topocm / topocm_content

Course on topology in condensed matter

The source materials for the EdX course "Topology in Condensed Matter: Tying Quantum Knots"

The latest stable version of the course is located at http://topocondmat.org.

Open these notebooks in Binder: such that you can experiment with the code.

Development tip: shallow clone

Initially, this repository contained all the output of all computations, which means that its size is somewhat big (~300MB). In order to not download all the data, you can use the shallow cloning feature of git (at least v1.9) by using these or analogous commands:

mkdir topocm && cd topocm
git init
git remote add origin https://github.com/topocm/topocm_content.git # (Or the location of your fork)
git fetch --depth 1 origin +refs/tags/cleaned # Here we get the first commit that doesn't contain cruft
git fetch origin
git checkout master

The cleaned tag corresponds to the beginning of development that stores no output. Then you get a repository that does not contain any cruft data, and has a size of ~15MB.

Topocondmat.org

The notebooks in this repo are executed and converted to html and available on topocondmat.org. For building the website we use the pelican static site generator, available under AGPL license.

topocm_content's Issues

Typo/grammar W3: last chapter - Quantization of Hall Conductance from edge states

We want to consider the case that the system is in equilibrium, and we want to ask what are the current flowing in the system as a consequence of the small applied voltage V.

Is the current or are the currents.

4.1 QHE lattice typo in eq

In the equation in the text:

Now let us make this formal by first labeling the chains by an index ny, which takes integer values. Let us also replace k→kx to denote the wave-vector along a chain. Hence a single chain has a Hamiltonian [−(2tcosk+μ)τz+Δsinkxτy]⊗∣∣ny⟩⟨ny∣∣. The projector ∣∣ny⟩⟨ny∣∣ is needed to single out one chain out of the stack.

The (2tcosk+μ) needs to be (2t cos(k_x)+μ)

5.1 add space in question

First question:

Consider the simple case where spin is conserved.In the quantum spin Hall bar system above, what happens if, instead of applying a voltage between terminals 1 and 2, you manage to apply a spin-polarized current between terminals 1 and 2?

Fix sentence in 2.1

As christophedebeule noticed answering a question on the forum, the logical structure of the following sentence should be fixed:

"We should just keep the Hamiltonian the same and change the sign of all terms that break time reversal symmetry, for instance the term proportional to the magnetic field B."

In fact it's the reverse: all terms change sign, except those that change sign under T.

Mislabeled band structures

Fig. 1 of the nanowire notebook has labels interchanged.

Fix sentence in 5.1, 'A simplification: inversion symmetry'

See discussion at https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/8671e5f3171da6cf67799b515263298a/threads/55187f1f2a472d0b4b00278c#

7.2 master/w7_defects/figures/vacancy.png doesn't exist on the server.

404:

http://nbviewer.ipython.org/github/topocm/topocm_content/blob/master/w7_defects/figures/vacancy.png

typo in 1.2

The sentence

"The value k=0 is always present in the chain with boundary conditions "

is crucially missing periodic.

Clarifications for pumping

See https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/i4x-DelftX-TOPOCMx-course-1T2015/threads/54e618192a472dd33300072e for discussion.

Notation: make sign of mass parameters consistent in the courseware

In w4, w5, w6 the sign of the mass corresponding to the topological phase in the effective Dirac Hamiltonian and in the BHZ Hamiltonian changes from being positive to negative. It would be nice to have this consistent.

See https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/40b7703b3c6bf29312ce60adc187a6cc/threads/55248ab678f733803e002fce

From Kitaev chain to a nanowire, Small parameters, graph at wrong position.

It's a minor thing, but the graphs that show the Trivial and the topological band stucture are somewhere between cells that don't mention the figure.

It should be below: "The effective electron mass m is just the coefficient of the expansion. Let's take a look at the band structure in this regime, both in the topological regime and in the trivial regime:"

W2 Signatures.ipynb - Detecting the fermion parity switch using the Josephson effect

Typo in the question:

"Suppose that in your topological nanowire junction, positive energy quasiparticles can escapeinto a reservoir very quickly ..."

Typos w7_defects/ti_majoranas.ipynb

And the only other possible place where we could have a zero-energy soltuion is the core of the vortex.

Fix parameter values in 2.1

In 2.1 "Putting everything together" Delta is formatted as %1.2f, while it has three digits.

Also alpha misses a prefactor of 1/2 to match continuum.

Typo in 2.2

The relation between superconducting phase difference and flux is missing a factor of $2\pi$.

Mention magnetic field in Hall cylinder 3.2

The fact that the surface of the cylinder is subject to a radial field is not mentioned in text or drawings. See https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/274a4f6a5ece6fa10cf0c5e3d8ca2085/threads/54eb5cbe78f733e0ea000a84.

1D - Topological phases from the bulk spectrum, typo

Indeed, for these values of μ the two bands at negative and positive energy touch at E=0, for k=π and k=0 respectiely.

possible typo

topocm_content/w1_topointro/0d.ipynb

CHPT: The concept of a topological invariant
Below, we plot the energy levels along our path from H to H′ again, together with our topological invariant, the number of filled energy levels. You can see that this number changes from 2 to 3. Hence we can say that H and H′ are not topologically equivalent.

should probably state "...changes from 5 to 6."

Typo in equation in 3.1

There is an $i$ missing in front of the phase shift $\phi$.

4.2 Mistake in image

In this image:

N = 0 should be W = 0.

Fix typos in 5.1

See https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/i4x-DelftX-TOPOCMx-course-1T2015/threads/551061f12a472d801200243c

1D - Bulk topological invariant and the bulk-edge correspondence

"Following the procedure that we learned the previous week, we can therefore put H(0) and H(π) in antisymmetric form."

"we must encounter a zero-energy level crossing in the energy spectrum, what last week we called a fermion parity switch."

Previous week the course did not even start yet...

Typos in 2.3

As mentioned by christophe debeule on the forum:

1. In the section "Majorana zero modes in nanowire networks". In the equation underneath

   We can obtain the total fermion parity by multiplying all the operators Pn…

there should be an i instead of −i according to the convention of pairing Majoranas in the text.

1. In the section "Non-Abelian statistics of Majoranas". The braiding matrix U23 is not unitary. In the basis where ∣∣11⟩=c†1c†2∣∣00⟩, there should be all −i's. And the matrix U12 and U34 should be conjugated with the convention γ2n−1γ2n=−iPn. Also in the figure at the beginning of this section, the second and third Majorana at the top seem to have the wrong color.
2. In the section "Majoranas and quantum computation: basic ideas" at the end it says γn and γn instead of γn and γm. Also, "deatils".

Typo in 1.2

The same question has been posted in the Disusion group part of the edx.

I found a typo under the Bulk-edge correspondence, section "Bulk topological Invariant", the paragraph start with "This means that for these two Hamiltonians we should compute a Pfaffian.":
Should μ=−2t corresponding to a gap closing at k=0, and μ=2t with k=π?

I mean there is a minus sign difference when the original text referring to different situations.

Typo w7_defects/ti_majoranas.ipynb

So far we have consider a uniform superconducting pairing Δ, with constant amplitude and phase. This is an idealized situation, which would correspond to a perfect superconductor with no defects.

Clarify momentum structure of PHS in Kitaev chain

Explain how to keep track of momentum structure.

strong topological invariant in first unit of 6.1 needs explanation

The first unit in 6.1 states that $Q(k_x=0)Q(k_x=\pi)=\dots$ as a constraint on the strong invariant without an explanation. This needs to either be explained here or mentioned that this is something that will be explained later.

Also the product rule is confusing because we multiply 0 and 1.

Typo: QHEedgestates.ipynb - The harmless anomaly of the chiral edges

(In the Hall cylinder, this can be done by threading a time-dependent flux through the cylinder, as you have seen in the previou spart of the lecture).

5.1 consistent use of spelling

Sometimes you write counter-propagating and other times counterpropagating.

Selfcheck solutions visible in ipython

Not sure if this is intentionally or a bug (depends on the grading policy), but the correct solutions for the selfchecks are cleary accessible in the ipython notebooks.

Expand 'Study of a domain wall' unit in 5.1 with a figure

See discussion at https://courses.edx.org/courses/DelftX/TOPOCMx/1T2015/discussion/forum/140ee7bb5b67c2f92a0ab6352436b267/threads/551053022a472dd3330027bf.

missing quizes in 6.2

The two quizes in 6.2 seem to have disappeared

Typo's question, Braiding.ipynb

Typo in first question:
Consider an isolated system with N=7 pairs of Majorana, and whose that total fermion parity is evenWhat is the ground state degeneracy of the system?

Second question:
Consider a system with only one pair of Majorana modes, thus with just two degenerate states with different fermion parity.What happens when we exchange the pair of Majorana modes, starting from a given fermion parity eigenstate?

Grace period is removed on import

It is probably set incorrectly in the stored jsons.

Graph position: Nanowire.ipynb Putting everything together

The graph is above the text and immediately after the title.

The more natural position would be after the text:

"Let's calculate the gap as a function of all of the relevant parameters.

Here the vertical line denotes the critical value of Zeeman field at which the wire becomes topological."

Typo question w7_defects/ti_majoranas.ipynb

by adding a secind vortex, so the edge has no Majoranas now

and add a space between: Majorana.And

Typo haldane_model.ipynb

As you heard in the video, there might be zero energy states for some specific termination of the lattice, but these are not particularly interested

Typo in 1.2

Jay's ending video at 4:21, subtitle: particle whole symmetry -> particle hole symmetry
4:26, subtitle: i while --> While

Bug in 1.2 sectioning

In the unit "Topological phases from the bulk spectrum", the title of the paragraph "Band structure" is lost in markdown and is not displayed correctly.

Rephrase the question in 1.1

The first question in sec 1.1 extremely hard and confusing, which is especially bad since it's the first encounter of the questions. It should be formulated clearer and/or simplified.

From Kitaev chain to a nanowire, incorrect formulation of question.

In From Kitaev chain to a nanowire, The need for spin.

First is says: "The simplest thing which we can do is to ...", then that's not the way to go, because: "Just kidding, this would be very bad!"

Then it says: "What's the next best thing to do?", while it's not a next best thing, but mere the best thing or a correct thing to do.

Next best would imply there was already a thing that worked.

Expand domain-wall section in 1.2

Following a remark by zhongweizi, in 1.2 the solution of the Dirac equation with a domain wall should be expanded to explain the spinor structure.

Expand derivation of Kitaev topological invariant

Derivation of Q = sgn Pf[H(0)]*Pf[H(π)] would benefit from more details.

Typos in 1.2

Reported by SamMugel

• in the first paragraph of Fermion operators and Majorana operators, the last c operator should be a c^\dagger
• in the section The Kitaev chain model, I think that when you give the formula you use to transform to the BdG Hamiltonian there is an extra factor of 2 in front of the (c^\dagger c) term

Explain the fragility of chiral classes

In 8.1 add a remark that D, DIII, C, CI appear naturally, while AIII, BDI, CII require fine-tuning.

5.1 complex conjugation sign

The $S^\dagger$ should be $S^T$ in:

5.1 typo

the cylinder geometry is really equvalent to the Corbino disk, except that it easier to study.

possible typo

topocm_content/w1_topointro/0d.ipynb

"This immediately means that if $(\psi_A, \psi_B)^T$ is an eigenvalue of the Hamiltonian with energy $\varepsilon$"

should possibly read

"... eigenvector..."

5.2 video stereo problem

In the intro and the summery videos by Michael, there is only sound from the right speaker which makes it a bit uncomfortable to listen to on headphones.

https://www.youtube.com/watch?v=Gx-A0dWAjBg&t=244

https://www.youtube.com/watch?v=7M1vI9PhM6I&t=38

4.2 typo in last question

The Berry curvature and momentum change sign under time-reversal, so that the Berry curvatureat one momentum becomes opposite to the Berry curvature at opposite momentum.

add a space before "at one.....