Chapter 36 Exercises: Topological Phases of Matter

Part A: Conceptual Questions (*)

These questions test your understanding of the core ideas. No calculations required.

A.1 Explain in your own words why a topological invariant must be an integer. Why does this integer nature automatically imply robustness against perturbations?

A.2 The genus (number of holes) of a surface is a topological invariant. Give three examples of physical objects with genus 0, genus 1, and genus 2. Then explain why you cannot smoothly deform an object of genus 1 into an object of genus 0 without tearing.

A.3 A friend claims: "The quantum Hall effect is just a very precise measurement of the Hall resistance — it is not qualitatively different from the classical Hall effect." Explain why your friend is wrong. What is fundamentally different about the quantum Hall plateau from the classical linear Hall resistance?

A.4 Explain the bulk-boundary correspondence in one sentence. Then give an intuitive argument for why it must be true: why must gapless states appear at the interface between two regions with different topological invariants?

A.5 What is the essential difference between a topological insulator and an ordinary insulator? Both have a bulk band gap and both can have surface states — so what makes the topological insulator's surface states special?

A.6 Why do non-abelian anyons require two dimensions? What is it about the topology of particle exchange paths in 2D that differs from 3D?

A.7 Explain the "highway with no U-turns" analogy for chiral edge states in your own words. Then identify one way in which this analogy breaks down — what aspect of the quantum physics is not captured by the highway metaphor?

A.8 The 2016 Nobel Prize in Physics was awarded for "theoretical discoveries of topological phase transitions and topological phases of matter." Why is the phrase "topological phase transition" somewhat paradoxical? What must happen at a topological phase transition that cannot happen during ordinary smooth deformations?

Part B: Applied Problems (**)

These problems require direct application of the chapter's key equations.

B.1: Landau Level Basics

Consider a 2D electron gas in a magnetic field $B = 8.0$ T. The effective mass in GaAs is $m^* = 0.067 m_e$.

(a) Calculate the cyclotron frequency $\omega_c = eB/m^*$.

(b) Calculate the Landau level spacing $\hbar\omega_c$ in meV.

(c) What temperature corresponds to this energy scale? (i.e., at what temperature $T$ is $k_BT = \hbar\omega_c$?) Why must quantum Hall experiments be performed at temperatures well below this?

(d) If the sample area is $A = 4\,\text{mm}^2$, how many states are in each Landau level?

(e) If the 2D electron density is $n_s = 3.0 \times 10^{15}\,\text{m}^{-2}$, how many Landau levels are filled at this magnetic field? (The filling fraction is $\nu = n_s h/(eB)$.)

B.2: Quantum Hall Conductance

(a) Calculate the von Klitzing constant $R_K = h/e^2$. Express your answer in ohms.

(b) If exactly $n = 3$ Landau levels are filled, what is the Hall resistance $R_{xy}$? What is the longitudinal resistance $R_{xx}$?

(c) The quantum Hall resistance has been measured to agree with $h/(ne^2)$ to a relative uncertainty of $10^{-10}$. In absolute terms, how many ohms of uncertainty does this correspond to for the $n = 1$ plateau?

(d) Explain why the longitudinal resistance $R_{xx}$ vanishes when the Hall resistance is on a plateau. (Hint: think about what "localized states at the Fermi energy" means for longitudinal transport.)

B.3: Berry Curvature for a Two-Level System

Consider a two-band Hamiltonian $\hat{H}(\mathbf{k}) = \mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma}$ with:

$$\mathbf{d}(\mathbf{k}) = (\sin k_x,\, \sin k_y,\, 2 - \cos k_x - \cos k_y)$$

(a) Show that the gap closes (i.e., $|\mathbf{d}| = 0$) only when $d_x = d_y = d_z = 0$ simultaneously. At what point(s) in the Brillouin zone does this occur, and for what value of the parameter "2" (if it were variable: $m - \cos k_x - \cos k_y$)?

(b) Compute $|\mathbf{d}(\mathbf{k})|$ at the high-symmetry points $\Gamma = (0,0)$, $X = (\pi, 0)$, $M = (\pi, \pi)$.

(c) The Berry curvature formula for a two-band model is: $$\mathcal{F}(\mathbf{k}) = \frac{1}{2}\hat{\mathbf{d}} \cdot \left(\frac{\partial \hat{\mathbf{d}}}{\partial k_x} \times \frac{\partial \hat{\mathbf{d}}}{\partial k_y}\right)$$

Evaluate $\mathcal{F}$ at $\mathbf{k} = (0, 0)$ for the model above.

(d) Using the code from this chapter (or writing your own), compute the Chern number numerically. You should get $C = 1$.

B.4: Haldane Model Phase Diagram

The Chern number of the Haldane model is:

$$C = \frac{1}{2}\left[\text{sgn}(M - 3\sqrt{3}t_2\sin\phi) - \text{sgn}(M + 3\sqrt{3}t_2\sin\phi)\right]$$

(a) For $t_2 = 0.1$ and $\phi = \pi/2$, determine the range of $M$ values for which $C = +1$, $C = -1$, and $C = 0$.

(b) Sketch the phase diagram in the $(M/t_2, \phi)$ plane, labeling each region with its Chern number.

(c) What happens exactly on the phase boundary? What physical quantity goes to zero?

(d) The original Haldane model has $\phi$ as a tunable parameter. In a real material, what physical mechanism would play the role of $\phi$? Why did it take until 2014 to realize the Haldane model experimentally?

B.5: Filling Fractions and Plateaus

In a quantum Hall experiment, the filling fraction is $\nu = n_s h / (eB) = n_s / n_\phi$, where $n_\phi = eB/h$ is the flux density (number of flux quanta per unit area).

(a) For $n_s = 2.5 \times 10^{15}\,\text{m}^{-2}$, at what magnetic field values do you expect plateaus at $\nu = 1, 2, 3$?

(b) As $B$ increases from $2$ T to $15$ T, in what order do you expect to see the integer Hall plateaus?

(c) The width of each plateau (in magnetic field) depends on the amount of disorder. Explain qualitatively why more disorder makes the plateaus wider, not narrower. (This is the opposite of what you might naively expect!)

Part C: Computational Problems (-*)

These problems involve quantitative analysis that benefits from numerical computation.

C.1: Chern Number Calculator (**)

Using the chern_number() function from code/example-01-topology.py (or writing your own implementation):

(a) Verify that the two-band model $\mathbf{d}(\mathbf{k}) = (\sin k_x, \sin k_y, m - \cos k_x - \cos k_y)$ has: - $C = +1$ for $0 < m < 2$ - $C = -1$ for $-2 < m < 0$ - $C = 0$ for $|m| > 2$

Compute the Chern number for $m = -3, -1.5, -0.5, 0.5, 1.5, 3$.

(b) Plot the Berry curvature $\mathcal{F}(k_x, k_y)$ as a heat map over the Brillouin zone for $m = 1.0$ and $m = 0.1$. How does the Berry curvature distribution change as $m$ approaches the phase transition at $m = 0$?

(c) Plot the Chern number as a function of $m$ for $m \in [-4, 4]$. Verify that it is a step function with jumps at $m = -2, 0, +2$.

C.2: Edge States in a Strip Geometry (***)

Consider the two-band model from C.1 on a strip geometry: periodic boundary conditions in $y$ (so $k_y$ is a good quantum number) but open boundary conditions in $x$ (a strip of width $N_x$ sites).

(a) For each $k_y$, diagonalize the $2N_x \times 2N_x$ Hamiltonian matrix. Plot the energy spectrum $E(k_y)$ for $m = 1.0$ (topological phase, $C = 1$) with $N_x = 30$.

(b) Identify the edge states — bands that cross the bulk gap. How many edge states cross the gap from valence to conduction band? How does this number relate to the bulk Chern number?

(c) Repeat for $m = 3.0$ (trivial phase, $C = 0$). Verify that no edge states cross the gap.

(d) For the topological case ($m = 1.0$), compute the probability density $|\psi(x)|^2$ of an edge state as a function of position across the strip. Verify that it is localized at one edge (exponentially decaying into the bulk).

C.3: Haldane Model Phase Diagram (***)

Implement the full Haldane model Hamiltonian (see the code module or Haldane's original paper) and numerically compute the Chern number across the phase diagram.

(a) Create a 2D grid of $(M/t_2, \phi)$ values with $M/t_2 \in [-10, 10]$ and $\phi \in [0, 2\pi]$. Compute $C$ at each grid point.

(b) Plot the phase diagram as a color map, with different colors for $C = -1, 0, +1$.

(c) Overlay the analytical phase boundaries $M = \pm 3\sqrt{3} t_2 \sin\phi$ and verify agreement with your numerical results.

(d) Along the line $\phi = \pi/2$, plot the band gap as a function of $M/t_2$. Verify that the gap closes at the topological phase transitions.

Part D: Synthesis Problems (***)

These problems require integrating multiple concepts and thinking beyond direct formulas.

D.1: Topology, Symmetry, and Classification

(a) The "periodic table of topological insulators" classifies all possible topological phases based on dimensionality and symmetry class. Without going into full detail, explain why the classification depends on three symmetries: time-reversal ($T$), particle-hole ($C$), and chiral ($S = TC$). Why are these three symmetries special?

(b) The integer quantum Hall state exists in symmetry class A (no symmetries). The topological insulator exists in class AII (time-reversal symmetry with $T^2 = -1$). Explain physically why $T^2 = -1$ (rather than $T^2 = +1$) is required for spin-1/2 topological insulators. (Hint: Kramers theorem.)

(c) Why can't a topological insulator exist in one dimension with time-reversal symmetry? (Think about what "edge states" would mean for a 1D system.)

(d) The quantum Hall effect requires a magnetic field, which breaks time-reversal symmetry. The topological insulator requires time-reversal symmetry. Can both exist in the same material at the same time? Explain.

D.2: From Berry Phase to Chern Number

(a) In Chapter 32, we computed the Berry phase $\gamma = \oint \mathcal{A} \cdot d\mathbf{k}$ for a spin-1/2 in a rotating magnetic field. Show that the Berry phase for a complete cycle ($\theta$ from $0$ to $\pi$, $\phi$ from $0$ to $2\pi$) is $\gamma = \pi$, and relate this to the Chern number of the spin-1/2 Bloch sphere.

(b) The Chern number can be written as a sum of Berry phases around the boundary of the Brillouin zone. Using Stokes' theorem, show that $C = \frac{1}{2\pi}\int_{\text{BZ}} \mathcal{F}\,d^2k = \frac{1}{2\pi}\oint_{\partial\text{BZ}} \mathcal{A}\cdot d\mathbf{k}$. But the BZ is a torus, which has no boundary! Resolve this apparent contradiction. (Hint: gauge singularities.)

(c) Explain why the Chern number of a time-reversal invariant system must vanish. (Hint: under time-reversal, $\mathcal{F}(\mathbf{k}) \to \mathcal{F}(-\mathbf{k})$ with a sign change.)

D.3: Topological Protection and Error Rates

Consider a topological qubit encoded in two Majorana zero modes separated by a distance $L$ in a topological superconductor with coherence length $\xi$.

(a) The error rate due to thermal excitations scales as $\Gamma_{\text{thermal}} \sim e^{-\Delta/(k_BT)}$, where $\Delta$ is the bulk gap. For $\Delta = 0.3$ meV and $T = 20$ mK, estimate the thermal error rate relative to the attempt frequency ($\sim \Delta/\hbar$).

(b) The error rate due to Majorana splitting (overlap between the two modes) scales as $\Gamma_{\text{split}} \sim e^{-L/\xi}$. For $\xi = 200$ nm and $L = 2\,\mu\text{m}$, estimate this error rate.

(c) Compare these topological error rates to the typical decoherence rates of superconducting qubits ($T_1 \sim 100\,\mu$s) and trapped-ion qubits ($T_1 \sim 1$ s). Is topological protection expected to outperform these standard approaches?

(d) Discuss the tradeoffs: topological qubits promise lower error rates but require exotic materials and have not yet been demonstrated. Standard qubits have higher error rates but are well-understood and commercially available. What would need to change for topological qubits to become competitive?

D.4: Thought Experiment — Adiabatically Threading Flux

Consider a quantum Hall system on a torus (periodic boundary conditions in both $x$ and $y$). Imagine slowly threading one quantum of magnetic flux $\Phi_0 = h/e$ through the hole of the torus (in the $x$-direction).

(a) By gauge invariance, the system must return to an eigenstate after one flux quantum is threaded. But it need not return to the same eigenstate. Use the adiabatic theorem to argue that one electron is transferred from one edge of the system to the other during this process. (This is Laughlin's argument.)

(b) Laughlin's argument shows that the Hall conductance is $\sigma_{xy} = ne^2/h$ where $n$ is the number of electrons transferred per flux quantum. Explain why $n$ must be an integer.

(c) This argument does not use translation symmetry or Bloch's theorem — it works even in a disordered system. Why is this significant?

Part E: Research and Exploration (****)

These problems go beyond the chapter's direct content and require independent investigation.

E.1: The Fractional Quantum Hall Effect

Research the fractional quantum Hall effect (FQHE), discovered by Tsui, Stormer, and Gossard in 1982.

(a) What filling fractions $\nu$ have been observed experimentally? Why do they form a specific pattern (the Jain sequence)?

(b) The FQHE cannot be explained by non-interacting electrons. What role do electron-electron interactions play? What is the Laughlin wave function?

(c) The quasiparticles of the FQHE carry fractional electric charge. How has this been verified experimentally?

(d) Which fractional quantum Hall states are believed to host non-abelian anyons? What experimental evidence exists?

E.2: Experimental Topological Insulators

Research the experimental discovery and characterization of topological insulators.

(a) Describe the König et al. (2007) experiment on HgTe quantum wells that provided the first evidence for a 2D topological insulator. What was measured, and why did the result confirm topological behavior?

(b) How is ARPES (angle-resolved photoemission spectroscopy) used to identify 3D topological insulators? What signature distinguishes a topological from a trivial insulator in ARPES data?

(c) What are the potential applications of topological insulators in technology (spintronics, thermoelectrics, quantum computing)?

E.3: The 2016 Nobel Prize

Research the contributions of Thouless, Haldane, and Kosterlitz that led to their 2016 Nobel Prize.

(a) Thouless' contribution was the TKNN formula. Explain it in your own words.

(b) Haldane's contribution was the Haldane model. Why was a model of a "quantum Hall effect without Landau levels" theoretically important, even before it was experimentally realized?

(c) Kosterlitz and Thouless (KT) discovered a topological phase transition in 2D systems that does not involve symmetry breaking. Describe the KT transition and explain how it differs from conventional phase transitions.

Part M: Mixed Practice

These problems mix concepts from Chapter 36 with material from earlier chapters.

M.1: A spin-1/2 particle is placed in a magnetic field $\mathbf{B} = B(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$. (a) From Chapter 32, the Berry phase for a complete circuit at polar angle $\theta$ is $\gamma = -\Omega/2$ where $\Omega$ is the solid angle. Compute $\gamma$ for $\theta = \pi/3$. (b) The Berry curvature on the Bloch sphere is $\mathcal{F} = \frac{1}{2}\sin\theta$. Verify that $\int_0^{2\pi}\int_0^\pi \mathcal{F}\,\sin\theta\,d\theta\,d\phi = 2\pi$, confirming $C = 1$.

M.2: From Chapter 26, the tight-binding model on a 1D chain gives the band structure $E(k) = -2t\cos(ka)$. (a) For a chain with alternating bond strengths $t_1$ and $t_2$ (the SSH model), show that the band structure is $E(k) = \pm\sqrt{t_1^2 + t_2^2 + 2t_1 t_2\cos(ka)}$. (b) When $t_1 > t_2$, the winding number is $w = 0$ (trivial). When $t_1 < t_2$, $w = 1$ (topological). Explain qualitatively why the topological phase has zero-energy edge states at the ends of a finite chain.

M.3: From Chapter 33, decoherence destroys quantum information at a rate $\Gamma = 1/T_2$. (a) For a standard superconducting qubit with $T_2 = 50\,\mu$s, how many gate operations (each taking $20$ ns) can be performed before decoherence? (b) If a topological qubit achieves an effective $T_2 = 1$ s (due to topological protection), how does this change the answer? (c) Why might the gate operations themselves be slower for topological qubits? (Think about the braiding process.)

M.4: From Chapter 4, the quantum harmonic oscillator has energy levels $E_n = \hbar\omega(n + 1/2)$. The Landau levels also have this structure with $\omega = \omega_c = eB/m$. (a) Show that the magnetic length $\ell_B = \sqrt{\hbar/(eB)}$ plays the role of the oscillator length $\sqrt{\hbar/(m\omega)}$. (b) Calculate $\ell_B$ for $B = 10$ T. (c) The number of states per Landau level equals the number of non-overlapping oscillator ground states that fit in the sample. Verify that this gives $N_\phi = A/(2\pi\ell_B^2) = eBA/h$.