Condensed Matter Physics

Fall 2025 · Graduate Level

Course Overview

Key topics covered in this course

This graduate-level course explores fundamental concepts in condensed matter physics, from magnetic phenomena to superconductivity. As the teaching assistant, I design the weekly exercise sheets and lead the tutorial sessions where students actively engage with the material.

Course Format: Exercises are not graded and do not need to be handed in. The only requirement is attendance at a minimum of 10 out of 14 tutorial sessions throughout the semester.

Hall Effect Superconductivity Quantum Oscillation Magnetism Ginzburg-Landau Theory

Tutorial Session

A flipped classroom approach

This tutorial takes a non-conventional approach: instead of the TA presenting solutions while students listen, we flip the roles. Students volunteer to present their solutions and thoughts at the blackboard, while I listen, provide feedback, clarify misconceptions, and offer deeper insights.

The Feynman Technique: This format is inspired by Richard Feynman's learning method—you truly understand something when you can explain it to others. By presenting to your peers and fielding their questions, you're forced to think deeply about the material and identify gaps in your understanding.

This approach benefits everyone in the room:

→ For presenters: Explaining concepts out loud solidifies your understanding. Anticipating questions forces you to think critically about what you've learned.
→ For listeners: Hearing different approaches and viewpoints from your peers often reveals alternative methods and fresh perspectives you might not have considered.
→ For exam preparation: Since the final exam involves presenting an exercise solution, this is the perfect opportunity to practice in a low-stakes environment throughout the semester.

This is your stage—feel free to volunteer and step up to the blackboard!

Examination

Oral exam format

The course concludes with an oral examination, where I serve as part of the examiner team alongside the course instructor. The exam consists of two components:

1 Topic Presentation (~10 min) — The student presents a randomly selected topic from the course material, demonstrating their understanding and ability to explain the concepts clearly.
2 Exercise Walk-through — An exercise is selected from the semester's exercise sheets, and the student presents their approach and solution on the spot.

Why practice matters: The more familiar you are with the exercises throughout the semester, the more confident and prepared you'll be during the exam. Our flipped tutorial format is designed precisely for this—by the end of the semester, presenting in front of the examiners will feel natural.

Exercise Sheets

A curated mix of classic and new problems

The exercises draw from an existing pool from last year's course, supplemented with new problems I designed myself. This hybrid approach was necessary for two reasons:

First, this year's course arrangement differs from last year, so some adjustments were needed. Second, I wanted to create exercises that help students better bridge the conceptual gaps between different topics, while also providing them with extra insights that go beyond standard textbook problems.

Tutorial Reflections & Exercise Details

Detailed commentary on each exercise sheet, student feedback, and workload statistics

Side note: Each exercise is marked with a "NEW" or "OLD" tag. "NEW" means the exercise was created by myself this year, while "OLD" indicates it was taken from last year's exercise pool.

1 Exercise Sheet 1

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Commentary

OLD

Standard exercise. Very light-weighted.

2 Exercise Sheet 2

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Commentary

MIXED

The first exercise comes from a textbook The Oxford Solid State Basics.

The second exercise is made by myself. The idea is to guide the student through the process of continue-ize (make the system continuous, opposite to discretize) the Heisenberg model. This provides a different perspective to understand the Heisenberg model. Also it is way easier to get the dispersion from a classical continuous system.

The third and the fourth exercises are from last year's exercises pool.

3 Exercise Sheet 3

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Commentary

OLD

Standard exercise. Nothing special.

4 Exercise Sheet 4

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Commentary

OLD

Standard exercise. Nothing special.

5 Exercise Sheet 5

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Commentary

NEW

The two exercises are new and made by myself.

The first exercise builds up the gap between the Landau theory and the Ginzburg-Landau theory. It helps the student to transit from a uniform system to a non-uniform system. Like in the exercise sheet 6, I compare the non-uniform Ginzburg-Landau theory with the simple Ising model. The idea behind it is that I want to illustrate the physical meaning of the gradient term \(\nabla \eta(x)\) in free energy.

In fact, if you continue-ize the Ising model, a gradient term also pops up in the Hamiltonian. Since the students are already familiar with the Ising model, they can use this model to understand and interpret the gradient term in the Ginzburg-Landau theory — this is nothing but a nearest-neighbor interaction.

The second exercise is for the students to get a deeper understanding of the susceptibility. As I always do in the exercise sheet, I always like to start from a toy model or from a minimal system that is simple enough to grasp but still captures all the key features of the problem.

In this exercise, I start from a simplest \(\delta\) function, and extend it to a more general function later on. This step-by-step approach could help the student to slowly build up their own understanding of the concept of susceptibility.

Once they get to a more general form of the susceptibility, there comes the math. They are asked to derive the susceptibility in a more specific but more complex system. This involves a bit of Fourier transform and residue theorem.

And then the exercise stops here. Ideally, there could be a follow-up exercise on the application of the susceptibility. This follows the learning theory: you understand, then you apply. However, the standard time the students should spend on each exercise sheet is 3–5 hours. It would be too much for the students if I squeeze in too many exercises in one sheet.

6 Exercise Sheet 6

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Commentary

OLD

Old exercise, not so much to say about it.

Time Spent (13 responses)

1 hour (2, 15%)

2 hours (7, 54%)

3 hours (4, 31%)

Student Feedback

Exercise 1b is the hardest, since the only method to do the fit is by brute force with enough data points.

I really liked it cause I like to fit data. Hardest was probably b)

I had difficulty with estimating the exponents on the non-log plots (fig 2 and 3), perhaps there's a useful tool for that.

Biggest challenge was extracting the data from the plots for fitting.

I did not find any part particularly difficult, but after talking with others I would imagine if I did try to upload the data instead of just using numbers from the plot it would have been harder.

7 Exercise Sheet 7

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Commentary

MIXED

The first exercise is taken from last year (2024) exercises pool. Very standard exercise. The interesting part is the use of a programming language to plot out the dispersion of the electron.

The second and the third exercises are new. As stated in the preface, I intended to build up a very simple toy model in quantum mechanics but still capture the feature of state hopping. Therefore, the setup of the second exercise is a two level system with a static magnetic field. The students were asked to write out the Hamiltonian and derive everything in matrix form. Given that only the Dirac notation was used in the lecture, I think it would be beneficial to think of the same problem from a different perspective (matrix representation).

Try-out: Another try-out is the use of ChatGPT. This doesn't mean that I used ChatGPT to help me, but rather, I actively ask the student to use ChatGPT to explain the concepts covered in the exercise sheet. In the second exercise, GPT will be used by the student to interpret the off-diagonal elements in the Hamiltonian.

The third exercise is also made by myself. This is about an electron in a periodic potential, and it has more connections to the lecture. Since close to the BZ boundary, the electron experiences a strong perturbation and effectively the Hamiltonian becomes 2-dimensional, and thus the degenerate perturbation theory was applied in the lecture.

I want the student to connect this effective 2-dimensional Hamiltonian with the 2-level Hamiltonian in the last exercise. Mathematically they are quite similar, so we can simply use the same method to solve them, or interpret them in the same way. Especially the off diagonal elements in the Hamiltonian: in the two-level system, they represent the hopping between the two levels. In this exercise, the "hopping" interpretation can again be applied.

Philosophy: I want to help the student build up a connection between two seemingly different problems which actually share the same mathematical structure under the hood.

Time Spent (11 responses)

3 hours (6, 55%)

4 hours (3, 27%)

5 hours (2, 18%)

Student Feedback

i liked the sheet very much, especially ex2 and ex3 were very instructive. i think you're doing a good job getting us to think about the physics behind the math. difficulty was fine for all exercises.

I really appreciated exercise 2 and 3, I think those concepts were never explained to me in that way and it really helped my understanding and intuition. The hardest, I thought, was probably 3 even though I did not get around to finishing it. And I spent the most time on 1 because I really wanted to review some concepts and make sure I understood it a bit better, though I think more discussion today will be fruitful.

This week I was very busy so I could only do one exersice. But The optional ones do seem more relevant to the lecture

No time to really work on it

8 Exercise Sheet 8

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Commentary

MIXED

The first exercise on the quantum Hall effect is partially taken from last year (2024) exercises pool. Very standard. The students were asked to use the Bohr-Sommerfeld quantization to calculate the energy levels of an electron in a magnetic field.

At first sight, the \(p\) in the quantization scheme looks like a mechanical momentum, but it is actually a canonical momentum. Inside the magnetic field, the students need to be careful about which momentum to use: the additional vector potential \(A\) will modify the canonical momentum.

Therefore, I slightly changed the exercise from last year, added in some hints on the usage of the canonical momentum. This helps the student to better understand the quantization scheme.

The second exercise on the Landau gauge in the Hall effect is new and made by myself. In the first part, the students were asked to do some math to fill the missing steps in the derivation in the lecture.

Questions raised:

(1) How do we understand the shift in position and why does it depend on \(k\)?

(2) How come the energy does not depend on \(k\)?

This leads to a further discussion on the usage of canonical momentum and the mechanical momentum.

In the last part of the exercise, another question was raised: Does the electron behave like a free particle in y direction? Actually, I don't have an answer to this question. However, when I asked this question in the tutorial session, a student came up to me with some of his rough ideas about it. I proposed that he can try to write a note and share his ideas with the class.

His note: (to be added)

I really appreciate his enthusiasm and effort to think about the problem. His ideas and thoughts will be very beneficial to everyone including me.

Time Spent (13 responses)

3 hours (5, 38%)

4 hours (4, 31%)

5 hours (3, 23%)

5+ hours (1, 8%)

Student Feedback

difficulty was good. ex2 a3 took me the longest. an intuitive explanation of ex2 a4 and b4 would be nice in exercise class

It was long for the fact wasn't uploaded in the last "working" week. Otherwise the exercises were good :)

I somehow did 2(b1) with Lagrangian mechanics instead and classical mechanics is hard as usual :)

good exercise sheet!

I appreciate the step by step process in the sheet and how it guides us through the thinking, but I found it a bit difficult with the changing of the sheet that every time I went back it seemed like something was new or changed.

1b) was not clear for me

9 Exercise Sheet 9

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Commentary

MIXED

The first exercise was taken from last year (2024) exercises pool. The students were asked to use a gauge-fixed London equation to prove that the magnetic field needs to be zero deep inside the superconductor. However, there is a mystery here: the exercise did not mention the gauge at all, which naturally leads to a question:

How come the current seems to be gauge dependent?

This becomes a question (raised by me) in the tutorial session. I asked the students to think about it by themselves for a week. I will come back to this question in the next tutorial session.

Actually, this "gauge-fixing" is implicitly done throughout the entire exercise sheet. The students will realize that everything seems to be gauge dependent. The difficult part is finding where this gauge fixing was done and starting from where the gauge cannot be changed any more.

In the second exercise, I guided the student through the process of constructing a free energy using only the symmetry consideration, together with some other natural assumptions (see the exercise sheet for details). This should provide some insights into why the free energy looks the way it does.

However, if the students take a closer look at the free energy, they will realize that the free energy is not gauge invariant... This means that the gauge is again implicitly fixed somewhere, but this is not stated. This would again be a question for the students, and will be discussed in the next tutorial.

The third exercise, examining the persistent current, is pretty standard. But in my opinion this is very important to understand superconductor physics: The biggest mystery in superconductors is why the current can flow without resistance. However, a lot of the times it is not discussed properly. Zero resistance is always not the main focus, but the Meissner effect is. Therefore, this exercise lets the student get a taste of the "zero resistance" phenomenon.

Follow-up: In the next tutorial, I answer the gauge mystery. Here is the note I wrote:

Gauge Mystery Note (PDF)

Time Spent (10 responses)

3 hours (4, 40%)

4 hours (3, 30%)

5 hours (2, 20%)

5+ hours (1, 10%)

Student Feedback

You could explain the more detailed derivation of flux quantization, because the way to treat the mechanical momentum is not obvious.

Variational derivative is challenging in terms of math, but overall straight forward.

Completely fine.

Really enjoyed the sheet a lot of very insightfull exercises that really helped digesting the lecture

Personally short on time but great exercise!

Very nice sheet!

I enjoyed the sheet, especially the derivation of the london equation using Landau theory was very nice

10 Exercise Sheet 10

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Commentary

OLD

Very standard exercise sheet. Nothing special. This is just to build up the gaps in the derivation in the lecture. There is a bit of maths involved in the first exercise, not difficult though.

In the second exercise, I asked the student to calculate or to find the frequency/energy/wavelength scale of some typical phenomena in nature. This is to help the student get the intuition of the order of magnitude of the phenomena.

Apparently, the AC Josephson effect and the visible light have similar frequency, and the electron cyclotron frequency is also similar to the microwave frequency.

System / mode	Typical parameter	Frequency f	Wavelength λ	Energy E
AC Josephson (1 V)	V = 1 V	≈ 480 THz	≈ 620 nm	2.0 eV
Visible light	λ = 550 nm	≈ 550 THz	550 nm	≈ 2.25 eV
Soft X-ray	E = 500 eV	≈ 120 PHz	≈ 2.5 nm	500 eV
Microwave	f = 10 GHz	10 GHz	≈ 3.0 cm	≈ 41 µeV
Plasma frequency (metal)	E_p ≈ 10 eV	≈ 2.4 PHz	≈ 124 nm	10 eV
Electron cyclotron	B = 1 T	≈ 28 GHz	—	—
Optical phonon	E ≈ 20 meV	≈ 4.8 THz	—	20 meV

Time Spent (12 responses)

3 hours (7, 58%)

4 hours (3, 25%)

5 hours (2, 17%)

Student Feedback

Can we look at SQUIDS and how one measurement works with all its physics behind?

Very good accordance to lecture. really helped deepen my understanding of the lecture material

exercise 3a was the most difficult, but over all difficulty was good

unfortunatley no time to work on it

11 Exercise Sheet 11

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Commentary

OLD

Standard exercise. Nothing special.

Overall Workload Summary

Aggregated data from sheets 6–10 (59 total responses)

59
responses

1 hour (2 responses, 3%)

2 hours (7 responses, 12%)

3 hours (26 responses, 44%)

4 hours (13 responses, 22%)

5 hours (9 responses, 15%)

5+ hours (2 responses, 3%)

Most students spend 3–4 hours on each exercise sheet, which aligns well with the target workload of 3–5 hours.