Commit 0f60badf authored by Pablo Piskunow's avatar Pablo Piskunow

update kpm tutorial

parent 3b75da63
......@@ -40,36 +40,89 @@
return syst
#HIDDEN_END_sys1
#HIDDEN_BEGIN_sys2
# define a Haldane system
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()
#HIDDEN_END_sys2
#HIDDEN_BEGIN_sys3
# define the system
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
#HIDDEN_END_sys3
# Plot several density of states curves on the same axes.
-def plot_dos(labels_to_data):
+def plot_dos(labels_to_data, file_name=None):
+def plot_dos(labels_to_data, file_name=None, ylabel="DoS [a.u.]"):
plt.figure(figsize=(5,4))
for label, (x, y) in labels_to_data:
plt.plot(x, y, label=label, linewidth=2)
plt.plot(x, y.real, label=label, linewidth=2)
plt.legend(loc=2, framealpha=0.5)
plt.xlabel("energy [t]")
plt.ylabel("DoS [a.u.]")
plt.ylabel(ylabel)
- plt.show()
+ save_figure(file_name)
plt.clf()
# Plot fill density of states plus curves on the same axes.
-def plot_dos_and_curves(dos labels_to_data):
+def plot_dos_and_curves(dos, labels_to_data, file_name=None, ylabel="DoS [a.u.]"):
plt.figure(figsize=(5,4))
plt.fill_between(dos[0], dos[1], label="DoS [a.u.]",
alpha=0.5, color='gray')
for label, (x, y) in labels_to_data:
plt.plot(x, y, label=label, linewidth=2)
plt.legend(loc=2, framealpha=0.5)
plt.xlabel("energy [t]")
plt.ylabel(ylabel)
- plt.show()
+ save_figure(file_name)
plt.clf()
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(fsyst, axes, titles_to_data, file_name=None):
for ax, (title, ldos) in zip(axes, titles_to_data):
site_size = site_size_conversion(ldos) # convert LDoS to sizes
kwant.plot(fsyst, site_size=site_size, site_color=(0, 0, 1, 0.3), ax=ax)
def plot_ldos(syst, densities, file_name=None):
fig, axes = plt.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')
- plt.show()
+ save_figure(file_name)
plt.clf()
+def save_figure(file_name):
+ if not file_name:
+ return
......@@ -187,11 +240,9 @@
#HIDDEN_BEGIN_op4
zero_energy_ldos = local_dos(energy=0)
finite_energy_ldos = local_dos(energy=1)
_, axes = plt.subplots(1, 2, figsize=(12, 7))
plot_ldos(fsyst, axes,[
plot_ldos(fsyst, [
('energy = 0', zero_energy_ldos),
('energy = 1', finite_energy_ldos),
('energy = 1', finite_energy_ldos)
- ])
+ ],
+ file_name='kpm_ldos'
......@@ -199,15 +250,43 @@
#HIDDEN_END_op4
def ldos_sites_example():
fsyst = make_syst_staggered().finalized()
#HIDDEN_BEGIN_op5
# 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, where)
#HIDDEN_END_op5
#HIDDEN_BEGIN_op6
# 'num_vectors' can be unspecified when using 'LocalVectors'
local_dos = kwant.kpm.SpectralDensity(fsyst, num_vectors=None,
vector_factory=vector_factory,
mean=False)
energies, densities = local_dos()
plot_dos([
('A sublattice', (energies, densities[:, 0])),
('B sublattice', (energies, densities[:, 1])),
- ])
+ ],
+ file_name='kpm_ldos_sites'
+ )
#HIDDEN_END_op6
def vector_factory_example(fsyst):
spectrum = kwant.kpm.SpectralDensity(fsyst)
#HIDDEN_BEGIN_fact1
# construct vectors with n random elements -1 or +1.
def binary_vectors(n):
return np.rint(np.random.random_sample(n)) * 2 - 1
# 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)
vector_factory=binary_vectors())
#HIDDEN_END_fact1
plot_dos([
('default vector factory', spectrum()),
......@@ -232,6 +311,47 @@
('bilinear operator', rho_alt_spectrum()),
])
def conductivity_example():
#HIDDEN_BEGIN_cond
# construct the Haldane model
lat, fsyst = 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) < 3
# component 'xx'
s_factory = kwant.kpm.LocalVectors(fsyst, where)
cond_xx = kwant.kpm.conductivity(fsyst, alpha='x', beta='x', mean=True,
num_vectors=None, vector_factory=s_factory)
# component 'xy'
s_factory = kwant.kpm.LocalVectors(fsyst, where)
cond_xy = kwant.kpm.conductivity(fsyst, alpha='x', beta='y', mean=True,
num_vectors=None, vector_factory=s_factory)
energies = cond_xx.energies
cond_array_xx = np.array([cond_xx(e, temp=0.01) for e in energies])
cond_array_xy = np.array([cond_xy(e, temp=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
#HIDDEN_END_cond
# ldos
s_factory = kwant.kpm.LocalVectors(fsyst, where)
spectrum = kwant.kpm.SpectralDensity(fsyst, num_vectors=None,
vector_factory=s_factory)
plot_dos_and_curves(
(spectrum.energies, spectrum.densities * 8),
[
(r'Longitudinal conductivity $\sigma_{xx} / 4$',
(energies, cond_array_xx / 4)),
(r'Hall conductivity $\sigma_{xy}$',
(energies, cond_array_xy))],
ylabel=r'$\sigma [e^2/h]$',
file_name='kpm_cond'
)
def main():
simple_dos_example()
......@@ -242,8 +362,10 @@
increasing_accuracy_example(fsyst)
operator_example(fsyst)
ldos_example(fsyst)
ldos_sites_example()
vector_factory_example(fsyst)
bilinear_map_operator_example(fsyst)
conductivity_example()
# Call the main function if the script gets executed (as opposed to imported).
......
......@@ -13,19 +13,49 @@ eigenenergies and eigenstates, but more in the *density of states*.
The kernel polynomial method (KPM), is an algorithm to obtain a polynomial
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 that is based on the algorithms presented in Ref. [1]_.
Roughly speaking, KPM approximates the density of states (or any other spectral
density) by expanding the action of the Hamiltonian (and operator of interest)
on a (small) set of *random vectors* as a sum of Chebyshev polynomials up to
some order, and then averaging. The accuracy of the method can be tuned by
modifying the order of the Chebyshev expansion and the number of random
vectors. See notes on accuracy_ below for details.
KPM method `kwant.kpm`, that is based on the algorithms presented in Ref. [1]_.
.. seealso::
The complete source code of this example can be found in
:download:`kernel_polynomial_method.py </code/download/kernel_polynomial_method.py>`
Introduction
************
Our aim is to use the kernel polynomial method to obtain the spectral density
:math:`ρ_A(E)`, as a function of the energy :math:`E`, of some Hilbert space
operator :math:`A`. We define
.. math::
ρ_A(E) = ρ(E) A(E),
where :math:`A(E)` is the expectation value of :math:`A` for all the
eigenstates of the Hamiltonian with energy :math:`E`, and the density of
states is
.. math::
ρ(E) = \sum_{k=0}^{D-1} δ(E-E_k) = \mathrm{Tr}\left(\delta(E-H)\right),
where :math:`H` is the Hamiltonian of the system, :math:`D` the
Hilbert space dimension, and :math:`E_k` the eigenvalues.
In the special case when :math:`A` is the identity, then :math:`ρ_A(E)` is
simply :math:`ρ(E)`, the density of states.
Calculating the density of states
*********************************
Roughly speaking, KPM approximates the density of states, or any other spectral
density, by expanding the action of the Hamiltonian and operator of interest
on a small set of *random vectors* (or *local vectors* for local density of
states), as a sum of Chebyshev polynomials up to some order, and then averaging.
The accuracy of the method can be tuned by modifying the order of the Chebyshev
expansion and the number of vectors. See notes on accuracy_ below for details.
.. _accuracy:
.. specialnote:: Performance and accuracy
......@@ -48,37 +78,28 @@ vectors. See notes on accuracy_ below for details.
number of random vectors is in general enough, and increasing this number
will not result in a visible improvement of the approximation.
Introduction
************
Our aim is to use the kernel polynomial method to obtain the spectral density
:math:`ρ_A(E)`, as a function of the energy :math:`E`, of some Hilbert space
operator :math:`A`. We define
The global *spectral density* :math:`ρ_A(E)` is approximated by the stochastic
trace, the average expectation value of random vectors :math:`r`
.. math::
ρ_A(E) = ρ(E) A(E),
ρ_A(E) = \mathrm{Tr}\left(A\delta(E-H)\right) \sim \frac{1}{R}
\sum_r \langle r \lvert A \delta(E-H) \rvert r \rangle,
where :math:`A(E)` is the expectation value of :math:`A` for all the
eigenstates of the Hamiltonian with energy :math:`E`, and the density of
states is
while the *local spectral density* for a site :math:`i` is
.. math::
ρ(E) = \sum_{k=0}^{D-1} δ(E-E_k),
ρ^i_A(E) = \langle i \lvert A \delta(E-H) \rvert i \rangle,
:math:`D` being the Hilbert space dimension, and :math:`E_k` the eigenvalues.
which is an exact expression.
In the special case when :math:`A` is the identity, then :math:`ρ_A(E)` is
simply :math:`ρ(E)`, the density of states.
Calculating the density of states
*********************************
Global spectral densities using random vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In the following example, we will use the KPM implementation in Kwant
to obtain the density of states of a graphene disk.
to obtain the (global) density of states of a graphene disk.
We start by importing kwant and defining our system.
......@@ -138,7 +159,7 @@ respect to a Fermi-Dirac distribution:
.. specialnote:: Stability and performance: spectral bounds
The KPM method internally rescales the spectrum of the Hamiltonian to the
interval ``(-1, 1)`` (see Ref [1]_ for details), which requires calculating
interval ``(-1, 1)`` (see Ref. [1]_ for details), which requires calculating
the boundaries of the spectrum (using ``scipy.sparse.linalg.eigsh``). This
can be very costly for large systems, so it is possible to pass this
explicitly as via the ``bounds`` parameter when instantiating the
......@@ -151,6 +172,49 @@ respect to a Fermi-Dirac distribution:
``scipy.sparse.linalg.eigsh`` is set to ``epsilon/2``.
Local spectral densities using local vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The *local density of states* can be obtained without
using random vectors, and using local vectors instead. This approach is best
when we want to estimate the local density on a small number of sites of
the system. The accuracy of this approach depends only on the number of
moments, but the computational cost increases linearly with the number of
sites sampled.
To output local densities for each local vector, and not the average,
we set the parameter ``mean=False``, and the local vectors will be created
with the `~kwant.kpm.LocalVectors` generator (see advanced_topics_ for details).
The spectral density can be restricted to the expectation value inside
a region of the system by passing a ``where`` function or list of sites
to the `~kwant.kpm.RandomVectors` or `~kwant.kpm.LocalVectors` generators.
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
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
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
.. image:: /code/figure/kpm_ldos_sites.*
Note that there is no noise comming from the random vectors.
Increasing the accuracy of the approximation
********************************************
......@@ -169,10 +233,9 @@ This will update the number of calculated moments and also the default
number of sampling points such that the maximum distance between successive
energy points is ``energy_resolution`` (see notes on accuracy_).
.. image:: /code/figure/kpm_dos_acc.*
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
......@@ -183,6 +246,7 @@ with ``add_moments``, or the number of random vectors with ``add_vectors``.
.. _operator_spectral_density:
Calculating the spectral density of an operator
***********************************************
......@@ -211,6 +275,10 @@ Or, to do the same calculation using `kwant.operator.Density`:
:start-after: #HIDDEN_BEGIN_op2
:end-before: #HIDDEN_END_op2
Spectral density with random vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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:
......@@ -220,8 +288,11 @@ sum over all the sites of the system:
:end-before: #HIDDEN_END_op3
`~kwant.kpm.SpectralDensity` will properly handle this vector output,
which allows us to plot the local density of states at different
point in the spectrum:
and will average the local density obtained with random vectors.
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
......@@ -229,18 +300,69 @@ point in the spectrum:
.. image:: /code/figure/kpm_ldos.*
This nicely illustrates the edge states of the graphene dot at zero
energy, and the bulk states at higher energy.
Calculating Kubo conductivity
*****************************
The Kubo conductivity can be calculated for a closed system with two
KPM expansions. In `~kwant.kpm.Conductivity` we implemented the
Kubo-Bastin formula of the conductivity and any temperature (see Ref. [2]_).
With the help of `~kwant.kpm.Conductivity`,
we can calculate any element of the conductivity tensor :math:`σ_{αβ}`,
that relates the applied electric field to the expected current.
.. math::
j_α = σ_{α, β} E_β
In the following example, we will calculate the longitudinal
conductivity :math:`σ_{xx}` and the Hall conductivity
:math:`σ_{xy}`, for the Haldane model. This model is the first
and one of the most simple ones for a topological insulator.
The Haldane model consist of a honeycomb lattice, similar to graphene,
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
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
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
covers an area of half a unit cell, while the sum covers one unit cell.
It is possible to use random vectors to get an average spectation
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.*
.. _advanced_topics:
Advanced topics
***************
Custom distributions for random vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Custom distributions of vectors
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
By default `~kwant.kpm.SpectralDensity` will use random vectors
whose components are unit complex numbers with phases drawn
from a uniform distribution. There are several reasons why you may
from a uniform distribution. The generator is accesible through
`~kwant.kpm.RandomVectors`.
For large systems, one will generally resort to random vectors to sample the
Hilbert space of the system. There are several reasons why you may
wish to make a different choice of distribution for your random vectors,
for example to enforce certain symmetries or to only use real-valued vectors.
......@@ -252,6 +374,16 @@ and which returns a vector in that Hilbert space:
:start-after: #HIDDEN_BEGIN_fact1
:end-before: #HIDDEN_END_fact1
Aditionally, a `~kwant.kpm.LocalVectors` generator is also available, that
returns local vectors that correspond to the sites passed. Note that
the vectors generated match the sites in order, and will be exhausted
after all vectors are drawn from the factory.
Both `~kwant.kpm.RandomVectors` and `~kwant.kpm.LocalVectors` take the
argument ``where``, that restricts the non zero values of the vectors
generated to the sites in ``where``.
Reproducible calculations
^^^^^^^^^^^^^^^^^^^^^^^^^
Because KPM internally uses random vectors, running the same calculation
......@@ -286,3 +418,5 @@ __ operator_spectral_density_
.. [1] `Rev. Mod. Phys., Vol. 78, No. 1 (2006)
<https://arxiv.org/abs/cond-mat/0504627>`_.
.. [2] `Phys. Rev. Lett. 114, 116602 (2015)
<https://arxiv.org/abs/1410.8140>`_.
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