Chapter 36 Quiz: Topological Phases of Matter

Instructions: This quiz covers the core concepts from Chapter 36. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. A topological invariant is robust against small perturbations because:

(a) It is very large and therefore difficult to change significantly (b) It is an integer and cannot change continuously (c) It is protected by energy conservation (d) It is a quantum number associated with a conserved symmetry

Q2. In the integer quantum Hall effect, the Hall conductance $\sigma_{xy} = ne^2/h$ is exactly quantized because:

(a) The magnetic field is extremely uniform (b) The sample is extremely pure with no defects (c) The integer $n$ is a topological invariant (Chern number) of the filled bands (d) Superconductivity eliminates resistance

Q3. The energy levels of a 2D electron in a perpendicular magnetic field are:

(a) Continuous, as in the classical Hall effect (b) Landau levels: $E_n = \hbar\omega_c(n + 1/2)$, each massively degenerate (c) Hydrogen-like levels: $E_n = -13.6/n^2$ eV (d) Band structure with a gap proportional to the spin-orbit coupling

Q4. A topological insulator is distinguished from an ordinary insulator by:

(a) A larger band gap (b) The presence of metallic surface/edge states protected by topology and symmetry (c) The absence of any surface states (d) A conducting bulk with an insulating surface

Q5. The bulk-boundary correspondence states that:

(a) Bulk and boundary have the same energy spectrum (b) A mismatch in topological invariants between two regions implies gapless boundary states (c) Edge states exist only when the bulk is conducting (d) The boundary conductance equals the bulk conductance

Q6. Chiral edge states in the quantum Hall effect are immune to backscattering because:

(a) The magnetic field prevents electrons from turning around (b) There are no states propagating in the opposite direction on the same edge (c) The edge is perfectly smooth (d) Electrons on the edge are spin-polarized

Q7. The Chern number $C = \frac{1}{2\pi}\int_{\text{BZ}} \mathcal{F}(\mathbf{k})\,d^2k$ is analogous to which mathematical result?

(a) The Pythagorean theorem (b) The Gauss-Bonnet theorem relating curvature to topology (c) The Fourier transform (d) The central limit theorem

Q8. In topological quantum computing, quantum gates are implemented by:

(a) Applying laser pulses to atoms (b) Braiding (exchanging) non-abelian anyons around each other (c) Measuring qubits in different bases (d) Cooling the system to absolute zero

Q9. Non-abelian anyons differ from ordinary fermions and bosons in that:

(a) They have fractional charge (b) Exchanging two of them produces a unitary matrix transformation, not just a phase (c) They exist only at high temperatures (d) They have zero mass

Q10. Which symmetry protects the edge states of a two-dimensional topological insulator?

(a) Translational symmetry (b) Rotational symmetry (c) Time-reversal symmetry (d) Particle-hole symmetry

True/False with Justification (4 questions)

Q11. TRUE or FALSE: The integer quantum Hall effect requires extremely pure samples to observe the quantized conductance plateaus.

Justification:

Q12. TRUE or FALSE: The Chern number of any band in a system with time-reversal symmetry must be zero.

Justification:

Q13. TRUE or FALSE: Topological quantum computing has been experimentally demonstrated using Majorana zero modes in semiconductor nanowires.

Justification:

Q14. TRUE or FALSE: A topological insulator has the same bulk band structure as an ordinary insulator — the difference appears only at the surface.

Justification:

Short Answer (4 questions)

Q15. Explain Laughlin's gauge argument for the quantization of Hall conductance in 2-3 sentences. Why does threading one flux quantum through a Hall cylinder transfer an integer number of electrons from one edge to the other?

Q16. The Haldane model demonstrates a "quantum Hall effect without Landau levels." Explain how this is possible: what takes the place of the magnetic field in producing a non-zero Chern number?

Q17. Describe the difference between chiral edge states (quantum Hall effect) and helical edge states (topological insulator). What symmetry protects each, and what can destroy each?

Q18. Why is the search for non-abelian anyons in condensed matter systems so important for quantum computing? What advantage would topological qubits have over conventional superconducting or trapped-ion qubits?

Applied Scenarios (2 questions)

Q19. Designing a Quantum Hall Experiment

You are designing a quantum Hall experiment using a GaAs/AlGaAs heterostructure. The 2D electron gas has density $n_s = 2.0 \times 10^{15}\,\text{m}^{-2}$ and effective mass $m^* = 0.067\,m_e$.

(a) At what magnetic field $B$ will the filling fraction be exactly $\nu = 2$? (3 points)

(b) Calculate the Landau level spacing $\hbar\omega_c$ at this field. Express in meV. (3 points)

(c) What is the maximum temperature at which you can expect to see the $\nu = 2$ plateau? (Estimate as $T_{\max} \sim \hbar\omega_c / (3k_B)$, since the plateau disappears when thermal broadening exceeds the gap.) (3 points)

(d) What Hall resistance will you measure on the $\nu = 2$ plateau? Express in ohms. (3 points)

(e) If your voltmeter has a precision of $\pm 0.001\,\Omega$, to what relative precision can you measure $h/e^2$? Is this competitive with the best measurements? (3 points)

Q20. Topological vs. Trivial Phase

Consider the two-band Hamiltonian $\hat{H}(\mathbf{k}) = \mathbf{d}(\mathbf{k}) \cdot \boldsymbol{\sigma}$ with $\mathbf{d} = (\sin k_x, \sin k_y, m - \cos k_x - \cos k_y)$.

(a) For $m = 1$ (topological phase, $C = 1$), compute the band gap at $\mathbf{k} = (0, 0)$ and $\mathbf{k} = (\pi, \pi)$. Which high-symmetry point has the smaller gap? (4 points)

(b) For $m = 3$ (trivial phase, $C = 0$), compute the band gap at the same points. Compare with part (a). (4 points)

(c) As $m$ decreases from $3$ toward $2$, the gap at $(\pi, \pi)$ shrinks. At $m = 2$, the gap closes. Verify this by computing $|\mathbf{d}(\pi, \pi)|$ for $m = 2$. (4 points)

(d) Explain why the gap closing at $m = 2$ signals a topological phase transition. What happens to the Chern number as $m$ passes through $2$? (3 points)

Answer Key

Multiple Choice

1. (b) — An integer cannot change by a small amount. Any continuous deformation either leaves the invariant unchanged or requires a discontinuous (singular) event.
2. (c) — The TKNN formula shows that $\sigma_{xy} = Ce^2/h$ where $C$ is the Chern number, a topological invariant. Sample quality, field uniformity, etc. affect the Berry curvature locally but not its integral.
3. (b) — In a perpendicular magnetic field, the 2D kinetic energy quantizes into Landau levels with the harmonic oscillator spectrum $E_n = \hbar\omega_c(n + 1/2)$. Each level has degeneracy $N_\phi = eBA/h$.
4. (b) — A topological insulator has topologically protected surface/edge states that cannot be removed without closing the bulk gap or breaking the protecting symmetry.
5. (b) — This is the statement of the bulk-boundary correspondence: topological invariant mismatch implies gapless boundary modes.
6. (b) — Chiral edge states propagate in only one direction. Backscattering requires a counter-propagating state, which does not exist on the same edge. No symmetry is needed for this protection.
7. (b) — The Gauss-Bonnet theorem relates the integral of curvature over a surface to a topological invariant (the Euler characteristic). The Chern number formula is its $k$-space analog: the integral of Berry curvature over the BZ torus gives a topological integer.
8. (b) — In topological quantum computing, quantum information is encoded in the fusion channels of non-abelian anyons, and gates are performed by braiding them. The gate depends only on the topology of the braid, providing inherent protection against local perturbations.
9. (b) — Exchanging ordinary particles gives a phase ($+1$ for bosons, $-1$ for fermions). Exchanging non-abelian anyons produces a unitary matrix acting on a degenerate ground state manifold. The matrix depends on the order of exchanges, hence "non-abelian."
10. (c) — Time-reversal symmetry protects the helical edge states of a 2D topological insulator by forbidding backscattering between Kramers pairs. Magnetic impurities that break $T$ can gap the edge states.

True/False

11. FALSE. The remarkable feature of the quantum Hall effect is that plateaus exist because of disorder, not despite it. Disorder localizes states in the Landau level tails, creating a mobility gap that stabilizes the plateaus. An extremely pure sample would actually have narrower plateaus.
12. TRUE. Under time reversal, the Berry curvature satisfies $\mathcal{F}(-\mathbf{k}) = -\mathcal{F}(\mathbf{k})$. Integrating over the BZ, contributions from $\mathbf{k}$ and $-\mathbf{k}$ cancel, giving $C = 0$. This is why topological insulators are classified by $\mathbb{Z}_2$, not $\mathbb{Z}$.
13. FALSE. As of the mid-2020s, topological quantum computing has not been experimentally demonstrated. While evidence for Majorana zero modes has been reported (and debated), a functioning topological qubit performing braiding operations has not been realized. Earlier claims of Majorana detection at Microsoft/Delft were retracted.
14. FALSE. The bulk band structures are topologically distinct — they have different topological invariants ($\mathbb{Z}_2$ index $\nu = 0$ vs. $\nu = 1$). This difference is not visible in the energy spectrum alone but is encoded in the band topology (the twist of the Bloch wavefunctions over the Brillouin zone). The surface states are a consequence of this bulk difference, via the bulk-boundary correspondence.

Short Answer (Key Points)

15. Laughlin considered a Hall system on a cylinder and imagined adiabatically threading one magnetic flux quantum $\Phi_0 = h/e$ through the cylinder. By the Aharonov-Bohm effect, this shifts the momentum of each state by one quantum. Since the system returns to a gauge-equivalent state after one flux quantum, an integer number of electrons must have been transferred from one edge to the other. The number transferred equals the number of filled Landau levels, giving $\sigma_{xy} = ne^2/h$.

16. In the Haldane model, complex next-nearest-neighbor hoppings create an effective magnetic flux that is opposite for the two sublattices of the honeycomb lattice, so the net flux per unit cell is zero (no Landau levels form). However, the complex hoppings break time-reversal symmetry and create a non-zero Berry curvature, giving a Chern number $C = \pm 1$ in certain parameter ranges. The key insight is that a non-zero Chern number requires time-reversal symmetry breaking, which can be achieved by complex hoppings rather than an external magnetic field.

17. Chiral edge states (QHE) propagate in only one direction on each edge. No symmetry is needed for protection — backscattering is geometrically impossible because no counter-propagating state exists on the same edge. Any perturbation is irrelevant. Helical edge states (TI) come in time-reversal pairs: spin-up right-movers and spin-down left-movers on the same edge. Protection requires time-reversal symmetry — magnetic impurities or magnetic fields can open a gap by coupling the two counter-propagating channels.

18. Topological qubits encode information in non-local degrees of freedom (the fusion channels of non-abelian anyons). Local perturbations — the primary source of decoherence — cannot access this information, providing inherent (hardware-level) protection against errors. Conventional qubits store information locally and are vulnerable to any local noise, requiring extensive software-level error correction. Topological protection could dramatically reduce the overhead ratio of physical-to-logical qubits, but the engineering challenge of creating and manipulating non-abelian anyons has so far prevented experimental realization.

Applied Scenarios

19. (a) $\nu = n_s h/(eB) = 2$ gives $B = n_s h/(2e) = (2.0 \times 10^{15})(6.626 \times 10^{-34}) / (2 \times 1.602 \times 10^{-19}) = 4.14$ T. (b) $\omega_c = eB/m^* = (1.602 \times 10^{-19})(4.14) / (0.067 \times 9.109 \times 10^{-31}) = 1.087 \times 10^{13}$ rad/s. $\hbar\omega_c = (1.055 \times 10^{-34})(1.087 \times 10^{13}) = 1.147 \times 10^{-21}$ J $= 7.2$ meV. (c) $T_{\max} \sim \hbar\omega_c/(3k_B) = 7.2 \times 10^{-3}/(3 \times 8.617 \times 10^{-5}) = 28$ K. Experiments typically use $T < 4$ K for clear plateaus. (d) $R_{xy} = h/(2e^2) = 25812.807/2 = 12906.4\,\Omega$. (e) Relative precision: $0.001/12906.4 = 7.7 \times 10^{-8}$. This is respectable but not competitive with the best measurements ($\sim 10^{-10}$), which use specialized cryogenic current comparators.

20. (a) At $(0,0)$: $\mathbf{d} = (0, 0, 1-1-1) = (0,0,-1)$. Gap $= 2|\mathbf{d}| = 2$. At $(\pi,\pi)$: $\mathbf{d} = (0, 0, 1+1+1) = (0,0,3)$. Gap $= 6$. The smaller gap is at $(0,0)$. (b) For $m = 3$: At $(0,0)$: $\mathbf{d} = (0,0,1)$, gap $= 2$. At $(\pi,\pi)$: $\mathbf{d} = (0,0,5)$, gap $= 10$. (c) At $(\pi,\pi)$ with $m = 2$: $\mathbf{d} = (0, 0, 2+1+1) = (0,0,4)$. Wait — recalculating: $d_z = m - \cos\pi - \cos\pi = 2 - (-1) - (-1) = 4$. The gap closes at $\Gamma$: $d_z = m - 1 - 1 = m - 2 = 0$ for $m = 2$. At $\Gamma$ with $m = 2$: $\mathbf{d} = (0, 0, 0)$, so $|\mathbf{d}| = 0$, confirming the gap closes. (d) The gap closing signals a topological phase transition because the Chern number can only change when the gap closes. For $m < 2$ (but $m > 0$), $C = 1$; for $m > 2$, $C = 0$. At the transition, the system passes through a gapless (metallic) point where the topological invariant is undefined, allowing it to jump between integer values.