From 4035315d4cc5f76c7b342a9aeb9ccb8e8f62ee53 Mon Sep 17 00:00:00 2001
From: Joseph Weston <joseph@weston.cloud>
Date: Tue, 28 May 2019 16:44:12 +0200
Subject: [PATCH] add tutorial
---
doc/source/tutorial/discretize.rst | 2 +
doc/source/tutorial/index.rst | 1 +
doc/source/tutorial/magnetic_field.rst | 232 +++++
ll.ipynb | 1080 ------------------------
4 files changed, 235 insertions(+), 1080 deletions(-)
create mode 100644 doc/source/tutorial/magnetic_field.rst
delete mode 100644 ll.ipynb
diff --git a/doc/source/tutorial/discretize.rst b/doc/source/tutorial/discretize.rst
index 9417d20b..51f94b79 100644
--- a/doc/source/tutorial/discretize.rst
+++ b/doc/source/tutorial/discretize.rst
@@ -217,6 +217,8 @@ In addition, the function passed as ``V`` expects two input parameters ``x``
and ``y``, the same as in the initial continuum Hamiltonian.
+.. _discretize-bhz-model:
+
Models with more structure: Bernevig-Hughes-Zhang
.................................................
diff --git a/doc/source/tutorial/index.rst b/doc/source/tutorial/index.rst
index 4fd4d188..0c361fcc 100644
--- a/doc/source/tutorial/index.rst
+++ b/doc/source/tutorial/index.rst
@@ -12,4 +12,5 @@ Tutorial: learning Kwant through examples
plotting
kpm
discretize
+ magnetic_field
faq
diff --git a/doc/source/tutorial/magnetic_field.rst b/doc/source/tutorial/magnetic_field.rst
new file mode 100644
index 00000000..6bb6fdcd
--- /dev/null
+++ b/doc/source/tutorial/magnetic_field.rst
@@ -0,0 +1,232 @@
+Adding magnetic field
+---------------------
+
+Computing Landau levels in a harmonic oscillator basis
+......................................................
+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{1}{\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()
+ 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.eigvals(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.
\ No newline at end of file
diff --git a/ll.ipynb b/ll.ipynb
deleted file mode 100644
index b6f7dda7..00000000
--- a/ll.ipynb
+++ /dev/null
@@ -1,1080 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "code",
- "execution_count": 1,
- "metadata": {},
- "outputs": [
- {
- "name": "stderr",
- "output_type": "stream",
- "text": [
- "/home/tinkerer/kwant/kwant/solvers/default.py:18: RuntimeWarning: MUMPS is not available, SciPy built-in solver will be used as a fallback. Performance can be very poor in this case.\n",
- " \"Performance can be very poor in this case.\", RuntimeWarning)\n"
- ]
- }
- ],
- "source": [
- "import numpy as np\n",
- "import sympy\n",
- "import operator\n",
- "import inspect\n",
- "import itertools\n",
- "import functools\n",
- "from copy import deepcopy\n",
- "\n",
- "import kwant\n",
- "import kwant.continuum"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 2,
- "metadata": {},
- "outputs": [],
- "source": [
- "%matplotlib inline\n",
- "import sympy\n",
- "import matplotlib.pyplot as plt\n",
- "\n",
- "sympy.init_printing(print_builtin=True)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 3,
- "metadata": {},
- "outputs": [],
- "source": [
- "from kwant.continuum import landau_levels"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [],
- "source": [
- "builder = kwant.continuum.discretize('k_x**2 + k_y**2 + V(x, y) + sqrt(B)')"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 8,
- "metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "# Discrete coordinates: x y\n",
- "\n",
- "# Onsite element:\n",
- "def onsite(site, B, V):\n",
- " (x, y, ) = site.pos\n",
- " _const_0 = (V(x, y))\n",
- " return (B**(1/2) + _const_0 + 4.0)\n",
- "\n",
- "# Hopping from (1, 0):\n",
- "(-1+0j)\n",
- "\n",
- "# Hopping from (0, 1):\n",
- "(-1+0j)\n"
- ]
- }
- ],
- "source": [
- "print(builder)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "metadata": {},
- "outputs": [],
- "source": [
- "a_dag = sympy.symbols('a^\\dagger')"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "iVBORw0KGgoAAAANSUhEUgAAABIAAAATBAMAAABvvEDBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90yVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdUlEQVQIHWNggIDnUJqBAcEyg4mtWBQLY7olwFhvYAyGtVhYb+FiIPMYlVXCGBh0gKxSAY4PDAyGDAxskQxMCxgY3BgYmL4xcDswMAgwMLAuYMg/wAAC+QYM98EMhvwLDC94gXIMDFwOnPuYwYJsT3zk3EEsAMsTFcMrnqiMAAAAAElFTkSuQmCC\n",
- "text/latex": [
- "$$a^\\dagger$$"
- ],
- "text/plain": [
- "a__\\dagger"
- ]
- },
- "execution_count": 36,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "a_dag"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 24,
- "metadata": {},
- "outputs": [],
- "source": [
- "chain = kwant.lattice.chain()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 25,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "kwant.builder.Site"
- ]
- },
- "execution_count": 25,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "type(chain(1))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "Site(kwant.lattice.Monatomic([[1.0]], [0.0], '', None), array([2]))"
- ]
- },
- "execution_count": 26,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "chain(2)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 17,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "Site(kwant.lattice.Monatomic([[1.0]], [0.0], '', None), array([1]))"
- ]
- },
- "execution_count": 17,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "kwant.builder.Site(chain, (1,))"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "metadata": {},
- "outputs": [],
- "source": [
- "s = kwant.builder.Site(chain, (40,)) # Second argument is the tag, for the chain its the position."
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 30,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "array([40.0])"
- ]
- },
- "execution_count": 30,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "s.pos"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 15,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "\u001b[0;31mInit signature:\u001b[0m \u001b[0mkwant\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbuilder\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mSite\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfamily\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtag\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_i_know_what_i_do\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;31mDocstring:\u001b[0m \n",
- "A site, member of a `SiteFamily`.\n",
- "\n",
- "Sites are the vertices of the graph which describes the tight binding\n",
- "system in a `Builder`.\n",
- "\n",
- "A site is uniquely identified by its family and its tag.\n",
- "\n",
- "Parameters\n",
- "----------\n",
- "family : an instance of `SiteFamily`\n",
- " The 'type' of the site.\n",
- "tag : a hashable python object\n",
- " The unique identifier of the site within the site family, typically a\n",
- " vector of integers.\n",
- "\n",
- "Raises\n",
- "------\n",
- "ValueError\n",
- " If `tag` is not a proper tag for `family`.\n",
- "\n",
- "Notes\n",
- "-----\n",
- "For convenience, ``family(*tag)`` can be used instead of ``Site(family,\n",
- "tag)`` to create a site.\n",
- "\n",
- "The parameters of the constructor (see above) are stored as instance\n",
- "variables under the same names. Given a site ``site``, common things to\n",
- "query are thus ``site.family``, ``site.tag``, and ``site.pos``.\n",
- "\u001b[0;31mFile:\u001b[0m ~/kwant/kwant/builder.py\n",
- "\u001b[0;31mType:\u001b[0m type\n",
- "\u001b[0;31mSubclasses:\u001b[0m \n"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
- "source": [
- "kwant.builder.Site?"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 12,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "\u001b[0;31mSignature:\u001b[0m \u001b[0mchain\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mtag\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
- "\u001b[0;31mType:\u001b[0m Monatomic\n",
- "\u001b[0;31mString form:\u001b[0m <unnamed Monatomic lattice, vectors [1.0], origin [0.0]>\n",
- "\u001b[0;31mFile:\u001b[0m ~/kwant/kwant/lattice.py\n",
- "\u001b[0;31mDocstring:\u001b[0m \n",
- "A Bravais lattice with a single site in the basis.\n",
- "\n",
- "Instances of this class provide the `~kwant.builder.SiteFamily` interface.\n",
- "Site tags (see `~kwant.builder.SiteFamily`) are sequences of integers and\n",
- "describe the lattice coordinates of a site.\n",
- "\n",
- "``Monatomic`` instances are used as site families on their own or as\n",
- "sublattices of `Polyatomic` lattices.\n",
- "\n",
- "Parameters\n",
- "----------\n",
- "prim_vecs : 2d array-like of floats\n",
- " Primitive vectors of the Bravais lattice.\n",
- "\n",
- "offset : vector of floats\n",
- " Displacement of the lattice origin from the real space\n",
- " coordinates origin.\n",
- "\n",
- "Attributes\n",
- "----------\n",
- "``offset`` : vector\n",
- " Displacement of the lattice origin from the real space coordinates origin\n",
- "\u001b[0;31mCall docstring:\u001b[0m\n",
- "A convenience function.\n",
- "\n",
- "This function allows to write fam(1, 2) instead of Site(fam, (1, 2)).\n"
- ]
- },
- "metadata": {},
- "output_type": "display_data"
- }
- ],
- "source": [
- "chain?"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 4,
- "metadata": {},
- "outputs": [],
- "source": [
- "H1 = \"k_x**2 + 10*k_y**2\"\n",
- "H2 = \"t_1*k_x**2 + t_2*k_y**2 + t_3*k_z**2\"\n",
- "H3 = \"sigma_z*k_x**2 + 10*sigma_x*k_y**2\""
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 5,
- "metadata": {},
- "outputs": [],
- "source": [
- "ll = landau_levels.ll_hamiltonian(H3)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 6,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "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\n",
- "text/latex": [
- "$$\\left \\{ k_{x}^{2} : \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right], \\quad k_{y}^{2} : \\left[\\begin{matrix}0 & 10\\\\10 & 0\\end{matrix}\\right]\\right \\}$$"
- ],
- "text/plain": [
- "⎧ 2 ⎡1 0 ⎤ 2 ⎡0 10⎤⎫\n",
- "⎨kₓ : ⎢ ⎥, k_y : ⎢ ⎥⎬\n",
- "⎩ ⎣0 -1⎦ ⎣10 0 ⎦âŽ"
- ]
- },
- "execution_count": 6,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.monomials"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "True"
- ]
- },
- "execution_count": 7,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.normal_coordinate == None"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 15,
- "metadata": {},
- "outputs": [],
- "source": [
- "ll.set_ll_momenta([1, 2])"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 16,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "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\n",
- "text/latex": [
- "$$\\left [ k_{y}, \\quad k_{z}\\right ]$$"
- ],
- "text/plain": [
- "[k_y, k_z]"
- ]
- },
- "execution_count": 16,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.ll_momenta"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 17,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "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\n",
- "text/latex": [
- "$$\\left \\{ 1 : \\left[\\begin{matrix}k_{x}^{2} & 0\\\\0 & - k_{x}^{2}\\end{matrix}\\right], \\quad k_{y}^{2} : \\left[\\begin{matrix}0 & 10\\\\10 & 0\\end{matrix}\\right]\\right \\}$$"
- ],
- "text/plain": [
- "⎧ ⎡ 2 ⎤ ⎫\n",
- "⎪ ⎢kₓ 0 ⎥ 2 ⎡0 10⎤⎪\n",
- "⎨1: ⎢ ⎥, k_y : ⎢ ⎥⎬\n",
- "⎪ ⎢ 2⎥ ⎣10 0 ⎦⎪\n",
- "⎩ ⎣ 0 -kâ‚“ ⎦ âŽ"
- ]
- },
- "execution_count": 17,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.monomials"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 18,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NuzJUmW693xMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDVgYGRgWsBAwjwKYCpfRuAFI+AHgOTAEPcgXUM7gwMwkpC1wsYABfXCcn8wW65AAAAAElFTkSuQmCC\n",
- "text/latex": [
- "$$x$$"
- ],
- "text/plain": [
- "x"
- ]
- },
- "execution_count": 18,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.normal_coordinate"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 19,
- "metadata": {},
- "outputs": [],
- "source": [
- "# ll.set_ll_momenta(['k_x', 'k_y'])"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 20,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NuzJUmW693xMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDVgYGRgWsBAwjwKYCpfRuAFI+AHgOTAEPcgXUM7gwMwkpC1wsYABfXCcn8wW65AAAAAElFTkSuQmCC\n",
- "text/latex": [
- "$$x$$"
- ],
- "text/plain": [
- "x"
- ]
- },
- "execution_count": 20,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.normal_coordinate"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 21,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "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\n",
- "text/latex": [
- "$$\\left \\{ 1 : \\left[\\begin{matrix}k_{x}^{2} & 0\\\\0 & - k_{x}^{2}\\end{matrix}\\right], \\quad k_{y}^{2} : \\left[\\begin{matrix}0 & 10\\\\10 & 0\\end{matrix}\\right]\\right \\}$$"
- ],
- "text/plain": [
- "⎧ ⎡ 2 ⎤ ⎫\n",
- "⎪ ⎢kₓ 0 ⎥ 2 ⎡0 10⎤⎪\n",
- "⎨1: ⎢ ⎥, k_y : ⎢ ⎥⎬\n",
- "⎪ ⎢ 2⎥ ⎣10 0 ⎦⎪\n",
- "⎩ ⎣ 0 -kâ‚“ ⎦ âŽ"
- ]
- },
- "execution_count": 21,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.monomials"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 22,
- "metadata": {},
- "outputs": [],
- "source": [
- "ll.shape_func"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 23,
- "metadata": {},
- "outputs": [],
- "source": [
- "N = 3"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 24,
- "metadata": {},
- "outputs": [],
- "source": [
- "ham = ll.hamiltonian_matrix(N)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 25,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "image/png": "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\n",
- "text/latex": [
- "$$\\left \\{ 1 : \\left[\\begin{matrix}k_{x}^{2} & 0\\\\0 & - k_{x}^{2}\\end{matrix}\\right], \\quad k_{y}^{2} : \\left[\\begin{matrix}0 & 10\\\\10 & 0\\end{matrix}\\right]\\right \\}$$"
- ],
- "text/plain": [
- "⎧ ⎡ 2 ⎤ ⎫\n",
- "⎪ ⎢kₓ 0 ⎥ 2 ⎡0 10⎤⎪\n",
- "⎨1: ⎢ ⎥, k_y : ⎢ ⎥⎬\n",
- "⎪ ⎢ 2⎥ ⎣10 0 ⎦⎪\n",
- "⎩ ⎣ 0 -kâ‚“ ⎦ âŽ"
- ]
- },
- "execution_count": 25,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ll.monomials"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "metadata": {},
- "outputs": [],
- "source": [
- "params = dict(t_1 = 1, t_2 = 2, t_3 = 3, k_x=0)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 29,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "matrix([[ 0. +0.j, 5. +0.j, 0. +0.j,\n",
- " 0. +0.j, 0. +0.j, 7.07106781+0.j],\n",
- " [ 5. +0.j, 0. +0.j, 0. +0.j,\n",
- " 0. +0.j, 7.07106781+0.j, 0. +0.j],\n",
- " [ 0. +0.j, 0. +0.j, 0. +0.j,\n",
- " 15. +0.j, 0. +0.j, 0. +0.j],\n",
- " [ 0. +0.j, 0. +0.j, 15. +0.j,\n",
- " 0. +0.j, 0. +0.j, 0. +0.j],\n",
- " [ 0. +0.j, 7.07106781+0.j, 0. +0.j,\n",
- " 0. +0.j, 0. +0.j, 25. +0.j],\n",
- " [ 7.07106781+0.j, 0. +0.j, 0. +0.j,\n",
- " 0. +0.j, 25. +0.j, 0. +0.j]])"
- ]
- },
- "execution_count": 29,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ham(B=1, params=params).todense()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [],
- "source": [
- "H1 = \"k_x**2 + 10*k_y**2\"\n",
- "H2 = \"t_1*k_x**2 + t_2*k_y**2 + t_3*k_z**2\"\n",
- "H3 = \"sigma_z*k_x**2 + 10*sigma_x*k_y**2\""
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 6,
- "metadata": {},
- "outputs": [],
- "source": [
- "N = 2\n",
- "ham, monomials = ll_hamiltonian(H3, N, momenta=None)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 7,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "<function kwant.continuum.landau_levels.combine.<locals>.final_wrapper(*, B, params=None)>"
- ]
- },
- "execution_count": 7,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ham"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 8,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "matrix([[ 0.5+0.j, 5. +0.j, 0. +0.j, 0. +0.j],\n",
- " [ 5. +0.j, -0.5+0.j, 0. +0.j, 0. +0.j],\n",
- " [ 0. +0.j, 0. +0.j, 1.5+0.j, 15. +0.j],\n",
- " [ 0. +0.j, 0. +0.j, 15. +0.j, -1.5+0.j]])"
- ]
- },
- "execution_count": 8,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "ham(B=1, params=dict(t_1=1, t_2=1, t_3=1)).todense()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 9,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "{k_x**2: <function __main__.ll_hamiltonian.<locals>.<lambda>.<locals>.<lambda>(params=None)>,\n",
- " k_y**2: <function __main__.ll_hamiltonian.<locals>.<lambda>.<locals>.<lambda>(params=None)>}"
- ]
- },
- "execution_count": 9,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "monomials"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 10,
- "metadata": {},
- "outputs": [],
- "source": [
- "funcs = list(monomials.values())"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 11,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "<function __main__.ll_hamiltonian.<locals>.<lambda>.<locals>.<lambda>(params=None)>"
- ]
- },
- "execution_count": 11,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "funcs[0]"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 12,
- "metadata": {},
- "outputs": [],
- "source": [
- "# funcs[0]().todense()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 13,
- "metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "k_x**2 [[ 1.+0.j 0.+0.j]\n",
- " [ 0.+0.j -1.+0.j]]\n",
- "k_y**2 [[ 0.+0.j 10.+0.j]\n",
- " [10.+0.j 0.+0.j]]\n"
- ]
- }
- ],
- "source": [
- "params = dict(t_1 = 1, t_2=1, t_3=1)\n",
- "for key, value in monomials.items():\n",
- " print(key, value(params=params).todense())"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 60,
- "metadata": {},
- "outputs": [],
- "source": [
- "f1 = lambda x, y=1, z=2: x*y + z\n",
- "f2 = lambda l, y=1, z=2: l*y + z"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 61,
- "metadata": {},
- "outputs": [],
- "source": [
- "funcs = [f1, f2]"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 62,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "<function __main__.combine.<locals>.final_wrapper(*, l, x, y=1, z=2)>"
- ]
- },
- "execution_count": 62,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "combine(operator.add, funcs, args_names=None)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": 15,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "False"
- ]
- },
- "execution_count": 15,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "('N_1', None) == ('N_2', None)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 81,
- "metadata": {},
- "outputs": [],
- "source": [
- "H = \"k_x**2 + k_y**2 + k_z**2\""
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 87,
- "metadata": {},
- "outputs": [],
- "source": [
- "disc = kwant.continuum.discretize(H, coords=['y', 'z'])"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 88,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "# Discrete coordinates: y z\n",
- "\n",
- "# Onsite element:\n",
- "def onsite(site, k_x):\n",
- " return (4.0 + k_x**2)\n",
- "\n",
- "# Hopping from (1, 0):\n",
- "(-1+0j)\n",
- "\n",
- "# Hopping from (0, 1):\n",
- "(-1+0j)"
- ],
- "text/plain": [
- "<kwant.continuum.discretizer._DiscretizedBuilder at 0x7f8e4a178630>"
- ]
- },
- "execution_count": 88,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "disc"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 89,
- "metadata": {},
- "outputs": [
- {
- "name": "stdout",
- "output_type": "stream",
- "text": [
- "[0.0 0.0]\n"
- ]
- }
- ],
- "source": [
- "for site in disc.sites():\n",
- " print(site.pos)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 71,
- "metadata": {},
- "outputs": [],
- "source": [
- "fsyst = disc.finalized()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 74,
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/plain": [
- "array([[ 2.+0.j, -1.-0.j],\n",
- " [-1.+0.j, 2.+0.j]])"
- ]
- },
- "execution_count": 74,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "fsyst.hamiltonian_submatrix()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 16,
- "metadata": {},
- "outputs": [],
- "source": [
- "l = ['a', 'b', 'c', 'd']\n",
- "inds = [0, 3]"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 22,
- "metadata": {},
- "outputs": [
- {
- "ename": "TypeError",
- "evalue": "zip argument #1 must support iteration",
- "output_type": "error",
- "traceback": [
- "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
- "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
- "\u001b[0;32m<ipython-input-22-39657fc954cd>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0ml\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mzip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0minds\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
- "\u001b[0;31mTypeError\u001b[0m: zip argument #1 must support iteration"
- ]
- }
- ],
- "source": []
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "work_env",
- "language": "python",
- "name": "work_env"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.1"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
--
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