Verified Commit 8b8e4647 authored by Anton Akhmerov's avatar Anton Akhmerov
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remove links to cocalc, switch url scheme to https

parent c6eeaf24
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......@@ -53,7 +53,7 @@ rather than this:
![Bored audience](figures/bored_audience_bruno_touschek.png)
*(Images by Bruno Touschek, © 1981 [CERN](http://cds.cern.ch/record/135949) CC-BY-3.0)*
*(Images by Bruno Touschek, © 1981 [CERN](https://cds.cern.ch/record/135949) CC-BY-3.0)*
### What you get
......@@ -105,16 +105,16 @@ Specifically you need:
* Python 3.5 or 3.6
* Python scientific stack (SciPy, NumPy, Matplotlib, Jupyter notebook)
* Holoviews 1.7, a Python library for plotting data.
* [Kwant](http://kwant-project.org) 1.3, a package for quantum transport simulations.
* [Kwant](https://kwant-project.org) 1.3, a package for quantum transport simulations.
The installation should be straightforward.
Installation of most of the requirements is described [here](http://kwant-project.org/install).
Installation of most of the requirements is described [here](https://kwant-project.org/install).
If you are using Windows, you are all set after following the above instructions.
The easiest way to install `kwant` when you are using Linux or OS X is with `conda` which comes with Miniconda, a Python distribution.
1. Open a terminal and download [Miniconda](http://conda.pydata.org/miniconda.html>)
1. Open a terminal and download [Miniconda](https://conda.pydata.org/miniconda.html>)
(or [Anaconda](https://www.continuum.io/downloads)) by running:
wget https://repo.continuum.io/miniconda/Miniconda3-latest-MacOSX-x86_64.sh
......@@ -143,27 +143,23 @@ These notebooks are extremely handy, they allow:
* To use an interactive computing environment where you can see what your simulation does right as you are creating it.
* To combine nicely formatted text (with Latex equations and images) together with code in any language and the output of that code.
* To easily share that same code: here, take a look at the source of [the notebook](http://nbviewer.ipython.org/github/topocm/topocm_content/blob/master/w0_background/intro.ipynb) that you are currently reading.
* To easily share that same code: here, take a look at the source of [the notebook](https://nbviewer.ipython.org/github/topocm/topocm_content/blob/master/w0_background/intro.ipynb) that you are currently reading.
* To convert it to a presentation, blog post, or an EdX course.
The combination of the above nice properties with many more and with Jupyter being free software lead to the notebooks being [highlighted](http://www.nature.com/news/interactive-notebooks-sharing-the-code-1.16261) in Nature.
The combination of the above nice properties with many more and with Jupyter being free software lead to the notebooks being [highlighted](https://www.nature.com/news/interactive-notebooks-sharing-the-code-1.16261) in Nature.
For a short presentation of Jupyter notebooks just use `Help -> User Interface Tour` inside the notebook.
### Sharing notebooks
Showing the results of your work is very easy.
If you are using Sage Cloud, you can just click the "share the notebook" button when you have it opened, and copy the URL.
Otherwise you can make the notebook visible online (for example by putting it in your Dropbox public folder or something similar), copy link, and paste it into [http://nbviewer.ipython.org](http://nbviewer.ipython.org).
You can make the notebook visible online (for example by putting it in your Dropbox public folder or something similar), copy link, and paste it into https://nbviewer.jupyter.org.
### Kwant, Python, and Python scientific software
For most of the simulations of condensed matter systems we are going to use the Kwant package. You can learn Kwant in more detail by following the [tutorial](http://kwant-project.org/doc/1.0/tutorial/), however we aim that for most of the exercises you will be able to learn by doing. The starting point of the exercises are the notebooks used in the lectures, and you should be able to solve them by only modifying the contents not too much.
For most of the simulations of condensed matter systems we are going to use the Kwant package. You can learn Kwant in more detail by following the [tutorial](https://kwant-project.org/doc/1.0/tutorial/), however we aim that for most of the exercises you will be able to learn by doing. The starting point of the exercises are the notebooks used in the lectures, and you should be able to solve them by only modifying the contents not too much.
The same applies to Python and the Python scientific stack (NumPy, SciPy, Matplotlib): these are easy to use, especially when you have code examples. If you are new to programming and wish to get acquainted with Python,
[here](http://www.learnpython.org/) are [several](http://www.python-course.eu/) [example courses](https://www.codecademy.com/learn/python) that start from the basics and slowly go into advanced topics. There are of course several MOOCs as well, but you will likely not need as much programming skill.
[here](https://www.learnpython.org/) are [several](https://www.python-course.eu/) [example courses](https://www.codecademy.com/learn/python) that start from the basics and slowly go into advanced topics. There are of course several MOOCs as well, but you will likely not need as much programming skill.
**Do you have questions about installation? Use this discussion:**
......
......@@ -39,11 +39,11 @@ Video("i-eNPei2zMg")
The materials of this lecture are kindly provided by the Topological Mechanics lab at Leiden University: Vincenzo Vitelli (PI), Bryan Chen, Anne Meeussen, Jayson Paulose, Benny van Zuiden, and Yujie Zhou. They are copyright of their creators, and are available under a
<a rel="license"
href="http://creativecommons.org/licenses/by-sa/4.0/"
href="https://creativecommons.org/licenses/by-sa/4.0/"
target="_blank">Creative Commons Attribution-ShareAlike 4.0
International License.</a>
<a rel="license"
href="http://creativecommons.org/licenses/by-sa/4.0/">
href="https://creativecommons.org/licenses/by-sa/4.0/">
<img
alt="Creative Commons License" style="border-width:0"
src="https://i.creativecommons.org/l/by-sa/4.0/88x31.png" /></a>
......@@ -72,7 +72,7 @@ We shall now show that the zero mode localized at the edge is a time snaphot of
![](figures/kink.png)
To see that, it is easier to consider the constraint equation $l^2_{n,n+1}=\bar{l}^2$ that demands that the length of the springs $l_{n, n+1}$ between any two rotors labeled by $n$ and $n+1$ is not stretched. This is certainly the case for the rigid bar systems shown in the video which correspond to the limit of infinite spring constant. Such systems are called [linkages](http://en.wikipedia.org/wiki/Linkage_%28mechanical%29).
To see that, it is easier to consider the constraint equation $l^2_{n,n+1}=\bar{l}^2$ that demands that the length of the springs $l_{n, n+1}$ between any two rotors labeled by $n$ and $n+1$ is not stretched. This is certainly the case for the rigid bar systems shown in the video which correspond to the limit of infinite spring constant. Such systems are called [linkages](https://en.wikipedia.org/wiki/Linkage_%28mechanical%29).
In terms of the parameters defined in the first figure, the constraint equation $l^2_{n,n+1}=\bar{l}^2$ reads
......@@ -135,7 +135,7 @@ x}\right)^2 -\frac{1}{2}K(\bar u^2-u^2)^2 - \frac{1}{2} K \frac{a^2}{2}(\bar
u^2-u^2)\frac{\partial u}{\partial x}\right].
$$
The first term is a linearized kinetic energy, while the second and third terms are the ordinary [$\phi^4$ field theory](http://en.wikipedia.org/wiki/Quartic_interaction) that you might have encountered for instance while studying the Ising model.
The first term is a linearized kinetic energy, while the second and third terms are the ordinary [$\phi^4$ field theory](https://en.wikipedia.org/wiki/Quartic_interaction) that you might have encountered for instance while studying the Ising model.
The final term linear in $\partial_x u$ is topological in the sense that it integrates to the boundary. It ensures that the static kink has zero energy, since the last three terms in the Lagrangian can be combined into a perfect square, that vanishes for the static kink solution (this is sometimes called a BPS state). The topological boundary term does not affect the equations of motions in the bulk, but it breaks the $\partial_x u \rightarrow -\partial_x u$ symmetry of the $\phi^4$ theory. That's why the two edges are not equivalent in terms of initiating the kink motion.
......@@ -164,7 +164,7 @@ If you are curious about the mathematical details and a systematic study of the
* arXiv:1404.2263
Note that with different geometrical parameters, the same chain above can realize the [sine-Gordon Lagrangian](http://en.wikipedia.org/wiki/Sine-Gordon_equation). As before, check out the following movie:
Note that with different geometrical parameters, the same chain above can realize the [sine-Gordon Lagrangian](https://en.wikipedia.org/wiki/Sine-Gordon_equation). As before, check out the following movie:
```{code-cell} ipython3
:tags: [remove-input]
......@@ -236,7 +236,7 @@ This relation between states of self stress and zero modes can be made more prec
The zero energy modes are members of the null space of $R_{ij}$, which in an isostatic system are also the (right) eigenvectors of $R_{ij}$ with eigenvalue 0. (Convince yourself that the rigidity matrix is square for an isostatic system.)
Conversely the states of self stress are the zero eigenvectors of the transpose matrix $R^{T}$ that relates the forces on the nodes to the tensions in the bonds. The Maxwell count equation follows from the rank-nullity theorem and can be expressed as an index theorem. (We suggest the reader interested in a careful proof and statement of these results to consult [this review](http://arxiv.org/abs/1503.01324).)
Conversely the states of self stress are the zero eigenvectors of the transpose matrix $R^{T}$ that relates the forces on the nodes to the tensions in the bonds. The Maxwell count equation follows from the rank-nullity theorem and can be expressed as an index theorem. (We suggest the reader interested in a careful proof and statement of these results to consult [this review](https://arxiv.org/abs/1503.01324).)
Here we merely remark that if one smoothly deforms the network without cutting bonds or adding sites then the left hand side of the Maxwell count does not change. This means that after such a smooth deformation, the difference between the number of zero modes and states of self stress must remain invariant even if $n_m$ and $n_{ss}$ change individually. In this sense the isostatic condition can be viewed as a charge neutrality condition. As we shall see, this electrostatic analogy is far reaching. We will now turn our attention to periodic mechanical structures that are isostatic, hence ‘‘charge neutral’’, but mechanically polarized so that edge modes can appear at the sample boundary like charges in an electrically polarized medium.
......
......@@ -185,7 +185,7 @@ So the basic rule of the game is to ask what are all possible braiding rules for
### Symmetries
To approach the classification of interacting topological insulators with symmetries, we can start by playing the same game as Kane and Mele, and combine two fractional quantum Hall states (instead of integer ones) to make a fractional topological insulator. If we choose a pair of FQH states which are related by time-reversal and stack them together for spin-up and spin-down electrons, that technically leads to a time-reversal invariant state. The key question that needs to be asked is whether one can gap out the edge states by adding time-reversal invariant perturbations. If one can do that, then unlike the Kane-Mele quantum spin Hall state, the state is not a phase that is protected by just time-reversal symmetry. This question is not too mathematically involved, though still beyond this course and can be answered by the bosonization technique as was done by [Stern and Levin](http://arxiv.org/abs/0906.2769).
To approach the classification of interacting topological insulators with symmetries, we can start by playing the same game as Kane and Mele, and combine two fractional quantum Hall states (instead of integer ones) to make a fractional topological insulator. If we choose a pair of FQH states which are related by time-reversal and stack them together for spin-up and spin-down electrons, that technically leads to a time-reversal invariant state. The key question that needs to be asked is whether one can gap out the edge states by adding time-reversal invariant perturbations. If one can do that, then unlike the Kane-Mele quantum spin Hall state, the state is not a phase that is protected by just time-reversal symmetry. This question is not too mathematically involved, though still beyond this course and can be answered by the bosonization technique as was done by [Stern and Levin](https://arxiv.org/abs/0906.2769).
We can however explain the result. If one stacks the two FQH states one obtains a spin-Hall conductance $\sigma_{sh}$, which is equal to the Hall conductance of each layer. Let the smallest charge of an excitation of our phase be $e^*$, some fraction of electron charge. It turns out that the edge states are protected from gapping by time-reversal invariant perturbations if and only if $\sigma_{sh}/e^*$ is an odd integer. This gives some idea as to what kind of interacting analogues of quantum spin Hall states one may get. But again, as with the non-symmetric case, the general classification is still up in the air. More importantly, we don't really have realistic candidates for such states yet.
......
......@@ -581,7 +581,7 @@ $$
We already know that the pairing matrix $\Delta$ is antisymmetric. Since $H$ is Hermitian $H-H^*$ is also antisymmetric and $H+H^*$ is symmetric. Then we can see that $\tilde{H}_\textrm{BdG}$ is antisymmetric.
There is a special number that we can compute for antisymmetric matrices, the [Pfaffian](http://en.wikipedia.org/wiki/Pfaffian). Its rigorous definition is not important for our course. The basic idea is simple: The eigenvalues of antisymmetric matrices always come in pairs. In the case of our $\tilde{H}_{BdG}$, these are the energy eigenvalues $\pm E_n$. By taking their product we obtain the determinant of the matrix, equal to $\prod_n (-E_n^2)$. The key property of the Pfaffian is that it allows to take a square root of the determinant, equal to $\pm \prod_n iE_n$, in such a way that the sign of the product is uniquely defined. At a fermion parity switch a single $E_n$ changes sign, so the Pfaffian changes sign as well (while the determinant stays the same).
There is a special number that we can compute for antisymmetric matrices, the [Pfaffian](https://en.wikipedia.org/wiki/Pfaffian). Its rigorous definition is not important for our course. The basic idea is simple: The eigenvalues of antisymmetric matrices always come in pairs. In the case of our $\tilde{H}_{BdG}$, these are the energy eigenvalues $\pm E_n$. By taking their product we obtain the determinant of the matrix, equal to $\prod_n (-E_n^2)$. The key property of the Pfaffian is that it allows to take a square root of the determinant, equal to $\pm \prod_n iE_n$, in such a way that the sign of the product is uniquely defined. At a fermion parity switch a single $E_n$ changes sign, so the Pfaffian changes sign as well (while the determinant stays the same).
This feature of the Pfaffian really makes it what we are looking for. Let's try out the sign of the Pfaffian as our topological invariant $Q_\textrm{BdG}$:
......@@ -591,7 +591,7 @@ $$
We have included a factor of $i$ just for convenience, so that the Pfaffian is a real number.
Whenever we need to compute a Pfaffian we just use the [Pfapack](http://arxiv.org/abs/1102.3440) package that calculates Pfaffians for numerical matrices. Let's use that package and check that the Pfaffian indeed allows us to calculate the fermion parity of the ground state of $H_\textrm{BdG}$.
Whenever we need to compute a Pfaffian we just use the [Pfapack](https://arxiv.org/abs/1102.3440) package that calculates Pfaffians for numerical matrices. Let's use that package and check that the Pfaffian indeed allows us to calculate the fermion parity of the ground state of $H_\textrm{BdG}$.
```{code-cell} ipython3
:tags: [remove-input]
......
......@@ -249,7 +249,7 @@ MultipleChoice(
### Going to momentum space
The Majoranas are edge excitations that arise from the bulk-edge correspondence. You might wonder if there is a way to deduce the existence of Majorana modes by looking at the bulk? To answer this, eliminate the boundaries from the study of the Kitaev chain. You can imagine that the last site of the chain is reconnected to the first, so that the chain is closed in a ring (a “Kitaev ring”). In the absence of boundaries, the Bogoliubov-de Gennes Hamiltonian has a translational symmetry $\left|n\right\rangle\,\to\,\left|n+1\right\rangle$, since all parameters $t, \Delta$ and $\mu$ do not depend on the chain site $n$. In the presence of translational symmetries, it is always convenient to use [Bloch's theorem](http://en.wikipedia.org/wiki/Bloch_wave#Preliminaries:_Crystal_symmetries.2C_lattice.2C_and_reciprocal_lattice) and write down the Hamiltonian in momentum space rather than in real space. In our case, a state with momentum $k$ is given by
The Majoranas are edge excitations that arise from the bulk-edge correspondence. You might wonder if there is a way to deduce the existence of Majorana modes by looking at the bulk? To answer this, eliminate the boundaries from the study of the Kitaev chain. You can imagine that the last site of the chain is reconnected to the first, so that the chain is closed in a ring (a “Kitaev ring”). In the absence of boundaries, the Bogoliubov-de Gennes Hamiltonian has a translational symmetry $\left|n\right\rangle\,\to\,\left|n+1\right\rangle$, since all parameters $t, \Delta$ and $\mu$ do not depend on the chain site $n$. In the presence of translational symmetries, it is always convenient to use [Bloch's theorem](https://en.wikipedia.org/wiki/Bloch_wave#Preliminaries:_Crystal_symmetries.2C_lattice.2C_and_reciprocal_lattice) and write down the Hamiltonian in momentum space rather than in real space. In our case, a state with momentum $k$ is given by
$$
\left|k\right\rangle =(N)^{-1/2} \sum_{n=1}^{N} e^{-ikn} \left|n\right\rangle.
......
......@@ -30,10 +30,6 @@ So you've made it through the content of the first week. Congratulations!
Now let's get our hands dirty.
Let's begin by grabbing the notebooks of this week and the extra code we use to run these notebooks over [here](http://tiny.cc/topocm_smc). (Click the [i] button to the left of the folder that you want to copy.)
You need to copy the `code` folder and the `w1_topointro` folder. Let's look into what's inside.
+++
### First task: combination of particle-hole and time-reversal symmetries
......
......@@ -165,7 +165,7 @@ Let's now imagine that experimentalists are not only able to build such a networ
![](figures/nanowire_network_exchange.svg)
Let's suppose that the trajectory takes a time $T$. During the trajectory, the system is described by a time-dependent Hamiltonian $H(t)$, $0\leq t \leq T$. This Hamiltonian contains all the details of the system, such as the positions of the domain walls where the Majoranas are located. Because the final configuration of the system is identical to the initial one, for instance all the domain walls are in the same positions as in the beginning, we have that $H(0)=H(T)$. In other words, we are considering a *closed trajectory* which brings the Hamiltonian back into itself. To ensure that the wave-function for the system does not leave the ground state manifold of states $|\Psi\rangle$, we need to change the Hamiltonian $H(t)$ slowly enough to obey the [adiabatic theorem](http://en.wikipedia.org/wiki/Adiabatic_theorem).
Let's suppose that the trajectory takes a time $T$. During the trajectory, the system is described by a time-dependent Hamiltonian $H(t)$, $0\leq t \leq T$. This Hamiltonian contains all the details of the system, such as the positions of the domain walls where the Majoranas are located. Because the final configuration of the system is identical to the initial one, for instance all the domain walls are in the same positions as in the beginning, we have that $H(0)=H(T)$. In other words, we are considering a *closed trajectory* which brings the Hamiltonian back into itself. To ensure that the wave-function for the system does not leave the ground state manifold of states $|\Psi\rangle$, we need to change the Hamiltonian $H(t)$ slowly enough to obey the [adiabatic theorem](https://en.wikipedia.org/wiki/Adiabatic_theorem).
So let's imagine that we are in the adiabatic limit and that we exchange two Majoranas $\gamma_n$ and $\gamma_m$. As usual in quantum mechanics, the initial and final quantum states are connected by a unitary operator $U$ ($U^{-1}=U^\dagger$),
......@@ -181,7 +181,7 @@ $$
U\equiv\exp(\beta \gamma_n \gamma_m) = \cos(\beta) + \gamma_n\gamma_m \sin(\beta)\,,
$$
*up to an overall phase*. Here, $\beta$ is a real coefficient to be determined, and in the last equality we have used the fact that $(\gamma_n\gamma_m)^2=-1$. To determine $\beta$, it is convenient to go to the [Heisenberg picture](http://en.wikipedia.org/wiki/Heisenberg_picture) and look at the evolution of the Majorana operators in time. We have that
*up to an overall phase*. Here, $\beta$ is a real coefficient to be determined, and in the last equality we have used the fact that $(\gamma_n\gamma_m)^2=-1$. To determine $\beta$, it is convenient to go to the [Heisenberg picture](https://en.wikipedia.org/wiki/Heisenberg_picture) and look at the evolution of the Majorana operators in time. We have that
$$
\gamma_n\,\to\, U\,\gamma_n\,U^\dagger\,,\\
......@@ -285,7 +285,7 @@ MultipleChoice(
## Majoranas and quantum computation: basic ideas
The non-Abelian statistics of Majorana modes is a very special property. Furthermore, it has some practical interest, since it could be used to realize a robust **quantum computer**. (If you are not yet interested in quantum computation, you can skip this part, even though we suggest that you get interested in it! Quantum computation is a huge topic of research, but [this](http://arxiv.org/abs/quant-ph/9708022) is a good place to start learning.)
The non-Abelian statistics of Majorana modes is a very special property. Furthermore, it has some practical interest, since it could be used to realize a robust **quantum computer**. (If you are not yet interested in quantum computation, you can skip this part, even though we suggest that you get interested in it! Quantum computation is a huge topic of research, but [this](https://arxiv.org/abs/quant-ph/9708022) is a good place to start learning.)
Let's discuss very briefly how this can be done.
......
......@@ -268,7 +268,7 @@ The final stretch is straightforward.
We know that there is no gap because of conservation of one of the spin projections, so we need to break the spin conservation.
If we don't want to create an inhomogeneous magnetic field, we have to use a different term that couples to spin. That term is spin-orbit interaction. In it's [simplest form](http://en.wikipedia.org/wiki/Rashba_effect) this interaction appears in our wire as
If we don't want to create an inhomogeneous magnetic field, we have to use a different term that couples to spin. That term is spin-orbit interaction. In it's [simplest form](https://en.wikipedia.org/wiki/Rashba_effect) this interaction appears in our wire as
$$
H_{SO} = \alpha \sigma_y k,
......
......@@ -320,11 +320,11 @@ $$
t\rightarrow -t.
$$
It turns out that the sign change in the hopping across the junction might also be obtained by introducing a magnetic flux through the superconducting ring (similar to the [Aharonov-Bohm effect](http://en.wikipedia.org/wiki/Aharonov-Bohm_effect)). The role of the special bond is now played by a Josephson junction, which is just an insulating barrier interrupting the ring, as in the following sketch:
It turns out that the sign change in the hopping across the junction might also be obtained by introducing a magnetic flux through the superconducting ring (similar to the [Aharonov-Bohm effect](https://en.wikipedia.org/wiki/Aharonov-Bohm_effect)). The role of the special bond is now played by a Josephson junction, which is just an insulating barrier interrupting the ring, as in the following sketch:
![](figures/josephson_majorana_ring.svg)
How does the magnetic flux enter the Hamiltonian? By following the usual argument for introducing magnetic fields into lattice Hamiltonians using [Peierls substitution](http://topocondmat.org/w2_majorana/Peierls.html), the flux $\Phi$ can be accounted for simply by changing the phase of the hopping across the junction in the ring:
How does the magnetic flux enter the Hamiltonian? By following the usual argument for introducing magnetic fields into lattice Hamiltonians using [Peierls substitution](Peierls), the flux $\Phi$ can be accounted for simply by changing the phase of the hopping across the junction in the ring:
$$
t\,\to\,t\,\exp (i\phi/2).
......@@ -415,7 +415,7 @@ Staring at the spectrum we see that, lo and behold, the fermion parity switch ap
## Detecting the fermion parity switch using the Josephson effect
The change in fermion parity of the ground state can be detected using the so-called [Josephson effect](http://en.wikipedia.org/wiki/Josephson_effect). The Josephson current can be computed from the expectation value of the derivative of the energy operator with respect to flux,
The change in fermion parity of the ground state can be detected using the so-called [Josephson effect](https://en.wikipedia.org/wiki/Josephson_effect). The Josephson current can be computed from the expectation value of the derivative of the energy operator with respect to flux,
$$
I(\Phi)=\frac{1}{2}\frac{d E_\textrm{tot}(\Phi)}{d \Phi},
......
......@@ -42,7 +42,7 @@ Video("QC3tQT7MD00")
We now move on to the quantum Hall effect, the mother of all topological effects in condensed matter physics.
But let's start from the classical [Hall effect](http://en.wikipedia.org/wiki/Hall_effect), the famous phenomenon by which a current flows perpendicular to an applied voltage, or vice versa a voltage develops perpendicular to a flowing current.
But let's start from the classical [Hall effect](https://en.wikipedia.org/wiki/Hall_effect), the famous phenomenon by which a current flows perpendicular to an applied voltage, or vice versa a voltage develops perpendicular to a flowing current.
How does one get a Hall effect? The key is to break time-reversal symmetry. A flowing current breaks time-reversal symmetry, while an electric field doesn't. Hence, any system with a Hall effect must somehow break time-reversal symmetry.
......@@ -137,7 +137,7 @@ MultipleChoice(
## The quantum Hall effect: experimental data
Instead, a completely unexpected result was measured for the first time by Klaus von Klitzing. Typical experimental data looks like this (taken from M.E. Suddards, A. Baumgartner, M. Henini and C. J. Mellor, [New J. Phys. 14 083015](http://iopscience.iop.org/1367-2630/14/8/083015)):
Instead, a completely unexpected result was measured for the first time by Klaus von Klitzing. Typical experimental data looks like this (taken from M.E. Suddards, A. Baumgartner, M. Henini and C. J. Mellor, [New J. Phys. 14 083015](https://iopscience.iop.org/1367-2630/14/8/083015)):
![](figures/QHE.png)
......
......@@ -134,7 +134,7 @@ $$
Here $k_F$ is the Fermi momentum, which in the case of our ribbon is equal to $k_F = 2\pi N / L$, with $N$ the number of electrons in the system.
Because the slope of the potential is just the local electric field $\mathcal{E}_y=-\partial_y V(y)$ perpendicular to the edge of the sample, the velocity $v$ of the edge states can be simply interpreted as the [drift velocity](http://en.wikipedia.org/wiki/Guiding_center) of a skipping state,
Because the slope of the potential is just the local electric field $\mathcal{E}_y=-\partial_y V(y)$ perpendicular to the edge of the sample, the velocity $v$ of the edge states can be simply interpreted as the [drift velocity](https://en.wikipedia.org/wiki/Guiding_center) of a skipping state,
$$
v = \mathcal{E}_y/B\,.
......@@ -408,7 +408,7 @@ MultipleChoice(
The physical picture that we presented this week is very simple, and it is also somewhat simplified.
In the summary video of this week, Bert Halperin from Harvard University will discuss how disorder and interactions enter in the description of the quantum Hall effect, and where the electric current is really carried. In 1982, Bert was the [first to understand](http://sites.fas.harvard.edu/~phys191r/References/e3/halperin1982.pdf) that the quantum Hall effect could be explained by the existence of chiral edge states, so we are very happy that you can learn the story directly from him.
In the summary video of this week, Bert Halperin from Harvard University will discuss how disorder and interactions enter in the description of the quantum Hall effect, and where the electric current is really carried. In 1982, Bert was the [first to understand](https://sites.fas.harvard.edu/~phys191r/References/e3/halperin1982.pdf) that the quantum Hall effect could be explained by the existence of chiral edge states, so we are very happy that you can learn the story directly from him.
```{code-cell} ipython3
:tags: [remove-input]
......
......@@ -26,8 +26,6 @@ init_notebook()
## Simulations: Disorder, butterflies, and honeycombs
As usual, start by grabbing the notebooks of this week (`w3_pump_QHE`). They are once again over [here](http://tiny.cc/topocm_smc).
There are really plenty of things that one can study with the quantum Hall effect and pumps. Remember, that you don't need to do everything at once (but of course all of the simulations are quite fun!)
### Pumping with disorder
......@@ -38,13 +36,13 @@ Grab the simulations of the Thouless pump, and see what happens to the pump when
Take a look at how we calculate numerically the spectrum of Landau levels in the Laughlin argument chapter.
We were always careful to only take weak fields so that the flux per unit cell of the tight binding lattice is small.
This is done to avoid certain [notorious insects](http://en.wikipedia.org/wiki/Hofstadter%27s_butterfly), but nothing should prevent you from cranking up the magnetic field and seeing this beautiful phenomenon.
This is done to avoid certain [notorious insects](https://en.wikipedia.org/wiki/Hofstadter%27s_butterfly), but nothing should prevent you from cranking up the magnetic field and seeing this beautiful phenomenon.
Plot the spectrum of a quantum Hall layer rolled into a cylinder at a fixed momentum as a function of $B$ as $B$ goes to one flux quantum per unit cell, so in lattice units $B = 2\pi$. Bonus (requires more work): attach a lead to the cylinder, calculate pumping, and color the butterfly according to the pumped charge.
### Graphene
Take a look at how to implement a honeycomb lattice in Kwant [tutorials](http://kwant-project.org/doc/1.0/tutorial/tutorial4), and modify the Hall bar from the Laughlin argument notebook to be made of graphene. Observe the famous [unconventional quantum Hall effect](http://arxiv.org/abs/cond-mat/0602565).
Take a look at how to implement a honeycomb lattice in Kwant [tutorials](https://kwant-project.org/doc/1.0/tutorial/tutorial4), and modify the Hall bar from the Laughlin argument notebook to be made of graphene. Observe the famous [unconventional quantum Hall effect](https://arxiv.org/abs/cond-mat/0602565).
Bonus: See what happens to the edge states as you introduce a constriction in the middle of the Hall bar. This is an extremely useful experimental tool used in making quantum Hall interferometers (also check out the density of states using the code from the edge states notebook).
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......@@ -30,7 +30,7 @@ pi_ticks = [(-np.pi, r"$-\pi$"), (0, "0"), (np.pi, r"$\pi$")]
## Intro
Duncan Haldane from Princeton University will teach us about an interesting two dimensional toy-model which he [introduced](http://faculty.washington.edu/cobden/papers/haldane88.pdf) in 1988, and which has become a prototype for the anomalous quantum Hall effect.
Duncan Haldane from Princeton University will teach us about an interesting two dimensional toy-model which he introduced in 1988, and which has become a prototype for the anomalous quantum Hall effect.
```{code-cell} ipython3
:tags: [remove-input]
......@@ -44,7 +44,7 @@ We will now study the model in detail, starting from the beginning. Along the w
In the last chapter we saw how it is possible to obtain a quantum Hall state by coupling one-dimensional systems. At the end, our recipe was to first obtain a Dirac cone, add a mass term to it and finally to make this mass change sign. Following this recipe we were able to obtain chiral edge states without applying an external magnetic field.
There is a real (and a very important) two-dimensional system which has Dirac cones: [graphene](http://en.wikipedia.org/wiki/Graphene). So in this chapter we will take graphene and make it into a topological system with chiral edge states.
There is a real (and a very important) two-dimensional system which has Dirac cones: [graphene](https://en.wikipedia.org/wiki/Graphene). So in this chapter we will take graphene and make it into a topological system with chiral edge states.
Graphene is a single layer of carbon atoms arranged in a honeycomb lattice. It is a triangular lattice with two atoms per unit cell, type $A$ and type $B$, represented by red and blue sites in the figure:
......@@ -296,7 +296,7 @@ $$
\exp\,\left[i\gamma(k_x)\right]\,\exp\,\left(-i \int_0^T E[\mathbf{k}(t)]\,d t\right)\,\left|\psi(\mathbf{k})\right\rangle\,.
$$
We have made explicit the fact that $\gamma$ in our case may depend on $k_x$. We will not derive the formula for the Berry phase, something which can be done directly from the Schrödinger equation, see for instance [here](http://arxiv.org/abs/0907.2021). What is important to know about $\gamma$ is that it is a **geometric phase**: its value depends on the path $C$ but not on how the path is performed in time, so not on the particular expression for $\mathbf{k}(t)$. We'll soon see that sometimes it can have an even stronger, topological character.
We have made explicit the fact that $\gamma$ in our case may depend on $k_x$. We will not derive the formula for the Berry phase, something which can be done directly from the Schrödinger equation, see for instance [here](https://arxiv.org/abs/0907.2021). What is important to know about $\gamma$ is that it is a **geometric phase**: its value depends on the path $C$ but not on how the path is performed in time, so not on the particular expression for $\mathbf{k}(t)$. We'll soon see that sometimes it can have an even stronger, topological character.
### Flux pumping
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......@@ -493,7 +493,7 @@ From reading the paper, or just from the above discussion, you see that it takes
## A simplification: inversion symmetry
As just mentioned, topological invariants in higher dimensions are often difficult to write down and evaluate in the general case. Luckily, in the presence of [inversion symmetry](http://en.wikipedia.org/wiki/Parity_%28physics%29#Effect_of_spatial_inversion_on_some_variables_of_classical_physics) - which reverses the lattice coordinates with respect to a symmetry center - the topological condition can be stated in rather simple terms.
As just mentioned, topological invariants in higher dimensions are often difficult to write down and evaluate in the general case. Luckily, in the presence of [inversion symmetry](https://en.wikipedia.org/wiki/Parity_%28physics%29#Effect_of_spatial_inversion_on_some_variables_of_classical_physics) - which reverses the lattice coordinates with respect to a symmetry center - the topological condition can be stated in rather simple terms.
This turns out to be quite useful to describe most topological materials, which happen to have crystal structure with inversion symmetry.
From our earlier discussion, we know that a system is a time-reversal invariant topological insulator if it has an odd number of helical edge states. We will now see how we can find an expression for the bulk topological invariant, using inversion symmetry and bulk-boundary correspondence.
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......@@ -39,7 +39,7 @@ Video("-HRBuCgOUvs")
This topic is special, since in order to meaningfully discuss experimental progress we need to do something we didn't do before in the course: we will show you the measurements and compare them with the *simple* theoretical expectations. Like this we will see what agrees and what doesn't.
All the figures showing the experiments are copyright Physical Society of Japan (2008), published in [
J. Phys. Soc. Jpn. 77, 031007 (2008)](http://journals.jps.jp/doi/abs/10.1143/JPSJ.77.031007) by Markus König, Hartmut Buhmann, Laurens W. Molenkamp, Taylor Hughes, Chao-Xing Liu, Xiao-Liang Qi, and Shou-Cheng Zhang. They are available under CC-BY-NC-SA 4.0 International license.
J. Phys. Soc. Jpn. 77, 031007 (2008)](https://journals.jps.jp/doi/abs/10.1143/JPSJ.77.031007) by Markus König, Hartmut Buhmann, Laurens W. Molenkamp, Taylor Hughes, Chao-Xing Liu, Xiao-Liang Qi, and Shou-Cheng Zhang. They are available under CC-BY-NC-SA 4.0 International license.
## Two limits: Mexican hat and weak pairing
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......@@ -26,11 +26,9 @@ init_notebook()
## Simulations
As usual, start by grabbing the notebooks of this week (`w5_qshe`). They are once again over [here](http://tiny.cc/topocm_smc).
### Kane-Mele model
The first known implementation of quantum spin Hall effect is the Kane-Mele model, introduced in [this paper](http://arxiv.org/abs/cond-mat/0411737). It is a doubled copy of the Haldane model (get that one from the previous week's notebooks), with spin up and spin down having next-nearest neighbor hoppings complex conjugate of each other due to spin-orbit coupling.
The first known implementation of quantum spin Hall effect is the Kane-Mele model, introduced in [this paper](https://arxiv.org/abs/cond-mat/0411737). It is a doubled copy of the Haldane model (get that one from the previous week's notebooks), with spin up and spin down having next-nearest neighbor hoppings complex conjugate of each other due to spin-orbit coupling.
Implement the Kane-Mele model and add a staggered onsite potential to also be able to create a trivial gap. Calculate the scattering matrix topological invariant of that model.
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......@@ -200,7 +200,7 @@ MultipleChoice(
## Spectroscopy of the surface of a 3D topological insulator
Amazingly, the surface states of a 3D topological insulator can be directly imaged experimentally using the technique of Angle Resolved Photo Emission Spectroscopy - or, in short, [ARPES](http://en.wikipedia.org/wiki/Angle-resolved_photoemission_spectroscopy).
Amazingly, the surface states of a 3D topological insulator can be directly imaged experimentally using the technique of Angle Resolved Photo Emission Spectroscopy - or, in short, [ARPES](https://en.wikipedia.org/wiki/Angle-resolved_photoemission_spectroscopy).
In ARPES, X-ray photons are shot vertically into a metal surface in order to eject electrons out of it. Due to energy and momentum conservation, the emitted electrons have the same momentum (parallel to the surface) that they had in the crystal, and an energy which is related to their band energy in the crystal. Hence, ARPES is the ideal tool to measure the energy dispersion of the surface states of a solid, i.e. $E(k_x, k_y)$ as a function of momenta $\hbar k_x$ and $\hbar k_y$.
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......@@ -48,11 +48,11 @@ Let us now go through the main points that lead to this conclusion, and argue fo
## Crystallographic defects and topology
There are many different types of [defects in crystals](http://en.wikipedia.org/wiki/Crystallographic_defect): vacancies, substitutions, grain boundaries, dislocations, and many more.
There are many different types of [defects in crystals](https://en.wikipedia.org/wiki/Crystallographic_defect): vacancies, substitutions, grain boundaries, dislocations, and many more.
What kinds of defects are important for topology? Consider a vacancy for example:
![](figures/Formation_Point_Defect.png)
(By Safe cracker (Own work) [CC BY 3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons)
(By Safe cracker (Own work) [CC BY 3.0 (https://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons)
To create a vacancy, we need to remove a single atom (or all the atoms following one line). Can this type of defect carry a topologically protected state?
......@@ -101,7 +101,7 @@ To see how the weak topological invariant relates to the number of states in the
![](figures/dislocation_helical.svg)
(adapted from Cdang (Own work), via Wikimedia Commons, [CC BY-SA 3.0](http://creativecommons.org/licenses/by-sa/3.0).)
(adapted from Cdang (Own work), via Wikimedia Commons, [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0).)
Counting the number and the orientation of the crystal planes approaching the core of the dislocation is just the Burgers vector. Hence, the number of edge states entering the dislocation core is the Burgers vector times the number of states per crystal plane. This brings us to the conclusion:
......
......@@ -256,7 +256,7 @@ We are now ready to see how unpaired Majoranas can appear in a 2D $p$-wave super
So far we have considered a uniform superconducting pairing $\Delta$, with constant amplitude and phase. This is an idealized situation, which corresponds to a perfect superconductor with no defects.
If you apply a small magnetic field to a superconducting film, or if there are defects in the material, a [**vortex**](http://en.wikipedia.org/wiki/Abrikosov_vortex) of supercurrent can form to lower the free energy of the system.
If you apply a small magnetic field to a superconducting film, or if there are defects in the material, a [**vortex**](https://en.wikipedia.org/wiki/Abrikosov_vortex) of supercurrent can form to lower the free energy of the system.
In a vortex, there is a supercurrent circulating in a small area around the defect or the magnetic field lines penetrating the superconductor. The magnetic flux enclosed by the vortex supercurrent is equal to a superconducting flux quantum $h/2e$.
The amplitude $\Delta$ of the superconducting pairing is suppressed in the core of the vortex, going to zero in its center, and the superconducting phase winds by $2\pi$ around a closed path surrounding it. The situation is sketched below:
......@@ -367,9 +367,9 @@ MultipleChoice(
## Vortices in 3D topological insulator
Unfortunately, superconductors with $p$-wave pairing are very rare, with mainly [one material](http://en.wikipedia.org/wiki/Strontium_ruthenate) being a good candidate. But instead waiting for nature to help us, we can try to be ingenious.
Unfortunately, superconductors with $p$-wave pairing are very rare, with mainly [one material](https://en.wikipedia.org/wiki/Strontium_ruthenate) being a good candidate. But instead waiting for nature to help us, we can try to be ingenious.
As Carlo mentioned, Fu and Kane [realized](http://arxiv.org/abs/0707.1692) that one could obtain an effective $p$-wave superconductor and Majoranas on the surface of a 3D TI.
As Carlo mentioned, Fu and Kane [realized](https://arxiv.org/abs/0707.1692) that one could obtain an effective $p$-wave superconductor and Majoranas on the surface of a 3D TI.
We already know how to make Majoranas with a 2D topological insulator. Let us now consider an interface between a magnet and a superconductor on the surface of a 3D topological insulator. Since the surface of the 3D TI is two dimensional, such an interface will be a one dimensional structure and not a point defect as in the quantum spin-Hall case.
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......@@ -290,7 +290,7 @@ displaymd(table_header.format(colordefs=colordefs, fmt=format_string, body=block
The somewhat cryptic notations in the leftmost column are just the names of the different symmetry classes. (Also the 'I's appearing there are Roman cardinal numbers, so for instance BDI is 'B D one', and AIII is "A three".)
Their names come from an elegant mathematical classification of [symmetric spaces](http://en.wikipedia.org/wiki/Symmetric_space) worked out by [Elie Cartan](http://en.wikipedia.org/wiki/%C3%89lie_Cartan) in 1926. While it is definitely intriguing that a group theory result from 1926 reappears in a totally different context almost 80 years later, the origin of this nomenclature is not directly relevant to most of the theory done in the field.
Their names come from an elegant mathematical classification of [symmetric spaces](https://en.wikipedia.org/wiki/Symmetric_space) worked out by [Elie Cartan](https://en.wikipedia.org/wiki/%C3%89lie_Cartan) in 1926. While it is definitely intriguing that a group theory result from 1926 reappears in a totally different context almost 80 years later, the origin of this nomenclature is not directly relevant to most of the theory done in the field.
The two complex classes are A and AIII.
```{code-cell} ipython3
......@@ -477,7 +477,7 @@ For example, this means that in the SSH chain, which is a $d=1$ dimensional syst
The first thing to observe is that the complex classes only have $\mathbb{Z}$ invariant. We already know what these invariants are in low dimensions: the Chern number, which we encountered in quantum Hall systems (class A, $d=2$), and the winding number of the reflection matrix, which we encountered when we studied reflection from the Thouless pump.
The higher dimensional invariants are simple generalizations of these two. Their mathematical expression can be found in several papers, for instance [this one](http://arxiv.org/abs/1104.1602).
The higher dimensional invariants are simple generalizations of these two. Their mathematical expression can be found in several papers, for instance [this one](https://arxiv.org/abs/1104.1602).
```{code-cell} ipython3
:tags: [remove-input]
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