut
.. jupyter-execute::
def circle(site):
return np.linalg.norm(site.pos) < 20
.. literalinclude:: /code/figure/circle_dos.txt
rho_circle = kwant.operator.Density(syst, where=circle, sum=True)
all_states = np.vstack((wf(0), wf(1)))
dos_in_circle = sum(rho_circle(p) for p in all_states) / (2 * pi)
print('density of states in circle:', dos_in_circle)
note that we also provide ``sum=True``, which means that evaluating the
operator on a wavefunction will produce a single scalar. This is semantically
......@@ -325,11 +508,22 @@ Current flowing through a cut
Below we calculate the probability current and z-projected spin current near
the interfaces with the left and right leads.
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_current_cut
:end-before: #HIDDEN_END_current_cut
.. jupyter-execute::
def left_cut(site_to, site_from):
return site_from.pos[0] <= -39 and site_to.pos[0] > -39
def right_cut(site_to, site_from):
return site_from.pos[0] < 39 and site_to.pos[0] >= 39
.. literalinclude:: /code/figure/current_cut.txt
J_left = kwant.operator.Current(syst, where=left_cut, sum=True)
J_right = kwant.operator.Current(syst, where=right_cut, sum=True)
Jz_left = kwant.operator.Current(syst, sigma_z, where=left_cut, sum=True)
Jz_right = kwant.operator.Current(syst, sigma_z, where=right_cut, sum=True)
print('J_left:', J_left(psi), ' J_right:', J_right(psi))
print('Jz_left:', Jz_left(psi), ' Jz_right:', Jz_right(psi))
We see that the probability current is conserved across the scattering region,
but the z-projected spin current is not due to the fact that the Hamiltonian
......@@ -360,8 +554,23 @@ This can be achieved with `~kwant.operator.Current.bind`:
*different* set of parameters. This will almost certainly give
incorrect results.
.. literalinclude:: /code/include/magnetic_texture.py
:start-after: #HIDDEN_BEGIN_bind
:end-before: #HIDDEN_END_bind
.. jupyter-execute::
J_m = kwant.operator.Current(syst, following_m_i)
J_z = kwant.operator.Current(syst, sigma_z)
J_m_bound = J_m.bind(params=dict(r0=25, delta=10, J=1))
J_z_bound = J_z.bind(params=dict(r0=25, delta=10, J=1))
# Sum current local from all scattering states on the left at energy=-1
wf_left = wf(0)
J_m_left = sum(J_m_bound(p) for p in wf_left)
J_z_left = sum(J_z_bound(p) for p in wf_left)
.. jupyter-execute::
:hide-code:
.. image:: /code/figure/bound_current.*
plot_currents(syst, [
(r'$J_{\mathbf{m}_i}$ (from left)', J_m_left),
(r'$J_{σ_z}$ (from left)', J_z_left),
])
doc/source/tutorial/plotting.rst
View file @ bc9b2edf
......@@ -10,17 +10,66 @@ these options can be used to achieve various very different objectives.
2D example: graphene quantum dot
................................
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:download:`plot_graphene.py </code/download/plot_graphene.py>`
:jupyter-download:script:`plot_graphene`
.. jupyter-kernel::
:id: plot_graphene
.. jupyter-execute::
:hide-code:
# Tutorial 2.8.1. 2D example: graphene quantum dot
# ================================================
#
# Physics background
# ------------------
# - graphene edge states
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting
import warnings
warnings.simplefilter("ignore")
from matplotlib import pyplot
import kwant
.. jupyter-execute:: boilerplate.py
:hide-code:
We begin by first considering a circular graphene quantum dot (similar to what
has been used in parts of the tutorial :ref:`tutorial-graphene`.) In contrast
to previous examples, we will also use hoppings beyond next-nearest neighbors:
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_makesyst
:end-before: #HIDDEN_END_makesyst
.. jupyter-execute::
lat = kwant.lattice.honeycomb(norbs=1)
a, b = lat.sublattices
def make_system(r=8, t=-1, tp=-0.1):
def circle(pos):
x, y = pos
return x**2 + y**2 < r**2
syst = kwant.Builder()
syst[lat.shape(circle, (0, 0))] = 0
syst[lat.neighbors()] = t
syst.eradicate_dangling()
if tp:
syst[lat.neighbors(2)] = tp
return syst
Note that adding hoppings hoppings to the `n`-th nearest neighbors can be
simply done by passing `n` as an argument to
......@@ -29,16 +78,14 @@ simply done by passing `n` as an argument to
out of the shape. It is necessary to do so *before* adding the
next-nearest-neighbor hopping [#]_.
Of course, the system can be plotted simply with default settings:
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotsyst1
:end-before: #HIDDEN_END_plotsyst1
However, due to the richer structure of the lattice, this results in a rather
Of course, the system can be plotted simply with default settings,
however, due to the richer structure of the lattice, this results in a rather
busy plot:
.. image:: /code/figure/plot_graphene_syst1.*
.. jupyter-execute::
syst = make_system()
kwant.plot(syst);
A much clearer plot can be obtained by using different colors for both
sublattices, and by having different line widths for different hoppings. This
......@@ -47,15 +94,19 @@ can be achieved by passing a function to the arguments of
must be a function taking one site as argument, for hoppings a function taking
the start end end site of hopping as arguments:
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotsyst2
:end-before: #HIDDEN_END_plotsyst2
.. jupyter-execute::
def family_color(site):
return 'black' if site.family == a else 'white'
def hopping_lw(site1, site2):
return 0.04 if site1.family == site2.family else 0.1
kwant.plot(syst, site_lw=0.1, site_color=family_color, hop_lw=hopping_lw);
Note that since we are using an unfinalized Builder, a ``site`` is really an
instance of `~kwant.builder.Site`. With these adjustments we arrive at a plot
that carries the same information, but is much easier to interpret:
.. image:: /code/figure/plot_graphene_syst2.*
that carries the same information, but is much easier to interpret.
Apart from plotting the *system* itself, `~kwant.plotter.plot` can also be used
to plot *data* living on the system.
......@@ -67,23 +118,27 @@ nearest-neighbor hopping. Computing the wave functions is done in the usual
way (note that for a large-scale system, one would probably want to use sparse
linear algebra):
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotdata1
:end-before: #HIDDEN_END_plotdata1
In most cases, to plot the wave function probability, one wouldn't use
`~kwant.plotter.plot`, but rather `~kwant.plotter.map`. Here, we plot the
`n`-th wave function using it:
.. jupyter-execute::
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotdata2
:end-before: #HIDDEN_END_plotdata2
import scipy.linalg as la
syst = make_system(tp=0).finalized()
ham = syst.hamiltonian_submatrix()
evecs = la.eigh(ham)[1]
wf = abs(evecs[:, 225])**2
In most cases, to plot the wave function probability, one wouldn't use
`~kwant.plotter.plot`, but rather `~kwant.plotter.map`. Here, we plot the
`n`-th wave function using it.
This results in a standard pseudocolor plot, showing in this case (``n=225``) a
graphene edge state, i.e. a wave function mostly localized at the zigzag edges
of the quantum dot.
.. image:: /code/figure/plot_graphene_data1.*
.. jupyter-execute::
kwant.plotter.map(syst, wf, oversampling=10, cmap='gist_heat_r');
However although in general preferable, `~kwant.plotter.map` has a few
deficiencies for this small system: For example, there are a few distortions at
......@@ -99,9 +154,17 @@ the symbol shape depending on the sublattice. With a triangle pointing up and
down on the respective sublattice, the symbols used by plot fill the space
completely:
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotdata3
:end-before: #HIDDEN_END_plotdata3
.. jupyter-execute::
def family_shape(i):
site = syst.sites[i]
return ('p', 3, 180) if site.family == a else ('p', 3, 0)
def family_color(i):
return 'black' if syst.sites[i].family == a else 'white'
kwant.plot(syst, site_color=wf, site_symbol=family_shape,
site_size=0.5, hop_lw=0, cmap='gist_heat_r');
Note that with ``hop_lw=0`` we deactivate plotting the hoppings (that would not
serve any purpose here). Moreover, ``site_size=0.5`` guarantees that the two
......@@ -114,31 +177,23 @@ Finally, note that since we are dealing with a finalized system now, a site `i`
is represented by an integer. In order to obtain the original
`~kwant.builder.Site`, ``syst.sites[i]`` can be used.
With this we arrive at
.. image:: /code/figure/plot_graphene_data2.*
with the same information as `~kwant.plotter.map`, but with a cleaner look.
The way how data is presented of course influences what features of the data
are best visible in a given plot. With `~kwant.plotter.plot` one can easily go
beyond pseudocolor-like plots. For example, we can represent the wave function
probability using the symbols itself:
.. literalinclude:: /code/include/plot_graphene.py
:start-after: #HIDDEN_BEGIN_plotdata4
:end-before: #HIDDEN_END_plotdata4
.. jupyter-execute::
def site_size(i):
return 3 * wf[i] / wf.max()
kwant.plot(syst, site_size=site_size, site_color=(0, 0, 1, 0.3),
hop_lw=0.1);
Here, we choose the symbol size proportional to the wave function probability,
while the site color is transparent to also allow for overlapping symbols to be
visible. The hoppings are also plotted in order to show the underlying lattice.
With this, we arrive at
.. image:: /code/figure/plot_graphene_data3.*
which shows the edge state nature of the wave function most clearly.
.. rubric:: Footnotes
.. [#] A dangling site is defined as having only one hopping connecting it to
......@@ -150,7 +205,31 @@ which shows the edge state nature of the wave function most clearly.
.. seealso::
The complete source code of this example can be found in
:download:`plot_zincblende.py </code/download/plot_zincblende.py>`
:jupyter-download:script:`plot_zincblende`
.. jupyter-kernel::
:id: plot_zincblende
.. jupyter-execute::
:hide-code:
# Tutorial 2.8.2. 3D example: zincblende structure
# ================================================
#
# Physical background
# -------------------
# - 3D Bravais lattices
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting in 3D
from matplotlib import pyplot
import kwant
.. jupyter-execute:: boilerplate.py
:hide-code:
Zincblende is a very common crystal structure of semiconductors. It is a
face-centered cubic crystal with two inequivalent atoms in the unit cell
......@@ -159,18 +238,30 @@ structure, but two equivalent atoms per unit cell).
It is very easily generated in Kwant with `kwant.lattice.general`:
.. literalinclude:: /code/include/plot_zincblende.py
:start-after: #HIDDEN_BEGIN_zincblende1
:end-before: #HIDDEN_END_zincblende1
.. jupyter-execute::
lat = kwant.lattice.general([(0, 0.5, 0.5), (0.5, 0, 0.5), (0.5, 0.5, 0)],
[(0, 0, 0), (0.25, 0.25, 0.25)],
norbs=1)
a, b = lat.sublattices
Note how we keep references to the two different sublattices for later use.
A three-dimensional structure is created as easily as in two dimensions, by
using the `~kwant.lattice.PolyatomicLattice.shape`-functionality:
.. literalinclude:: /code/include/plot_zincblende.py
:start-after: #HIDDEN_BEGIN_zincblende2
:end-before: #HIDDEN_END_zincblende2
.. jupyter-execute::
def make_cuboid(a=15, b=10, c=5):
def cuboid_shape(pos):
x, y, z = pos
return 0 <= x < a and 0 <= y < b and 0 <= z < c
syst = kwant.Builder()
syst[lat.shape(cuboid_shape, (0, 0, 0))] = None
syst[lat.neighbors()] = None
return syst
We restrict ourselves here to a simple cuboid, and do not bother to add real
values for onsite and hopping energies, but only the placeholder ``None`` (in a
......@@ -179,13 +270,11 @@ real calculation, several atomic orbitals would have to be considered).
`~kwant.plotter.plot` can plot 3D systems just as easily as its two-dimensional
counterparts:
.. literalinclude:: /code/include/plot_zincblende.py
:start-after: #HIDDEN_BEGIN_plot1
:end-before: #HIDDEN_END_plot1
.. jupyter-execute::
resulting in
syst = make_cuboid()
.. image:: /code/figure/plot_zincblende_syst1.*
kwant.plot(syst);
You might notice that the standard options for plotting are quite different in
3D than in 2D. For example, by default hoppings are not printed, but sites are
......@@ -207,14 +296,18 @@ Also for 3D it is possible to customize the plot. For example, we
can explicitly plot the hoppings as lines, and color sites differently
depending on the sublattice:
.. literalinclude:: /code/include/plot_zincblende.py
:start-after: #HIDDEN_BEGIN_plot2
:end-before: #HIDDEN_END_plot2
.. jupyter-execute::
which results in a 3D plot that allows to interactively (when plotted
in a window) explore the crystal structure:
syst = make_cuboid(a=1.5, b=1.5, c=1.5)
.. image:: /code/figure/plot_zincblende_syst2.*
def family_colors(site):
return 'r' if site.family == a else 'g'
kwant.plot(syst, site_size=0.18, site_lw=0.01, hop_lw=0.05,
site_color=family_colors);
which results in a 3D plot that allows to interactively (when plotted
in a window) explore the crystal structure.
Hence, a few lines of code using Kwant allow to explore all the different
crystal lattices out there!
......
doc/source/tutorial/spectrum.rst
View file @ bc9b2edf
......@@ -4,9 +4,40 @@ Beyond transport: Band structure and closed systems
Band structure calculations
...........................
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:download:`band_structure.py </code/download/band_structure.py>`
:jupyter-download:script:`band_structure`
.. jupyter-kernel::
:id: band_structure
.. jupyter-execute::
:hide-code:
# Tutorial 2.4.1. Band structure calculations
# ===========================================
#
# Physics background
# ------------------
# band structure of a simple quantum wire in tight-binding approximation
#
# Kwant features highlighted
# --------------------------
# - Computing the band structure of a finalized lead.
import kwant
# For plotting
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
When doing transport simulations, one also often needs to know the band
structure of the leads, i.e. the energies of the propagating plane waves in the
......@@ -19,9 +50,26 @@ tight-binding wire.
Computing band structures in Kwant is easy. Just define a lead in the
usual way:
.. literalinclude:: /code/include/band_structure.py
:start-after: #HIDDEN_BEGIN_zxip
:end-before: #HIDDEN_END_zxip
.. jupyter-execute::
def make_lead(a=1, t=1.0, W=10):
# Start with an empty lead with a single square lattice
lat = kwant.lattice.square(a, norbs=1)
sym_lead = kwant.TranslationalSymmetry((-a, 0))
lead = kwant.Builder(sym_lead)
# build up one unit cell of the lead, and add the hoppings
# to the next unit cell
for j in range(W):
lead[lat(0, j)] = 4 * t
if j > 0:
lead[lat(0, j), lat(0, j - 1)] = -t
lead[lat(1, j), lat(0, j)] = -t
return lead
"Usual way" means defining a translational symmetry vector, as well
as one unit cell of the lead, and the hoppings to neighboring
......@@ -42,13 +90,24 @@ do something more profound with the dispersion relation these energies may be
calculated directly using `~kwant.physics.Bands`. For now we just plot the
bandstructure:
.. literalinclude:: /code/include/band_structure.py
:start-after: #HIDDEN_BEGIN_pejz
:end-before: #HIDDEN_END_pejz
.. jupyter-execute::
def main():
lead = make_lead().finalized()
kwant.plotter.bands(lead, show=False)
pyplot.xlabel("momentum [(lattice constant)^-1]")
pyplot.ylabel("energy [t]")
pyplot.show()
This gives the result:
.. image:: /code/figure/band_structure_result.*
.. jupyter-execute::
:hide-code:
# 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()
where we observe the cosine-like dispersion of the square lattice. Close
to ``k=0`` this agrees well with the quadratic dispersion this tight-binding
......@@ -61,7 +120,34 @@ Closed systems
.. seealso::
The complete source code of this example can be found in
:download:`closed_system.py </code/download/closed_system.py>`
:jupyter-download:script:`closed_system`
.. jupyter-kernel::
:id: closed_system
.. jupyter-execute::
:hide-code:
# Tutorial 2.4.2. Closed systems
# ==============================
#
# Physics background
# ------------------
# Fock-darwin spectrum of a quantum dot (energy spectrum in
# as a function of a magnetic field)
#
# Kwant features highlighted
# --------------------------
# - Use of `hamiltonian_submatrix` in order to obtain a Hamiltonian
# matrix.
from cmath import exp
import numpy as np
from matplotlib import pyplot
import kwant
.. jupyter-execute:: boilerplate.py
:hide-code:
Although Kwant is (currently) mainly aimed towards transport problems, it
can also easily be used to compute properties of closed systems -- after
......@@ -74,16 +160,49 @@ To compute the eigenenergies and eigenstates, we will make use of the sparse
linear algebra functionality of `SciPy <https://www.scipy.org>`_, which
interfaces the ARPACK package:
.. literalinclude:: /code/include/closed_system.py
:start-after: #HIDDEN_BEGIN_tibv
:end-before: #HIDDEN_END_tibv
.. jupyter-execute::
# For eigenvalue computation
import scipy.sparse.linalg as sla
We set up the system using the `shape`-function as in
:ref:`tutorial-abring`, but do not add any leads:
.. literalinclude:: /code/include/closed_system.py
:start-after: #HIDDEN_BEGIN_qlyd
:end-before: #HIDDEN_END_qlyd
.. jupyter-execute::
:hide-code:
a = 1
t = 1.0
r = 10
.. jupyter-execute::
def make_system(a=1, t=1.0, r=10):
lat = kwant.lattice.square(a, norbs=1)
syst = kwant.Builder()
# Define the quantum dot
def circle(pos):
(x, y) = pos
rsq = x ** 2 + y ** 2
return rsq < r ** 2
def hopx(site1, site2, B):
# The magnetic field is controlled by the parameter B
y = site1.pos[1]
return -t * exp(-1j * B * y)
syst[lat.shape(circle, (0, 0))] = 4 * t
# hoppings in x-direction
syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx
# hoppings in y-directions
syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t
# It's a closed system for a change, so no leads
return syst
We add the magnetic field using a function and a global variable as we
did in the two previous tutorial. (Here, the gauge is chosen such that
......@@ -93,14 +212,40 @@ The spectrum can be obtained by diagonalizing the Hamiltonian of the
system, which in turn can be obtained from the finalized
system using `~kwant.system.System.hamiltonian_submatrix`:
.. literalinclude:: /code/include/closed_system.py
:start-after: #HIDDEN_BEGIN_yvri
:end-before: #HIDDEN_END_yvri
.. jupyter-execute::
def plot_spectrum(syst, Bfields):
energies = []
for B in Bfields:
# Obtain the Hamiltonian as a sparse matrix
ham_mat = syst.hamiltonian_submatrix(params=dict(B=B), sparse=True)
# we only calculate the 15 lowest eigenvalues
ev = sla.eigsh(ham_mat.tocsc(), k=15, sigma=0,
return_eigenvectors=False)
energies.append(ev)
pyplot.figure()
pyplot.plot(Bfields, energies)
pyplot.xlabel("magnetic field [arbitrary units]")
pyplot.ylabel("energy [t]")
pyplot.show()
Note that we use sparse linear algebra to efficiently calculate only a
few lowest eigenvalues. Finally, we obtain the result:
.. image:: /code/figure/closed_system_result.*
.. jupyter-execute::
:hide-code:
syst = make_system()
syst = syst.finalized()
# We should observe energy levels that flow towards Landau
# level energies with increasing magnetic field.
plot_spectrum(syst, [iB * 0.002 for iB in range(100)])
At zero magnetic field several energy levels are degenerate (since our
quantum dot is rather symmetric). These degeneracies are split
......@@ -110,11 +255,34 @@ Landau level energies at higher magnetic fields [#]_.
The eigenvectors are obtained very similarly, and can be plotted directly
using `~kwant.plotter.map`:
.. literalinclude:: /code/include/closed_system.py
:start-after: #HIDDEN_BEGIN_wave
:end-before: #HIDDEN_END_wave
.. jupyter-execute::
:hide-code:
def sorted_eigs(ev):
evals, evecs = ev
evals, evecs = map(np.array, zip(*sorted(zip(evals, evecs.transpose()))))
return evals, evecs.transpose()
.. jupyter-execute::
def plot_wave_function(syst, B=0.001):
# Calculate the wave functions in the system.
ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))
# Plot the probability density of the 10th eigenmode.
kwant.plotter.map(syst, np.abs(evecs[:, 9])**2,
colorbar=False, oversampling=1)
.. image:: /code/figure/closed_system_eigenvector.*
.. jupyter-execute::
:hide-code:
syst = make_system(r=30)
# Plot an eigenmode of a circular dot. Here we create a larger system for
# better spatial resolution.
syst = make_system(r=30).finalized()
plot_wave_function(syst);
The last two arguments to `~kwant.plotter.map` are optional. The first prevents
a colorbar from appearing. The second, ``oversampling=1``, makes the image look
......@@ -127,11 +295,22 @@ that they can have *non-zero* local current. We can calculate the local
current due to a state by using `kwant.operator.Current` and plotting
it using `kwant.plotter.current`:
.. literalinclude:: /code/include/closed_system.py
:start-after: #HIDDEN_BEGIN_current
:end-before: #HIDDEN_END_current
.. jupyter-execute::
def plot_current(syst, B=0.001):
# Calculate the wave functions in the system.
ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))
# Calculate and plot the local current of the 10th eigenmode.
J = kwant.operator.Current(syst)
current = J(evecs[:, 9], params=dict(B=B))
kwant.plotter.current(syst, current, colorbar=False)
.. jupyter-execute::
:hide-code:
.. image:: /code/figure/closed_system_current.*
plot_current(syst);
.. specialnote:: Technical details
......
doc/source/tutorial/spin_potential_shape.rst
View file @ bc9b2edf
......@@ -9,9 +9,18 @@ very simple examples of the previous section.
Matrix structure of on-site and hopping elements
................................................
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:download:`spin_orbit.py </code/download/spin_orbit.py>`
:jupyter-download:script:`spin_orbit`
.. jupyter-kernel::
:id: spin_orbit
We begin by extending the simple 2DEG-Hamiltonian by a Rashba spin-orbit
coupling and a Zeeman splitting due to an external magnetic field:
......@@ -42,25 +51,82 @@ use matrices in our program, we import the Tinyarray package. (`NumPy
<http://www.numpy.org/>`_ would work as well, but Tinyarray is much faster
for small arrays.)
.. literalinclude:: /code/include/spin_orbit.py
:start-after: #HIDDEN_BEGIN_xumz
:end-before: #HIDDEN_END_xumz
.. jupyter-execute::
:hide-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
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
.. jupyter-execute::
# For matrix support
import tinyarray
For convenience, we define the Pauli-matrices first (with :math:`\sigma_0` the
unit matrix):
.. literalinclude:: /code/include/spin_orbit.py
:start-after: #HIDDEN_BEGIN_hwbt
:end-before: #HIDDEN_END_hwbt
.. jupyter-execute::
# 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]])
and we also define some other parameters useful for constructing our system:
.. jupyter-execute::
t = 1.0
alpha = 0.5
e_z = 0.08
W, L = 10, 30
Previously, we used numbers as the values of our matrix elements.
However, `~kwant.builder.Builder` also accepts matrices as values, and
we can simply write:
.. literalinclude:: /code/include/spin_orbit.py
:start-after: #HIDDEN_BEGIN_uxrm
:end-before: #HIDDEN_END_uxrm
.. jupyter-execute::
:hide-code:
lat = kwant.lattice.square(norbs=2)
syst = kwant.Builder()
.. jupyter-execute::
#### 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
Note that we specify ``norbs=2`` when creating the lattice, as each site
has 2 degrees of freedom associated with it, giving us 2x2 matrices as
onsite/hopping terms.
Note that the Zeeman energy adds to the onsite term, whereas the Rashba
spin-orbit term adds to the hoppings (due to the derivative operator).
Furthermore, the hoppings in x and y-direction have a different matrix
......@@ -85,14 +151,49 @@ when specifying `(1, 0)` it is not necessary to specify `(-1, 0)`),
The leads also allow for a matrix structure,
.. literalinclude:: /code/include/spin_orbit.py
:start-after: #HIDDEN_BEGIN_yliu
:end-before: #HIDDEN_END_yliu
.. jupyter-execute::
:hide-code:
#### Define the left lead. ####
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
.. jupyter-execute::
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
.. jupyter-execute::
:hide-code:
#### Attach the leads and finalize the system. ####
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
syst = syst.finalized()
The remainder of the code is unchanged, and as a result we should obtain
the following, clearly non-monotonic conductance steps:
.. image:: /code/figure/spin_orbit_result.*
.. jupyter-execute::
:hide-code:
# Compute conductance
energies=[0.01 * i - 0.3 for i in range(100)]
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, 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()
.. specialnote:: Technical details
......@@ -129,7 +230,32 @@ Spatially dependent values through functions
.. seealso::
The complete source code of this example can be found in
:download:`quantum_well.py </code/download/quantum_well.py>`
:jupyter-download:script:`quantum_well`
.. jupyter-kernel::
:id: quantum_well
.. jupyter-execute::
:hide-code:
# Tutorial 2.3.2. Spatially dependent values through functions
# ============================================================
#
# Physics background
# ------------------
# transmission through a quantum well
#
# Kwant features highlighted
# --------------------------
# - Functions as values in Builder
import kwant
# For plotting
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
Up to now, all examples had position-independent matrix-elements
(and thus translational invariance along the wire, which
......@@ -148,22 +274,50 @@ changing the potential then implies the need to build up the system again.
Instead, we use a python *function* to define the onsite energies. We
define the potential profile of a quantum well as:
.. literalinclude:: /code/include/quantum_well.py
:start-after: #HIDDEN_BEGIN_ehso
:end-before: #HIDDEN_END_ehso
.. jupyter-execute::
W, L, L_well = 10, 30, 10
def potential(site, pot):
(x, y) = site.pos
if (L - L_well) / 2 < x < (L + L_well) / 2:
return pot
else:
return 0
This function takes two arguments: the first of type `~kwant.builder.Site`,
from which you can get the real-space coordinates using ``site.pos``, and the
value of the potential as the second. Note that in `potential` we can access
variables of the surrounding function: `L` and `L_well` are taken from the
namespace of `make_system`.
variables `L` and `L_well` that are defined globally.
Kwant now allows us to pass a function as a value to
`~kwant.builder.Builder`:
.. literalinclude:: /code/include/quantum_well.py
:start-after: #HIDDEN_BEGIN_coid
:end-before: #HIDDEN_END_coid
.. jupyter-execute::
a = 1
t = 1.0
def onsite(site, pot):
return 4 * t + potential(site, pot)
lat = kwant.lattice.square(a, norbs=1)
syst = kwant.Builder()
syst[(lat(x, y) for x in range(L) for y in range(W))] = onsite
syst[lat.neighbors()] = -t
.. jupyter-execute::
:hide-code:
#### Define and attach the leads. ####
lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0)))
lead[(lat(0, j) for j in range(W))] = 4 * t
lead[lat.neighbors()] = -t
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
syst = syst.finalized()
For each lattice point, the corresponding site is then passed as the
first argument to the function `onsite`. The values of any additional
......@@ -180,9 +334,21 @@ of the lead -- this should be kept in mind.
Finally, we compute the transmission probability:
.. literalinclude:: /code/include/quantum_well.py
:start-after: #HIDDEN_BEGIN_sqvr
:end-before: #HIDDEN_END_sqvr
.. jupyter-execute::
def plot_conductance(syst, energy, welldepths):
# Compute conductance
data = []
for welldepth in welldepths:
smatrix = kwant.smatrix(syst, energy, params=dict(pot=-welldepth))
data.append(smatrix.transmission(1, 0))
pyplot.figure()
pyplot.plot(welldepths, data)
pyplot.xlabel("well depth [t]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
``kwant.smatrix`` allows us to specify a dictionary, `params`, that contains
the additional arguments required by the Hamiltonian matrix elements.
......@@ -191,7 +357,11 @@ of the potential well by passing the potential value (remember above
we defined our `onsite` function that takes a parameter named `pot`).
We obtain the result:
.. image:: /code/figure/quantum_well_result.*
.. jupyter-execute::
:hide-code:
plot_conductance(syst, energy=0.2,
welldepths=[0.01 * i for i in range(100)])
Starting from no potential (well depth = 0), we observe the typical
oscillatory transmission behavior through resonances in the quantum well.
......@@ -218,13 +388,44 @@ Nontrivial shapes
.. seealso::
The complete source code of this example can be found in
:download:`ab_ring.py </code/download/ab_ring.py>`
:jupyter-download:script:`ab_ring`
.. jupyter-kernel::
:id: ab_ring
.. jupyter-execute::
:hide-code:
# Tutorial 2.3.3. Nontrivial shapes
# =================================
#
# Physics background
# ------------------
# Flux-dependent transmission through a quantum ring
#
# Kwant features highlighted
# --------------------------
# - More complex shapes with lattices
# - Allows for discussion of subtleties of `attach_lead` (not in the
# example, but in the tutorial main text)
# - Modifcations of hoppings/sites after they have been added
from cmath import exp
from math import pi
import kwant
# For plotting
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
Up to now, we only dealt with simple wire geometries. Now we turn to the case
of a more complex geometry, namely transport through a quantum ring
that is pierced by a magnetic flux :math:`\Phi`:
.. image:: /code/figure/ab_ring_sketch.*
.. image:: /figure/ab_ring_sketch.*
For a flux line, it is possible to choose a gauge such that a
charged particle acquires a phase :math:`e\Phi/h` whenever it
......@@ -238,9 +439,14 @@ First, define a boolean function defining the desired shape, i.e. a function
that returns ``True`` whenever a point is inside the shape, and
``False`` otherwise:
.. literalinclude:: /code/include/ab_ring.py
:start-after: #HIDDEN_BEGIN_eusz
:end-before: #HIDDEN_END_eusz
.. jupyter-execute::
r1, r2 = 10, 20
def ring(pos):
(x, y) = pos
rsq = x ** 2 + y ** 2
return (r1 ** 2 < rsq < r2 ** 2)
Note that this function takes a real-space position as argument (not a
`~kwant.builder.Site`).
......@@ -249,9 +455,16 @@ We can now simply add all of the lattice points inside this shape at
once, using the function `~kwant.lattice.Square.shape`
provided by the lattice:
.. literalinclude:: /code/include/ab_ring.py
:start-after: #HIDDEN_BEGIN_lcak
:end-before: #HIDDEN_END_lcak
.. jupyter-execute::
a = 1
t = 1.0
lat = kwant.lattice.square(a, norbs=1)
syst = kwant.Builder()
syst[lat.shape(ring, (0, r1 + 1))] = 4 * t
syst[lat.neighbors()] = -t
Here, ``lat.shape`` takes as a second parameter a (real-space) point that is
inside the desired shape. The hoppings can still be added using
......@@ -264,9 +477,30 @@ along the branch cut in the lower arm of the ring. For this we select
all hoppings in x-direction that are of the form `(lat(1, j), lat(0, j))`
with ``j<0``:
.. literalinclude:: /code/include/ab_ring.py
:start-after: #HIDDEN_BEGIN_lvkt
:end-before: #HIDDEN_END_lvkt
.. jupyter-execute::
# In order to introduce a flux through the ring, we introduce a phase on
# the hoppings on the line cut through one of the arms. Since we want to
# change the flux without modifying the Builder instance repeatedly, we
# define the modified hoppings as a function that takes the flux as its
# parameter phi.
def hopping_phase(site1, site2, phi):
return -t * exp(1j * phi)
def crosses_branchcut(hop):
ix0, iy0 = hop[0].tag
# builder.HoppingKind with the argument (1, 0) below
# returns hoppings ordered as ((i+1, j), (i, j))
return iy0 < 0 and ix0 == 1 # ix1 == 0 then implied
# Modify only those hopings in x-direction that cross the branch cut
def hops_across_cut(syst):
for hop in kwant.builder.HoppingKind((1, 0), lat, lat)(syst):
if crosses_branchcut(hop):
yield hop
syst[hops_across_cut] = hopping_phase
Here, `crosses_branchcut` is a boolean function that returns ``True`` for
the desired hoppings. We then use again a generator (this time with
......@@ -279,9 +513,21 @@ by the parameter `phi`.
For the leads, we can also use the ``lat.shape``-functionality:
.. literalinclude:: /code/include/ab_ring.py
:start-after: #HIDDEN_BEGIN_qwgr
:end-before: #HIDDEN_END_qwgr
.. jupyter-execute::
#### Define the leads. ####
W = 10
sym_lead = kwant.TranslationalSymmetry((-a, 0))
lead = kwant.Builder(sym_lead)
def lead_shape(pos):
(x, y) = pos
return (-W / 2 < y < W / 2)
lead[lat.shape(lead_shape, (0, 0))] = 4 * t
lead[lat.neighbors()] = -t
Here, the shape must be compatible with the translational symmetry
of the lead ``sym_lead``. In particular, this means that it should extend to
......@@ -290,9 +536,12 @@ no restriction on ``x`` in ``lead_shape``) [#]_.
Attaching the leads is done as before:
.. literalinclude:: /code/include/ab_ring.py
:start-after: #HIDDEN_BEGIN_skbz
:end-before: #HIDDEN_END_skbz
.. jupyter-execute::
:hide-output:
#### Attach the leads ####
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
In fact, attaching leads seems not so simple any more for the current
structure with a scattering region very much different from the lead
......@@ -305,16 +554,40 @@ back (going opposite to the direction of the translational vector)
until it intersects the scattering region. At this intersection,
the lead is attached:
.. image:: /code/figure/ab_ring_sketch2.*
.. image:: /figure/ab_ring_sketch2.*
After the lead has been attached, the system should look like this:
.. image:: /code/figure/ab_ring_syst.*
.. jupyter-execute::
:hide-code:
kwant.plot(syst);
The computation of the conductance goes in the same fashion as before.
Finally you should get the following result:
.. image:: /code/figure/ab_ring_result.*
.. jupyter-execute::
:hide-code:
def plot_conductance(syst, energy, fluxes):
# compute conductance
normalized_fluxes = [flux / (2 * pi) for flux in fluxes]
data = []
for flux in fluxes:
smatrix = kwant.smatrix(syst, energy, params=dict(phi=flux))
data.append(smatrix.transmission(1, 0))
pyplot.figure()
pyplot.plot(normalized_fluxes, data)
pyplot.xlabel("flux [flux quantum]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
# We should see a conductance that is periodic with the flux quantum
plot_conductance(syst.finalized(), energy=0.15,
fluxes=[0.01 * i * 3 * 2 * pi for i in range(100)])
where one can observe the conductance oscillations with the
period of one flux quantum.
......@@ -330,7 +603,47 @@ period of one flux quantum.
becomes more apparent if we attach the leads a bit further away
from the central axis o the ring, as was done in this example:
.. image:: /code/figure/ab_ring_note1.*
.. jupyter-kernel::
:id: ab_ring_note1
.. jupyter-execute::
:hide-code:
import kwant
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
.. jupyter-execute::
:hide-code:
a = 1
t = 1.0
W = 10
r1, r2 = 10, 20
lat = kwant.lattice.square(norbs=1)
syst = kwant.Builder()
def ring(pos):
(x, y) = pos
rsq = x**2 + y**2
return ( r1**2 < rsq < r2**2)
syst[lat.shape(ring, (0, 11))] = 4 * t
syst[lat.neighbors()] = -t
sym_lead0 = kwant.TranslationalSymmetry((-a, 0))
lead0 = kwant.Builder(sym_lead0)
def lead_shape(pos):
(x, y) = pos
return (-1 < x < 1) and ( 0.5 * W < y < 1.5 * W )
lead0[lat.shape(lead_shape, (0, W))] = 4 * t
lead0[lat.neighbors()] = -t
lead1 = lead0.reversed()
syst.attach_lead(lead0)
syst.attach_lead(lead1)
kwant.plot(syst);
- Per default, `~kwant.builder.Builder.attach_lead` attaches
the lead to the "outside" of the structure, by tracing the
......@@ -345,7 +658,46 @@ period of one flux quantum.
starts the trace-back in the middle of the ring, resulting
in the lead being attached to the inner circle:
.. image:: /code/figure/ab_ring_note2.*
.. jupyter-kernel::
:id: ab_ring_note2
.. jupyter-execute::
:hide-code:
import kwant
from matplotlib import pyplot
.. jupyter-execute:: boilerplate.py
:hide-code:
.. jupyter-execute::
:hide-code:
a = 1
t = 1.0
W = 10
r1, r2 = 10, 20
lat = kwant.lattice.square(a, norbs=1)
syst = kwant.Builder()
def ring(pos):
(x, y) = pos
rsq = x**2 + y**2
return ( r1**2 < rsq < r2**2)
syst[lat.shape(ring, (0, 11))] = 4 * t
syst[lat.neighbors()] = -t
sym_lead0 = kwant.TranslationalSymmetry((-a, 0))
lead0 = kwant.Builder(sym_lead0)
def lead_shape(pos):
(x, y) = pos
return (-1 < x < 1) and ( -W/2 < y < W/2 )
lead0[lat.shape(lead_shape, (0, 0))] = 4 * t
lead0[lat.neighbors()] = -t
lead1 = lead0.reversed()
syst.attach_lead(lead0)
syst.attach_lead(lead1, lat(0, 0))
kwant.plot(syst);
Note that here the lead is treated as if it would pass over
the other arm of the ring, without intersecting it.
......
doc/source/tutorial/superconductors.rst
View file @ bc9b2edf
Superconductors: orbital degrees of freedom, conservation laws and symmetries
-----------------------------------------------------------------------------
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:download:`superconductor.py </code/download/superconductor.py>`
:jupyter-download:script:`superconductor`
.. jupyter-kernel::
:id: superconductor
.. jupyter-execute::
:hide-code:
# Tutorial 2.6. "Superconductors": orbitals, conservation laws and symmetries
# ===========================================================================
#
# Physics background
# ------------------
# - conductance of a NS-junction (Andreev reflection, superconducting gap)
#
# Kwant features highlighted
# --------------------------
# - Implementing electron and hole ("orbital") degrees of freedom
# using conservation laws.
# - Use of discrete symmetries to relate scattering states.
import kwant
from matplotlib import pyplot
import tinyarray
import numpy as np
tau_x = tinyarray.array([[0, 1], [1, 0]])
tau_y = tinyarray.array([[0, -1j], [1j, 0]])
tau_z = tinyarray.array([[1, 0], [0, -1]])
.. jupyter-execute:: boilerplate.py
:hide-code:
This example deals with superconductivity on the level of the
Bogoliubov-de Gennes (BdG) equation. In this framework, the Hamiltonian
......@@ -51,18 +90,50 @@ of freedom.
Let us consider a system that consists of a normal lead on the left,
a superconductor on the right, and a tunnel barrier in between:
.. image:: /code/figure/superconductor_transport_sketch.*
.. image:: /figure/superconductor_transport_sketch.*
We implement the BdG Hamiltonian in Kwant using a 2x2 matrix structure
for all Hamiltonian matrix elements, as we did
previously in the :ref:`spin example <tutorial_spinorbit>`. We declare
the square lattice and construct the scattering region with the following:
.. literalinclude:: /code/include/superconductor.py
:start-after: #HIDDEN_BEGIN_nbvn
:end-before: #HIDDEN_END_nbvn
Note the new argument ``norbs`` to `~kwant.lattice.square`. This is
previously in the :ref:`spin example <tutorial_spinorbit>`.
We start by declaring some parameters that will be used in the following code:
.. jupyter-execute::
a = 1
W, L = 10, 10
barrier = 1.5
barrierpos = (3, 4)
mu = 0.4
Delta = 0.1
Deltapos = 4
t = 1.0
and we declare the square lattice and construct the scattering region with the following:
.. jupyter-execute::
# Start with an empty tight-binding system. On each site, there
# are now electron and hole orbitals, so we must specify the
# number of orbitals per site. The orbital structure is the same
# as in the Hamiltonian.
lat = kwant.lattice.square(norbs=2)
syst = kwant.Builder()
#### Define the scattering region. ####
# The superconducting order parameter couples electron and hole orbitals
# on each site, and hence enters as an onsite potential.
# The pairing is only included beyond the point 'Deltapos' in the scattering region.
syst[(lat(x, y) for x in range(Deltapos) for y in range(W))] = (4 * t - mu) * tau_z
syst[(lat(x, y) for x in range(Deltapos, L) for y in range(W))] = (4 * t - mu) * tau_z + Delta * tau_x
# The tunnel barrier
syst[(lat(x, y) for x in range(barrierpos[0], barrierpos[1])
for y in range(W))] = (4 * t + barrier - mu) * tau_z
# Hoppings
syst[lat.neighbors()] = -t * tau_z
Note the argument ``norbs`` to `~kwant.lattice.square`. This is
the number of orbitals per site in the discretized BdG Hamiltonian - of course,
``norbs = 2``, since each site has one electron orbital and one hole orbital.
It is necessary to specify ``norbs`` here, such that we may later separate the
......@@ -80,9 +151,16 @@ of it). The next step towards computing conductance is to attach leads.
Let's attach two leads: a normal one to the left end, and a superconducting
one to the right end. Starting with the left lead, we have:
.. literalinclude:: /code/include/superconductor.py
:start-after: #HIDDEN_BEGIN_ttth
:end-before: #HIDDEN_END_ttth
.. jupyter-execute::
#### Define the leads. ####
# Left lead - normal, so the order parameter is zero.
sym_left = kwant.TranslationalSymmetry((-a, 0))
# Specify the conservation law used to treat electrons and holes separately.
# We only do this in the left lead, where the pairing is zero.
lead0 = kwant.Builder(sym_left, conservation_law=-tau_z, particle_hole=tau_y)
lead0[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z
lead0[lat.neighbors()] = -t * tau_z
Note the two new new arguments in `~kwant.builder.Builder`, ``conservation_law``
and ``particle_hole``. For the purpose of computing conductance, ``conservation_law``
......@@ -120,9 +198,19 @@ of ascending eigenvalues of the conservation law.
In order to move on with the conductance calculation, let's attach the second
lead to the right side of the scattering region:
.. literalinclude:: /code/include/superconductor.py
:start-after: #HIDDEN_BEGIN_zuuw
:end-before: #HIDDEN_END_zuuw
.. jupyter-execute::
# Right lead - superconducting, so the order parameter is included.
sym_right = kwant.TranslationalSymmetry((a, 0))
lead1 = kwant.Builder(sym_right)
lead1[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z + Delta * tau_x
lead1[lat.neighbors()] = -t * tau_z
#### Attach the leads and finalize the system. ####
syst.attach_lead(lead0)
syst.attach_lead(lead1)
syst = syst.finalized()
The second (right) lead is superconducting, such that the electron and hole
blocks are coupled. Of course, this means that we can not separate them into
......@@ -140,9 +228,23 @@ confused by the fact that it says ``transmission`` -- transmission
into the same lead is reflection), and reflection from electrons to holes
is ``smatrix.transmission((0, 1), (0, 0))``:
.. literalinclude:: /code/include/superconductor.py
:start-after: #HIDDEN_BEGIN_jbjt
:end-before: #HIDDEN_END_jbjt
.. jupyter-execute::
def plot_conductance(syst, energies):
# Compute conductance
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
# Conductance is N - R_ee + R_he
data.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] -
smatrix.transmission((0, 0), (0, 0)) +
smatrix.transmission((0, 1), (0, 0)))
pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
Note that ``smatrix.submatrix((0, 0), (0, 0))`` returns the block concerning
reflection of electrons to electrons, and from its size we can extract the number of modes
......@@ -150,7 +252,10 @@ reflection of electrons to electrons, and from its size we can extract the numbe
For the default parameters, we obtain the following conductance:
.. image:: /code/figure/superconductor_transport_result.*
.. jupyter-execute::
:hide-code:
plot_conductance(syst, energies=[0.002 * i for i in range(-10, 100)])
We a see a conductance that is proportional to the square of the tunneling
probability within the gap, and proportional to the tunneling probability
......@@ -176,13 +281,34 @@ the electron and hole blocks. The exception is of course at zero energy, in whic
case particle-hole symmetry transforms between the electron and hole blocks, resulting
in a symmetric scattering matrix. We can check the symmetry explicitly with
.. literalinclude:: /code/include/superconductor.py
:start-after: #HIDDEN_BEGIN_pqmp
:end-before: #HIDDEN_END_pqmp
.. jupyter-execute::
def check_PHS(syst):
# Scattering matrix
s = kwant.smatrix(syst, energy=0)
# Electron to electron block
s_ee = s.submatrix((0,0), (0,0))
# Hole to hole block
s_hh = s.submatrix((0,1), (0,1))
print('s_ee: \n', np.round(s_ee, 3))
print('s_hh: \n', np.round(s_hh[::-1, ::-1], 3))
print('s_ee - s_hh^*: \n',
np.round(s_ee - s_hh[::-1, ::-1].conj(), 3), '\n')
# Electron to hole block
s_he = s.submatrix((0,1), (0,0))
# Hole to electron block
s_eh = s.submatrix((0,0), (0,1))
print('s_he: \n', np.round(s_he, 3))
print('s_eh: \n', np.round(s_eh[::-1, ::-1], 3))
print('s_he + s_eh^*: \n',
np.round(s_he + s_eh[::-1, ::-1].conj(), 3))
which yields the output
.. literalinclude:: /code/figure/check_PHS_out.txt
.. jupyter-execute::
:hide-code:
check_PHS(syst)
Note that :math:`\mathcal{P}` flips the sign of momentum, and for the parameters
we consider here, there are two electron and two hole modes active at zero energy.
......
docker/Dockerfile.conda
View file @ bc9b2edf
......@@ -2,6 +2,7 @@ FROM conda/miniconda3:latest
ENV LANG C.UTF-8
ENV LC_ALL C.UTF-8
ENV DEBIAN_FRONTEND noninteractive
RUN apt-get update && apt-get install -y --no-install-recommends \
# all the hard non-Python dependencies
......@@ -9,7 +10,7 @@ RUN apt-get update && apt-get install -y --no-install-recommends \
# Additional tools for running CI
file rsync openssh-client \
# Documentation building
inkscape texlive-full zip \
inkscape texlive-full zip librsvg2-bin \
&& apt-get clean && \
rm -rf /var/lib/apt/lists/*
......@@ -18,3 +19,5 @@ COPY kwant-latest.yml kwant-stable.yml kwant-stable-no-extras.yml /
RUN conda env create -qf kwant-stable.yml
RUN conda env create -qf kwant-stable-no-extras.yml
RUN conda env create -qf kwant-latest.yml
RUN /usr/local/envs/kwant-latest/bin/python -m ipykernel install --user --name kwant-latest
docker/Dockerfile.debian
View file @ bc9b2edf
......@@ -2,11 +2,12 @@ FROM debian:latest
ENV LANG C.UTF-8
ENV LC_ALL C.UTF-8
ENV DEBIAN_FRONTEND noninteractive
RUN apt-get update && apt-get install -y --no-install-recommends \
gnupg dirmngr apt-transport-https ca-certificates curl software-properties-common
RUN echo "deb http://downloads.kwant-project.org/debian/ stretch-backports main" >> /etc/apt/sources.list && \
RUN echo "deb http://downloads.kwant-project.org/debian/ stable main" >> /etc/apt/sources.list && \
apt-key adv --no-tty --keyserver pool.sks-keyservers.net --recv-key C3F147F5980F3535 && \
apt-get update && apt-get install -y --no-install-recommends \
# all the hard non-Python dependencies
......@@ -20,6 +21,10 @@ RUN echo "deb http://downloads.kwant-project.org/debian/ stretch-backports main"
&& apt-get clean && \
rm -rf /var/lib/apt/lists/*
### install optional dependencies not available from the Debian repositories
RUN pip3 install \
qsymm==1.2.6
### install build and testing dependencies
RUN pip3 install \
cython \
......
docker/Dockerfile.ubuntu
View file @ bc9b2edf
FROM ubuntu:16.04
FROM ubuntu:18.04
ENV LANG C.UTF-8
ENV LC_ALL C.UTF-8
ENV DEBIAN_FRONTEND noninteractive
RUN apt-get update && apt-get install -y --no-install-recommends \
gnupg dirmngr apt-transport-https software-properties-common
......@@ -18,6 +19,10 @@ RUN apt-add-repository -s ppa:kwant-project/ppa && \
&& apt-get clean && \
rm -rf /var/lib/apt/lists/*
### install optional dependencies not available from the Debian repositories
RUN pip3 install \
qsymm==1.2.6
### install build and testing dependencies
RUN pip3 install \
cython \
......
docker/kwant-latest.yml
View file @ bc9b2edf
......@@ -8,6 +8,7 @@ dependencies:
- tinyarray
- sympy
- matplotlib
- qsymm
# Linear algebra libraraies
- mumps
- blas #=1.1 openblas
......@@ -23,6 +24,9 @@ dependencies:
- pytest-flakes
- pytest-pep8
# Documentation building
- sphinx=1.7.4 # later versions seem to have problems
- sphinx
- numpydoc
- requests
- jupyter_sphinx
- pip:
- sphinxcontrib-svg2pdfconverter
docker/kwant-stable-no-extras.yml
View file @ bc9b2edf
......@@ -2,9 +2,9 @@ name: kwant-stable-no-extras
channels:
- conda-forge
dependencies:
- python=3.5
- numpy=1.11.0
- scipy=0.17 # No minor version because conda-forge has no 0.17.0
- python=3.6
- numpy=1.13.1
- scipy=0.19.1
- tinyarray=1.2.0
# Linear algebra libraraies
- blas #=1.1 openblas
......@@ -19,7 +19,3 @@ dependencies:
- pytest-cov
- pytest-flakes
- pytest-pep8
# Documentation building
- sphinx=1.7.4 # later versions seem to have problems
- numpydoc
- requests
docker/kwant-stable.yml
View file @ bc9b2edf
......@@ -2,12 +2,13 @@ name: kwant-stable
channels:
- conda-forge
dependencies:
- python=3.5
- numpy=1.11.0
- scipy=0.17 # No minor version because conda-forge has no 0.17.0
- python=3.6
- numpy=1.13.3
- scipy=0.19.1
- tinyarray=1.2.0
- sympy=0.7.6
- matplotlib=1.5.1
- sympy=1.1.1
- matplotlib=2.1.1
- qsymm=1.2.6
# Linear algebra libraraies
- mumps
- blas #=1.1 openblas
......@@ -22,7 +23,3 @@ dependencies:
- pytest-cov
- pytest-flakes
- pytest-pep8
# Documentation building
- sphinx=1.7.4 # later versions seem to have problems
- numpydoc
- requests
kwant/__init__.py
View file @ bc9b2edf
......@@ -29,22 +29,29 @@ except ImportError:
raise
from ._common import KwantDeprecationWarning, UserCodeError
__all__.extend(['KwantDeprecationWarning', 'UserCodeError'])
for module in ['system', 'builder', 'lattice', 'solvers', 'digest', 'rmt',
'operator', 'kpm', 'wraparound']:
exec('from . import {0}'.format(module))
__all__.append(module)
# Pre-import most submodules. (We make exceptions for submodules that have
# special dependencies or where that would take too long.)
from . import system
from . import builder
from . import lattice
from . import solvers
from . import digest
from . import rmt
from . import operator
from . import kpm
from . import wraparound
__all__.extend(['system', 'builder', 'lattice', 'solvers', 'digest', 'rmt',
'operator', 'kpm', 'wraparound'])
# Make selected functionality available directly in the root namespace.
available = [('builder', ['Builder', 'HoppingKind']),
('lattice', ['TranslationalSymmetry']),
('solvers.default',
['smatrix', 'greens_function', 'ldos', 'wave_function'])]
for module, names in available:
exec('from .{0} import {1}'.format(module, ', '.join(names)))
__all__.extend(names)
from .builder import Builder, HoppingKind
__all__.extend(['Builder', 'HoppingKind'])
from .lattice import TranslationalSymmetry
__all__.extend(['TranslationalSymmetry'])
from .solvers.default import smatrix, greens_function, ldos, wave_function
__all__.extend(['smatrix', 'greens_function', 'ldos', 'wave_function'])
# Importing plotter might not work, but this does not have to be a problem --
# only no plotting will be available.
......
kwant/_common.py
View file @ bc9b2edf
......@@ -13,6 +13,7 @@ import inspect
import warnings
import importlib
import functools
import collections
from contextlib import contextmanager
__all__ = ['KwantDeprecationWarning', 'UserCodeError']
......@@ -191,3 +192,27 @@ class lazy_import:
package = sys.modules[self.__package]
setattr(package, self.__module, mod)
return getattr(mod, name)
def _hashable(obj):
return isinstance(obj, collections.abc.Hashable)
def memoize(f):
"""Decorator to memoize a function that works even with unhashable args.
This decorator will even work with functions whose args are not hashable.
The cache key is made up by the hashable arguments and the ids of the
non-hashable args. It is up to the user to make sure that non-hashable
args do not change during the lifetime of the decorator.
This decorator will keep reevaluating functions that return None.
"""
def lookup(*args):
key = tuple(arg if _hashable(arg) else id(arg) for arg in args)
result = cache.get(key)
if result is None:
cache[key] = result = f(*args)
return result
cache = {}
return lookup
kwant/builder.py
View file @ bc9b2edf
......@@ -13,6 +13,7 @@ import collections
import copy
from functools import total_ordering, wraps, update_wrapper
from itertools import islice, chain
import textwrap
import inspect
import tinyarray as ta
import numpy as np
......@@ -20,13 +21,14 @@ from scipy import sparse
from . import system, graph, KwantDeprecationWarning, UserCodeError
from .linalg import lll
from .operator import Density
from .physics import DiscreteSymmetry
from .physics import DiscreteSymmetry, magnetic_gauge
from ._common import (ensure_isinstance, get_parameters, reraise_warnings,
interleave, deprecate_args)
interleave, deprecate_args, memoize)
__all__ = ['Builder', 'Site', 'SiteFamily', 'SimpleSiteFamily', 'Symmetry',
'HoppingKind', 'Lead', 'BuilderLead', 'SelfEnergyLead', 'ModesLead']
'HoppingKind', 'Lead', 'BuilderLead', 'SelfEnergyLead', 'ModesLead',
'add_peierls_phase']
################ Sites and site families
......@@ -138,6 +140,10 @@ class SiteFamily(metaclass=abc.ABCMeta):
self.canonical_repr = canonical_repr
self.hash = hash(canonical_repr)
self.name = name
if norbs is None:
warnings.warn("Not specfying norbs is deprecated. Always specify "
"norbs when creating site families.",
KwantDeprecationWarning, stacklevel=3)
if norbs is not None:
if int(norbs) != norbs or norbs <= 0:
raise ValueError('The norbs parameter must be an integer > 0.')
......@@ -1807,6 +1813,169 @@ class Builder:
inter_cell_hopping = cell_hamiltonian = precalculated = \
_require_system
def _add_peierls_phase(syst, peierls_parameter):
@memoize
def wrap_hopping(hop):
def f(*args):
a, b, *args, phi = args
return hop(a, b, *args) * phi(a, b)
params = ('_site0', '_site1')
if callable(hop):
params += get_parameters(hop)[2:] # cut off both site parameters
params += (peierls_parameter,)
f.__signature__ = inspect.Signature(
inspect.Parameter(name, inspect.Parameter.POSITIONAL_ONLY)
for name in params)
return f
const_hoppings = {}
def const_hopping(a, b, phi):
return const_hoppings[a, b] * phi(a, b)
# fix the signature
params = ('_site0', '_site1', peierls_parameter)
const_hopping.__signature__ = inspect.Signature(
inspect.Parameter(name, inspect.Parameter.POSITIONAL_ONLY)
for name in params)
ret = Builder(
syst.symmetry,
# No time reversal, as magnetic fields break it
conservation_law=syst.conservation_law,
particle_hole=syst.particle_hole,
chiral=syst.chiral,
)
for i, lead in enumerate(syst.leads):
ret.leads.append(
BuilderLead(
_add_peierls_phase(
lead.builder,
peierls_parameter + '_lead{}'.format(i),
),
lead.interface,
lead.padding,
)
)
for site, val in syst.site_value_pairs():
ret[site] = val
for hop, val in syst.hopping_value_pairs():
if callable(val):
ret[hop] = wrap_hopping(val)
else:
const_hoppings[hop] = val
ret[hop] = const_hopping
return ret
def add_peierls_phase(syst, peierls_parameter='phi', fix_gauge=True):
"""Add a Peierls phase parameter to a Builder.
Parameters
----------
syst : kwant.Builder
The system to which to add a Peierls phase.
peierls_parameter : str
The name of the Peierls phase parameter to value functions.
fix_gauge : bool (default: True)
If True, fix the gauge of the system and return it also
Returns
-------
syst : kwant.builder.Finitesystem or kwant.builder.InfiniteSystem
A system where all hoppings are given by value functions
that have an additional parameter 'phi' that should be
the Peierls phase returned by `kwant.physics.magnetic_gauge`.
Any leads have similar parameters named 'phi_lead0',
'phi_lead1' etc.
gauge : callable
Only returned if 'fix_gauge' is True. Called with magnetic
field(s), and returns a parameter dictionary that can be
passed to the system as 'params' (see the example below).
Examples
--------
>>> import numpy as np
>>> import kwant
>>>
>>> syst = make_system()
>>> lead = make_lead()
>>> syst.attach_lead(lead)
>>> syst.attach_lead(lead.reversed())
>>>
>>> fsyst, phase = kwant.builder.add_peierls_phase(syst)
>>>
>>> def B_syst(pos):
>>> return np.exp(-np.sum(pos * pos))
>>>
>>> kwant.smatrix(fsyst, parameters=dict(t=-1, **phase(B_syst, 0, 0)))
"""
def wrap_gauge(gauge):
phase_names = [peierls_parameter]
phase_names += [peierls_parameter + '_lead{}'.format(i)
for i in range(len(syst.leads))]
doc = textwrap.dedent("""\
Return the Peierls phase for a magnetic field configuration.
Parameters
----------
syst_field : scalar, vector or callable
The magnetic field to apply to the scattering region.
If callable, takes a position and returns the
magnetic field at that position. Can be a scalar if
the system is 1D or 2D, otherwise must be a vector.
Magnetic field is expressed in units :math:`φ₀ / l²`,
where :math:`φ₀` is the magnetic flux quantum and
:math:`l` is the unit of length.
*lead_fields : scalar, vector or callable
The magnetic fields to apply to each of the leads, in
the same format as 'syst_field'. In addition, if a callable
is provided, then the magnetic field must have the symmetry
of the associated lead.
tol : float, default: 1E-8
The tolerance to which to calculate the flux through each
hopping loop in the system.
average : bool, default: False
If True, estimate the magnetic flux through each hopping loop
in the system by evaluating the magnetic field at a single
position inside the loop and multiplying it by the area of the
loop. If False, then ``scipy.integrate.quad`` is used to integrate
the magnetic field. This parameter is only used when 'syst_field'
or 'lead_fields' are callable.
Returns
-------
phases : dict of callables
To be passed directly as the parameters of a system wrapped with
'kwant.builder.add_peierls_phase'.
""")
@wraps(gauge)
def f(*args, **kwargs):
phases = gauge(*args, **kwargs)
return dict(zip(phase_names, phases))
f.__doc__ = doc
return f
ret = _add_peierls_phase(syst, peierls_parameter).finalized()
if fix_gauge:
return ret, wrap_gauge(magnetic_gauge(ret))
else:
return ret
################ Finalized systems
......
kwant/continuum/__init__.py
View file @ bc9b2edf
......@@ -10,11 +10,12 @@ try:
from .discretizer import discretize, discretize_symbolic, build_discretized
from ._common import sympify, lambdify
from ._common import momentum_operators, position_operators
from .landau_levels import to_landau_basis, discretize_landau, LandauLattice
except ImportError as error:
msg = ("'kwant.continuum' is not available because one or more of its "
"dependencies is not installed.")
raise ImportError(msg) from error
__all__ = ['discretize', 'discretize_symbolic', 'build_discretized',
'to_landau_basis', 'discretize_landau', 'LandauLattice',
'sympify', 'lambdify', 'momentum_operators', 'position_operators']
kwant/continuum/landau_levels.py 0 → 100644
View file @ bc9b2edf
# Copyright 2011-2019 Kwant authors.
#
# This file is part of Kwant. It is subject to the license terms in the file
# LICENSE.rst found in the top-level directory of this distribution and at
# http://kwant-project.org/license. A list of Kwant authors can be found in
# the file AUTHORS.rst at the top-level directory of this distribution and at
# http://kwant-project.org/authors.
from keyword import iskeyword
import functools
import operator
import collections
from math import sqrt
import tinyarray as ta
import numpy as np
import sympy
import kwant.lattice
import kwant.builder
import kwant.continuum
import kwant.continuum._common
import kwant.continuum.discretizer
from kwant.continuum import momentum_operators, position_operators
__all__ = ["to_landau_basis", "discretize_landau"]
coordinate_vectors = dict(zip("xyz", np.eye(3)))
ladder_lower, ladder_raise = sympy.symbols(r"a a^\dagger", commutative=False)
def to_landau_basis(hamiltonian, momenta=None):
r"""Replace two momenta by Landau level ladder operators.
Replaces:
k_0 -> sqrt(B/2) * (a + a^\dagger)
k_1 -> 1j * sqrt(B/2) * (a - a^\dagger)
Parameters
----------
hamiltonian : str or sympy expression
The Hamiltonian to which to apply the Landau level transformation.
momenta : sequence of str (optional)
The momenta to replace with Landau level ladder operators. If not
provided, 'k_x' and 'k_y' are used
Returns
-------
hamiltonian : sympy expression
momenta : sequence of sympy atoms
The momentum operators that have been replaced by ladder operators.
normal_coordinate : sympy atom
The remaining position coordinate. May or may not be present
in 'hamiltonian'.
"""
hamiltonian = kwant.continuum.sympify(hamiltonian)
momenta = _normalize_momenta(momenta)
normal_coordinate = _find_normal_coordinate(hamiltonian, momenta)
# Substitute ladder operators for Landau level momenta
B = sympy.symbols("B")
hamiltonian = hamiltonian.subs(
{
momenta[0]: sympy.sqrt(abs(B) / 2) * (ladder_raise + ladder_lower),
momenta[1]: sympy.I
* sympy.sqrt(abs(B) / 2)
* (ladder_lower - ladder_raise),
}
)
return hamiltonian, momenta, normal_coordinate
def discretize_landau(hamiltonian, N, momenta=None, grid_spacing=1):
"""Discretize a Hamiltonian in a basis of Landau levels.
Parameters
----------
hamiltonian : str or sympy expression
N : positive integer
The number of Landau levels in the basis.
momenta : sequence of str (optional)
The momenta defining the plane perpendicular to the magnetic field.
If not provided, "k_x" and "k_y" are used.
grid_spacing : float, default: 1
The grid spacing to use when discretizing the normal direction
(parallel to the magnetic field).
Returns
-------
builder : `~kwant.builder.Builder`
'hamiltonian' discretized in a basis of Landau levels in the plane
defined by 'momenta'. If a third coordinate is present in 'hamiltonian',
then this also has a translational symmetry in that coordinate direction.
The builder has a parameter 'B' in addition to any other parameters
present in the provided 'hamiltonian'.
Notes
-----
The units of magnetic field are :math:`ϕ₀ / 2 π a²` with :math:`ϕ₀ = h/e`
the magnetic flux quantum and :math:`a` the unit length.
"""
if N <= 0:
raise ValueError("N must be positive")
hamiltonian, momenta, normal_coordinate = to_landau_basis(hamiltonian, momenta)
# Discretize in normal direction and split terms for onsites/hoppings into
# monomials in ladder operators.
tb_hamiltonian, _ = kwant.continuum.discretize_symbolic(
hamiltonian, coords=[normal_coordinate.name]
)
tb_hamiltonian = {
key: kwant.continuum._common.monomials(value, gens=(ladder_lower, ladder_raise))
for key, value in tb_hamiltonian.items()
}
# Replace ladder operator monomials by tuple of integers:
# e.g. a^\dagger a^2 -> (+1, -2).
tb_hamiltonian = {
outer_key: {
_ladder_term(inner_key): inner_value
for inner_key, inner_value in outer_value.items()
}
for outer_key, outer_value in tb_hamiltonian.items()
}
# Construct map from LandauLattice HoppingKinds to a sequence of pairs
# that encode the ladder operators and multiplying expression.
tb_hoppings = collections.defaultdict(list)
for outer_key, outer_value in tb_hamiltonian.items():
for inner_key, inner_value in outer_value.items():
tb_hoppings[(*outer_key, sum(inner_key))] += [(inner_key, inner_value)]
# Extract the number of orbitals on each site/hopping
random_element = next(iter(tb_hoppings.values()))[0][1]
norbs = 1 if isinstance(random_element, sympy.Expr) else random_element.shape[0]
tb_onsite = tb_hoppings.pop((0, 0), None)
# Construct Builder
if _has_coordinate(normal_coordinate, hamiltonian):
sym = kwant.lattice.TranslationalSymmetry([grid_spacing, 0])
else:
sym = kwant.builder.NoSymmetry()
lat = LandauLattice(grid_spacing, norbs=norbs)
syst = kwant.Builder(sym)
# Add onsites
landau_sites = (lat(0, j) for j in range(N))
if tb_onsite is None:
syst[landau_sites] = ta.zeros((norbs, norbs))
else:
syst[landau_sites] = _builder_value(
tb_onsite, normal_coordinate.name, grid_spacing, is_onsite=True
)
# Add zero hoppings between adjacent Landau levels.
# Necessary to be able to use the Landau level builder
# to populate another builder using builder.fill().
syst[kwant.builder.HoppingKind((0, 1), lat)] = ta.zeros((norbs, norbs))
# Add the hoppings from the Hamiltonian
for hopping, parts in tb_hoppings.items():
syst[kwant.builder.HoppingKind(hopping, lat)] = _builder_value(
parts, normal_coordinate.name, grid_spacing, is_onsite=False
)
return syst
# This has to subclass lattice so that it will work with TranslationalSymmetry.
class LandauLattice(kwant.lattice.Monatomic):
"""
A `~kwant.lattice.Monatomic` lattice with a Landau level index per site.
Site tags (see `~kwant.builder.SiteFamily`) are pairs of integers, where
the first integer describes the real space position and the second the
Landau level index.
The real space Bravais lattice is one dimensional, oriented parallel
to the magnetic field. The Landau level index represents the harmonic
oscillator basis states used for the Landau quantization in the plane
normal to the field.
Parameters
----------
grid_spacing : float
Real space lattice spacing (parallel to the magnetic field).
offset : float (optional)
Displacement of the lattice origin from the real space
coordinates origin.
"""
def __init__(self, grid_spacing, offset=None, name="", norbs=None):
if offset is not None:
offset = [offset, 0]
# The second vector and second coordinate do not matter (they are
# not used in pos())
super().__init__([[grid_spacing, 0], [0, 1]], offset, name, norbs)
def pos(self, tag):
return ta.array((self.prim_vecs[0, 0] * tag[0] + self.offset[0],))
def landau_index(self, tag):
return tag[-1]
def _builder_value(terms, normal_coordinate, grid_spacing, is_onsite):
"""Construct an onsite/hopping value function from a list of terms
Parameters
----------
terms : list
Each element is a pair (ladder_term, sympy expression/matrix).
ladder_term is a tuple of integers that encodes a string of
Landau raising/lowering operators and the sympy expression
is the rest
normal_coordinate : str
grid_spacing : float
The grid spacing in the normal direction
is_onsite : bool
True if we are constructing an onsite value function
"""
ladder_term_symbols = [sympy.Symbol(_ladder_term_name(lt)) for lt, _ in terms]
# Construct a single expression from the terms, multiplying each part
# by the placeholder that represents the prefactor from the ladder operator term.
(ladder, (_, part)), *rest = zip(ladder_term_symbols, terms)
expr = ladder * part
for ladder, (_, part) in rest:
expr += ladder * part
expr = expr.subs(
{kwant.continuum.discretizer._displacements[normal_coordinate]: grid_spacing}
)
# Construct the return string and temporary variable names
# for function calls.
return_string, map_func_calls, const_symbols, _cache = kwant.continuum.discretizer._return_string(
expr, coords=[normal_coordinate]
)
# Remove the ladder term placeholders, as these are not parameters
const_symbols = set(const_symbols)
for ladder_term in ladder_term_symbols:
const_symbols.discard(ladder_term)
# Construct the argument part of signature. Arguments
# consist of any constants and functions in the return string.
arg_names = set.union(
{s.name for s in const_symbols}, {str(k.func) for k in map_func_calls}
)
arg_names.discard("Abs") # Abs function is not a parameter
for arg_name in arg_names:
if not (arg_name.isidentifier() and not iskeyword(arg_name)):
raise ValueError(
"Invalid name in used symbols: {}\n"
"Names of symbols used in Hamiltonian "
"must be valid Python identifiers and "
"may not be keywords".format(arg_name)
)
arg_names = ", ".join(sorted(arg_names))
# Construct site part of the function signature
site_string = "from_site" if is_onsite else "to_site, from_site"
# Construct function signature
if arg_names:
function_header = "def _({}, {}):".format(site_string, arg_names)
else:
function_header = "def _({}):".format(site_string)
# Construct function body
function_body = []
if "B" not in arg_names:
# B is not a parameter for terms with no ladder operators but we still
# need something to pass to _evaluate_ladder_term
function_body.append("B = +1")
function_body.extend(
[
"{}, = from_site.pos".format(normal_coordinate),
"_ll_index = from_site.family.landau_index(from_site.tag)",
]
)
# To get the correct hopping if B < 0, we need to set the Hermitian
# conjugate of the ladder operator matrix element, which swaps the
# from_site and to_site Landau level indices.
if not is_onsite:
function_body.extend(
["if B < 0:",
" _ll_index = to_site.family.landau_index(to_site.tag)"
]
)
function_body.extend(
"{} = _evaluate_ladder_term({}, _ll_index, B)".format(symb.name, lt)
for symb, (lt, _) in zip(ladder_term_symbols, terms)
)
function_body.extend(
"{} = {}".format(v, kwant.continuum.discretizer._print_sympy(k))
for k, v in map_func_calls.items()
)
function_body.append(return_string)
func_code = "\n ".join([function_header] + function_body)
# Add "I" to namespace just in case sympy again would miss to replace it
# with Python's 1j as it was the case with SymPy 1.2 when "I" was argument
# of some function.
namespace = {
"pi": np.pi,
"I": 1j,
"_evaluate_ladder_term": _evaluate_ladder_term,
"Abs": abs,
}
namespace.update(_cache)
# Construct full source, including cached arrays
source = []
for k, v in _cache.items():
source.append("{} = (\n{})\n".format(k, repr(np.array(v))))
source.append(func_code)
exec(func_code, namespace)
f = namespace["_"]
f._source = "".join(source)
return f
def _ladder_term(operator_string):
r"""Return a tuple of integers representing a string of ladder operators
Parameters
----------
operator_string : Sympy expression
Monomial in ladder operators, e.g. a^\dagger a^2 a^\dagger.
Returns
-------
ladder_term : tuple of int
e.g. a^\dagger a^2 -> (+1, -2)
"""
ret = []
for factor in operator_string.as_ordered_factors():
ladder_op, exponent = factor.as_base_exp()
if ladder_op == ladder_lower:
sign = -1
elif ladder_op == ladder_raise:
sign = +1
else:
sign = 0
ret.append(sign * int(exponent))
return tuple(ret)
def _ladder_term_name(ladder_term):
"""
Parameters
----------
ladder_term : tuple of int
Returns
-------
ladder_term_name : str
"""
def ns(i):
if i >= 0:
return str(i)
else:
return "_" + str(-i)
return "_ladder_{}".format("_".join(map(ns, ladder_term)))
def _evaluate_ladder_term(ladder_term, n, B):
r"""Evaluates the prefactor for a ladder operator on a landau level.
Example: a^\dagger a^2 -> (n - 1) * sqrt(n)
Parameters
----------
ladder_term : tuple of int
Represents a string of ladder operators. Positive
integers represent powers of the raising operator,
negative integers powers of the lowering operator.
n : non-negative int
Landau level index on which to act with ladder_term.
B : float
Magnetic field with sign
Returns
-------
ladder_term_prefactor : float
"""
assert n >= 0
# For negative B we swap a and a^\dagger.
if B < 0:
ladder_term = tuple(-i for i in ladder_term)
ret = 1
for m in reversed(ladder_term):
if m > 0:
factors = range(n + 1, n + m + 1)
elif m < 0:
factors = range(n + m + 1, n + 1)
if n == 0:
return 0 # a|0> = 0
else:
factors = (1,)
ret *= sqrt(functools.reduce(operator.mul, factors))
n += m
return ret
def _normalize_momenta(momenta=None):
"""Return Landau level momenta as Sympy atoms
Parameters
----------
momenta : None or pair of int or pair of str
The momenta to choose. If None then 'k_x' and 'k_y'
are chosen. If integers, then these are the indices
of the momenta: 0 → k_x, 1 → k_y, 2 → k_z. If strings,
then these name the momenta.
Returns
-------
The specified momenta as sympy atoms.
"""
# Specify which momenta to substitute for the Landau level basis.
if momenta is None:
# Use k_x and k_y by default
momenta = momentum_operators[:2]
else:
if len(momenta) != 2:
raise ValueError("Two momenta must be specified.")
k_names = [k.name for k in momentum_operators]
if all([type(i) is int for i in momenta]) and all(
[i >= 0 and i < 3 for i in momenta]
):
momenta = [momentum_operators[i] for i in momenta]
elif all([isinstance(momentum, str) for momentum in momenta]) and all(
[momentum in k_names for momentum in momenta]
):
momenta = [
momentum_operators[k_names.index(momentum)] for momentum in momenta
]
else:
raise ValueError("Momenta must all be integers or strings.")
return tuple(momenta)
def _find_normal_coordinate(hamiltonian, momenta):
discrete_momentum = next(
momentum for momentum in momentum_operators if momentum not in momenta
)
normal_coordinate = position_operators[momentum_operators.index(discrete_momentum)]
return normal_coordinate
def _has_coordinate(coord, expr):
momentum = momentum_operators[position_operators.index(coord)]
atoms = set(expr.atoms())
return coord in atoms or momentum in atoms
kwant/continuum/tests/test_discretizer.py
View file @ bc9b2edf
......@@ -548,14 +548,14 @@ def test_grid(ham, grid_spacing, grid):
@pytest.mark.parametrize('ham, grid_offset, offset, norbs', [
('k_x', None, 0, None),
('k_x', None, 0, 1),
('k_x', None, 0, 1),
('k_x * eye(2)', None, 0, 2),
('k_x', (0,), 0, None),
('k_x', (1,), 1, None),
('k_x + k_y', None, (0, 0), None),
('k_x + k_y', (0, 0), (0, 0), None),
('k_x + k_y', (1, 2), (1, 2), None),
('k_x', (0,), 0, 1),
('k_x', (1,), 1, 1),
('k_x + k_y', None, (0, 0), 1),
('k_x + k_y', (0, 0), (0, 0), 1),
('k_x + k_y', (1, 2), (1, 2), 1),
])
def test_grid_input(ham, grid_offset, offset, norbs):
# build appriopriate grid
......@@ -579,7 +579,7 @@ def test_grid_input(ham, grid_offset, offset, norbs):
def test_grid_offset_passed_to_functions():
V = lambda x: x
grid = Monatomic([[1, ]], offset=[0.5, ])
grid = Monatomic([[1, ]], offset=[0.5, ], norbs=1)
tb = discretize('V(x)', 'x', grid=grid)
onsite = tb[tb.lattice(0)]
bools = [np.allclose(onsite(tb.lattice(i), V), V(tb.lattice(i).pos))
......@@ -588,11 +588,11 @@ def test_grid_offset_passed_to_functions():
@pytest.mark.parametrize("ham, coords, grid", [
("k_x", None, Monatomic([[1, 0]])),
("k_x", 'xy', Monatomic([[1, 0]])),
("k_x", None, Monatomic([[1, 0]], norbs=1)),
("k_x", 'xy', Monatomic([[1, 0]], norbs=1)),
("k_x", None, Monatomic([[1, ]], norbs=2)),
("k_x * eye(2)", None, Monatomic([[1, ]], norbs=1)),
("k_x+k_y", None, Monatomic([[1, 0], [1, 1]])),
("k_x+k_y", None, Monatomic([[1, 0], [1, 1]], norbs=1)),
])
def test_grid_constraints(ham, coords, grid):
with pytest.raises(ValueError):
......@@ -606,7 +606,7 @@ def test_check_symbol_names(name):
def test_rectangular_grid():
lat = Monatomic([[1, 0], [0, 2]])
lat = Monatomic([[1, 0], [0, 2]], norbs=1)
tb = discretize("V(x, y)", 'xy', grid=lat)
assert np.allclose(tb[lat(0, 0)](lat(1, 0), lambda x, y: x), 1)
......
kwant/continuum/tests/test_landau_levels.py 0 → 100644
View file @ bc9b2edf
# Copyright 2011-2019 Kwant authors.
#
# This file is part of Kwant. It is subject to the license terms in the file
# LICENSE.rst found in the top-level directory of this distribution and at
# http://kwant-project.org/license. A list of Kwant authors can be found in
# the file AUTHORS.rst at the top-level directory of this distribution and at
# http://kwant-project.org/authors.
from math import sqrt
import numpy as np
import sympy
import pytest
import itertools
import kwant.builder
import kwant.lattice
from .._common import position_operators, momentum_operators, sympify
from ..landau_levels import (
to_landau_basis,
discretize_landau,
_ladder_term,
_evaluate_ladder_term,
ladder_lower,
ladder_raise,
LandauLattice,
)
x, y, z = position_operators
k_x, k_y, k_z = momentum_operators
B = sympy.symbols("B")
V = sympy.symbols("V", cls=sympy.Function)
a, a_dag = ladder_lower, ladder_raise
def test_ladder_term():
assert _ladder_term(a ** 2 * a_dag) == (-2, 1)
assert _ladder_term(a_dag ** 5 * a ** 3 * a_dag) == (5, -3, 1)
# non-ladder operators give a shift of 0
assert _ladder_term(B * a ** 2) == (0, -2)
def test_evaluate_ladder_term():
assert np.isclose(_evaluate_ladder_term((+1, -1), 1, +1), 1)
assert np.isclose(_evaluate_ladder_term((+1, -1), 2, +1), 2)
assert np.isclose(_evaluate_ladder_term((-2, +3, -2), 5, +1), 4 * 5 * 6 * sqrt(5))
# annihilating |0> is always 0
assert _evaluate_ladder_term((-1,), 0, +1) == 0
assert _evaluate_ladder_term((-2,), 1, +1) == 0
assert _evaluate_ladder_term((-3,), 1, +1) == 0
assert _evaluate_ladder_term((+3, -2), 1, +1) == 0
assert _evaluate_ladder_term((-3, -2, +3), 1, +1) == 0
# negative B swaps creation and annihilation operators
assert _evaluate_ladder_term((+1, -1), 2, +1) == _evaluate_ladder_term(
(-1, +1), 2, -1
)
assert _evaluate_ladder_term((-2, +3, -2), 5, +1) == _evaluate_ladder_term(
(+2, -3, +2), 5, -1
)
def test_to_landau_basis():
# test basic usage
ham, momenta, normal_coord = to_landau_basis("k_x**2 + k_y**2")
assert sympy.expand(ham) == abs(B) * a * a_dag + abs(B) * a_dag * a
assert momenta == (k_x, k_y)
assert normal_coord == z
# test that hamiltonian can be specified as a sympy expression
ham, momenta, normal_coord = to_landau_basis(sympify("k_x**2 + k_y**2"))
assert sympy.expand(ham) == abs(B) * a * a_dag + abs(B) * a_dag * a
assert momenta == (k_x, k_y)
assert normal_coord == z
# test that
ham, momenta, normal_coord = to_landau_basis("k_x**2 + k_y**2 + k_z**2 + V(z)")
assert sympy.expand(ham) == (
abs(B) * a * a_dag + abs(B) * a_dag * a + k_z ** 2 + V(z)
)
assert momenta == (k_x, k_y)
assert normal_coord == z
# test for momenta explicitly specified
ham, momenta, normal_coord = to_landau_basis(
"k_x**2 + k_y**2 + k_z**2 + k_x*k_y", momenta=("k_z", "k_y")
)
assert sympy.expand(ham) == (
abs(B) * a * a_dag
+ abs(B) * a_dag * a
+ k_x ** 2
+ sympy.I * sympy.sqrt(abs(B) / 2) * k_x * a
- sympy.I * sympy.sqrt(abs(B) / 2) * k_x * a_dag
)
assert momenta == (k_z, k_y)
assert normal_coord == x
def test_discretize_landau():
n_levels = 10
magnetic_field = 1 / 3 # a suitably arbitrary value
# Ensure that we can handle products of ladder operators by truncating
# several levels higher than the number of levels we actually want.
a = np.diag(np.sqrt(np.arange(1, n_levels + 5)), k=1)
a_dag = a.conjugate().transpose()
k_x = sqrt(magnetic_field / 2) * (a + a_dag)
k_y = 1j * sqrt(magnetic_field / 2) * (a - a_dag)
sigma_0 = np.eye(2)
sigma_x = np.array([[0, 1], [1, 0]])
sigma_y = np.array([[0, -1j], [1j, 0]])
# test that errors are raised on invalid input
with pytest.raises(ValueError):
discretize_landau("k_x**2 + k_y**2", N=0)
with pytest.raises(ValueError):
discretize_landau("k_x**2 + k_y**2", N=-1)
# test a basic Hamiltonian with no normal coordinate dependence
syst = discretize_landau("k_x**2 + k_y**2", N=n_levels)
lat = LandauLattice(1, norbs=1)
assert isinstance(syst.symmetry, kwant.builder.NoSymmetry)
syst = syst.finalized()
assert set(syst.sites) == {lat(0, j) for j in range(n_levels)}
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=0)), np.zeros((n_levels, n_levels))
)
should_be = k_x @ k_x + k_y @ k_y
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=magnetic_field)),
should_be[:n_levels, :n_levels],
)
# test negative magnetic field
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=-magnetic_field)),
should_be[:n_levels, :n_levels],
)
# test hamiltonian with no onsite elements
syst = discretize_landau("k_x", N=n_levels)
syst = syst.finalized()
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=magnetic_field)),
k_x[:n_levels, :n_levels],
)
# test a basic Hamiltonian with normal coordinate dependence
grid = 1 / 7 # a suitably arbitrary value
syst = discretize_landau(
"k_x**2 + k_y**2 + k_z**2 + V(z)", N=n_levels, grid_spacing=grid
)
assert isinstance(syst.symmetry, kwant.lattice.TranslationalSymmetry)
syst = syst.finalized()
zero_potential = syst.cell_hamiltonian(params=dict(B=magnetic_field, V=lambda z: 0))
with_potential = syst.cell_hamiltonian(params=dict(B=magnetic_field, V=lambda z: 1))
# extra +2/grid**2 from onsite kinetic energy
assert np.allclose(
zero_potential,
should_be[:n_levels, :n_levels] + (2 / grid ** 2) * np.eye(n_levels),
)
# V(z) just adds the same energy to each onsite
assert np.allclose(with_potential - zero_potential, np.eye(n_levels))
# hopping matrix does not exchange landau levels
assert np.allclose(
syst.inter_cell_hopping(params=dict(B=magnetic_field, V=lambda z: 0)),
-np.eye(n_levels) / grid ** 2,
)
# test a Hamiltonian with mixing between Landau levels
# and spatial degrees of freedom.
syst = discretize_landau("k_x**2 + k_y**2 + k_x*k_z", N=n_levels)
syst = syst.finalized()
assert np.allclose(
syst.inter_cell_hopping(params=dict(B=magnetic_field)),
-1j * k_x[:n_levels, :n_levels] / 2,
)
# test a Hamiltonian with extra degrees of freedom
syst = discretize_landau("sigma_0 * k_x**2 + sigma_x * k_y**2", N=n_levels)
syst = syst.finalized()
assert syst.sites[0].family.norbs == 2
should_be = np.kron(k_x @ k_x, sigma_0) + np.kron(k_y @ k_y, sigma_x)
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=magnetic_field)),
should_be[: 2 * n_levels, : 2 * n_levels],
)
# test a linear Hamiltonian
syst = discretize_landau("sigma_y * k_x - sigma_x * k_y", N=n_levels)
syst = syst.finalized()
should_be = np.kron(k_x, sigma_y) - np.kron(k_y, sigma_x)
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=magnetic_field)),
should_be[: 2 * n_levels, : 2 * n_levels],
)
assert np.allclose(
syst.hamiltonian_submatrix(params=dict(B=magnetic_field)),
syst.hamiltonian_submatrix(params=dict(B=-magnetic_field)),
)
def test_analytical_spectrum():
hamiltonian = """(k_x**2 + k_y**2) * sigma_0 +
alpha * (k_x * sigma_y - k_y * sigma_x) +
g * B * sigma_z"""
def exact_Es(n, B, alpha, g):
# See e.g. R. Winkler (2003), section 8.4.1
sign_B = np.sign(B)
B = np.abs(B)
Ep = 2*B*(n+1) - 0.5*np.sqrt((2*B - sign_B*2*g*B)**2 + 8*B*alpha**2*(n+1))
Em = 2*B*n + 0.5*np.sqrt((2*B - sign_B*2*g*B)**2 + 8*B*alpha**2*n)
return Ep, Em
N = 20
syst = discretize_landau(hamiltonian, N)
syst = syst.finalized()
params = dict(alpha=0.07, g=0.04)
for _ in range(5):
B = 0.01 + np.random.rand()*3
params['B'] = B
exact = [exact_Es(n, B, params['alpha'], params['g']) for n in range(N)]
# We only check the first N levels - the SOI couples adjacent levels,
# so the higher ones are not necessarily determined accurately in the
# discretization
exact = np.sort([energy for energies in exact for energy in energies])[:N]
ll_spect = np.linalg.eigvalsh(syst.hamiltonian_submatrix(params=params))[:len(exact)]
assert np.allclose(ll_spect, exact)
def test_fill():
def shape(site, lower, upper):
(z, )= site.pos
return lower <= z < upper
hamiltonian = "k_x**2 + k_y**2 + k_z**2"
N = 6
template = discretize_landau(hamiltonian, N)
syst = kwant.Builder()
width = 4
syst.fill(template, lambda site: shape(site, 0, width), (0, ));
correct_tags = [(coordinate, ll_index) for coordinate, ll_index
in itertools.product(range(width), range(N))]
syst_tags = [site.tag for site in syst.sites()]
assert len(syst_tags) == len(correct_tags)
assert all(tag in correct_tags for tag in syst_tags)