Topology and its Role in the Electronic Properties of Materials: 100
MCQ Questions
1. What is topology in the context of condensed matter physics?
a) The study of geometric properties that remain unchanged under continuous deformations
b) The study of crystal structures
c) The analysis of electronic band structures
d) The investigation of superconductivity
Answer: a) The study of geometric properties that remain unchanged under continuous
deformations
2. Which of the following is a topological invariant?
a) Band gap
b) Fermi energy
c) Chern number
d) Lattice constant
Answer: c) Chern number
3. What is the primary:
4. In the context of topological insulators, what is the Z2 invariant?
a) An integer that characterizes the topology of the band structure
b) A complex number describing the spin-orbit coupling
c) A binary (0 or 1) topological index for time-reversal invariant systems
d) The number of edge states in a 2D system
Answer: c) A binary (0 or 1) topological index for time-reversal invariant systems
5. What is the mathematical expression for the Berry curvature in k-space?
a) Ω(k) = × A(k)∇
b) Ω(k) = · A(k)∇
c) Ω(k) = A(k)∇
d) Ω(k) = ^2 A(k)∇
Answer: a) Ω(k) = × A(k)∇
6. The Chern number C for a 2D system is given by the integral of what quantity over the
Brillouin zone?
a) Berry phase
b) Berry curvature
c) Bloch wavefunctions
d) Density of states
Answer: b) Berry curvature
7. What is the mathematical condition for the existence of topologically protected edge states in
the Su-Schrieffer-Heeger (SSH) model?
a) |v/w| > 1
b) |v/w| < 1
c) |v/w| = 1
d) v = w
Where v and w are the alternating hopping amplitudes.
Answer: a) |v/w| > 1
8. In the Kane-Mele model for topological insulators, what is the form of the spin-orbit coupling
term?
a) iλSO Σ<i,j> νij c†i σz cj
b) λSO Σ<i,j> c†i σx cj
c) iλSO Σ<i,j> c†i σy cj
d) λSO Σ<i,j> νij c†i σz cj
Answer: a) iλSO Σ<i,j> νij c†i σz cj
9. What is the formula for the Hall conductivity σxy in terms of the Chern number C?
a) σxy = C(e^2/h)
b) σxy = C(h/e^2)
c) σxy = C(e/h)
d) σxy = C(h/e)
Answer: a) σxy = C(e^2/h)
10. In the Haldane model, what is the condition for the system to be in a Chern insulator phase?
a) |M/t2| < 3√3
b) |M/t2| > 3√3
c) |M/t2| = 3√3
d) M = t2
Where M is the mass term and t2 is the next-nearest-neighbor hopping amplitude.
Answer: a) |M/t2| < 3√3
11. What is the mathematical expression for the Berry phase γ along a closed path C in k-
space?
a) γ = C A(k) · dk∮
b) γ = ∫C A(k) × dk
c) γ = C A(k) × dk∮
d) γ = ∫C A(k) · dk
Answer: a) γ = C A(k) · dk∮
12. In the BHZ model for 2D topological insulators, what is the form of the Hamiltonian in k-
space?
a) H(k) = ε(k)I × + da(k)Γa₄ ₄
b) H(k) = ε(k)σz + da(k)σa
c) H(k) = ε(k)I × + d(k) · σ₂ ₂
d) H(k) = ε(k)σx + d(k) · Γ
Answer: a) H(k) = ε(k)I × + da(k)Γa₄ ₄
13. What is the mathematical expression for the Z2 invariant in terms of the Pfaffian of the time-
reversal operator?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Π_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 Pf(w(Λi))
d) ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
14. In the theory of topological crystalline insulators, what is the mathematical condition for the
existence of mirror Chern numbers?
a) C+ - C- ≠ 0
b) C+ + C- = 0
c) C+ × C- = 1
d) |C+| = |C-|
Where C+ and C- are the Chern numbers for different mirror eigenvalues.
Answer: a) C+ - C- ≠ 0
15. What is the formula for the Wannier center x
Yn in terms of the Berry connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
16. In the context of topological semimetals, what is the mathematical condition for the
existence of a Weyl point?
a) · B = 0∇
b) × B = 0∇
c) · B = ±δ(k - k0)∇
d) × B = ±δ(k - k0)∇
Where B is the Berry curvature and k0 is the position of the Weyl point.
Answer: c) · B = ±δ(k - k0)∇
17. What is the expression for the topological invariant N3 in a 3D topological insulator?
a) N3 = (1/24π^2) εijk ∫ d³k Tr[(∂iH^-1)(∂jH)(∂kH^-1)H]
b) N3 = (1/8π^2) εijk ∫ d³k Tr[(∂iH^-1)(∂jH)(∂kH^-1)H]
c) N3 = (1/12π^2) εijk ∫ d³k Tr[(∂iH)(∂jH)(∂kH)]
d) N3 = (1/16π^2) εijk ∫ d³k Tr[(∂iH^-1)(∂jH)(∂kH)]
Answer: b) N3 = (1/8π^2) εijk ∫ d³k Tr[(∂iH^-1)(∂jH)(∂kH^-1)H]
18. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
19. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
20. In the context of the quantum spin Hall effect, what is the mathematical expression for the
spin Chern number Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
21. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
22. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
23. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
24. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
25. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
What is the mathematical expression for the Berry phase γ in terms of the Chern number C for a
closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
26. What is the mathematical expression for the Zak phase in a 1D crystal?
a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
b) γ = ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
c) γ = i uk|∂k|uk dk∮ ⟨ ⟩
d) γ = uk|∂k|uk dk∮ ⟨ ⟩
Where G is the reciprocal lattice vector.
Answer: a) γ = i ∫ ^G uk|∂k|uk dk₀ ⟨ ⟩
27. In the context of topological defects, what is the mathematical form of the winding number
for a vortex?
a) n = (1/2π) θ · dl∮ ∇
b) n = (1/2π) ∫ θ · dS∇
c) n = (1/2π) θ dl∮
d) n = (1/2π) ∫ θ dS
Where θ is the phase of the order parameter.
Answer: a) n = (1/2π) θ · dl∮ ∇
28. What is the formula for the Kane-Mele Z2 invariant in terms of the Pfaffian?
a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
b) ν = Σ_{i=1}^4 Pf(w(Λi))
c) (-1)ν = Σ_{i=1}^4 sgn[Pf(w(Λi))]
d) ν = Π_{i=1}^4 Pf(w(Λi))
Where Λi are the time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^4 sgn[Pf(w(Λi))]
29. In the theory of topological quantum computation, what is the braiding relation for Majorana
fermions?
a) γiγj = -γjγi
b) γiγj = γjγi
c) γi²γj² = -γj²γi²
d) γi²γj² = γj²γi²
Where γi and γj are Majorana operators.
Answer: a) γiγj = -γjγi
30. What is the mathematical expression for the Chern-Simons action in (2+1) dimensions?
a) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ
b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
c) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ
d) SCS = (k/2π) ∫ d³x εμνρ Aμ∂νAρ + (1/3)AμAνAρ
Where k is the Chern-Simons level and Aμ is the gauge field.
Answer: b) SCS = (k/4π) ∫ d³x εμνρ Aμ∂νAρ + (2/3)AμAνAρ
31. What is the formula for the spin Hall conductivity σsxy in terms of the spin Chern number
Cs?
a) σsxy = Cs(e/4π)
b) σsxy = Cs(e/2π)
c) σsxy = Cs(e²/h)
d) σsxy = Cs(e²/2h)
Answer: a) σsxy = Cs(e/4π)
32. In the context of the quantum anomalous Hall effect, what is the expression for the Hall
conductance σxy?
a) σxy = (e²/h)C
b) σxy = (e²/2h)C
c) σxy = (e/h)C
d) σxy = (e/2h)C
Where C is the Chern number.
Answer: a) σxy = (e²/h)C
33. What is the mathematical condition for the existence of Weyl nodes in a 3D system?
a) det[H(k) - EI] = 0
b) Tr[H(k)] = 0
c) H(k) = H(-k)
d) H(k) = -H(-k)
Where H(k) is the Hamiltonian and E is the energy.
Answer: a) det[H(k) - EI] = 0
34. In the theory of topological superconductors, what is the formula for the topological invariant
M in class D systems?
a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
b) M = Pf(iHBdG(0)) + Pf(iHBdG(π))
c) M = |Pf(iHBdG(0)) - Pf(iHBdG(π))|
d) M = Pf(iHBdG(0)) × Pf(iHBdG(π))
Where HBdG is the Bogoliubov-de Gennes Hamiltonian.
Answer: a) M = sgn[Pf(iHBdG(0))Pf(iHBdG(π))]
35. What is the mathematical expression for the Berry curvature in terms of energy eigenstates?
a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
b) Ωn(k) = k un(k)| · | k un(k)⟨∇ ∇ ⟩
c) Ωn(k) = i un(k)| k| × | k un(k)⟨ ∇ ∇ ⟩
d) Ωn(k) = un(k)| k| · | k un(k)⟨ ∇ ∇ ⟩
Where un(k) is the periodic part of the Bloch function.
Answer: a) Ωn(k) = i k un(k)| × | k un(k)⟨∇ ∇ ⟩
36. In the context of topological band theory, what is the formula for the Wilson loop?
a) W[C] = P exp(i C A(k) · dk)∮
b) W[C] = P exp(-i C A(k) · dk)∮
c) W[C] = exp(i C A(k) · dk)∮
d) W[C] = exp(-i C A(k) · dk)∮
Where P is the path-ordering operator and A(k) is the Berry connection.
Answer: b) W[C] = P exp(-i C A(k) · dk)∮
37. What is the mathematical expression for the Z2 invariant in terms of the Chern-Simons 3-
form?
a) ν = (1/2π) ∫BZ ω3 mod 2
b) ν = (1/4π) ∫BZ ω3 mod 2
c) ν = (1/2π) ∂BZ ω3 mod 2∮
d) ν = (1/4π) ∂BZ ω3 mod 2∮
Where ω3 is the Chern-Simons 3-form and BZ is the Brillouin zone.
Answer: b) ν = (1/4π) ∫BZ ω3 mod 2
38. In the theory of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (C+ - C-) / 2
b) nM = C+ + C-
c) nM = |C+ - C-|
d) nM = (C+ × C-)^(1/2)
Where C+ and C- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (C+ - C-) / 2
39. What is the mathematical condition for the existence of Dirac points in graphene?
a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
b) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 0
c) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 1
d) sin(kxa) + 2sin(kxa/2)sin(√3kya/2) = 1
Where a is the lattice constant.
Answer: a) cos(kxa) + 2cos(kxa/2)cos(√3kya/2) = 0
40. In the context of the quantum spin Hall effect, what is the formula for the spin Chern number
Cs?
a) Cs = (C↑ - C↓) / 2
b) Cs = C↑ + C↓
c) Cs = |C↑ - C↓|
d) Cs = (C↑ × C↓)^(1/2)
Where C↑ and C↓ are the Chern numbers for spin-up and spin-down states, respectively.
Answer: a) Cs = (C↑ - C↓) / 2
41. What is the mathematical expression for the Wannier center x
Yn in terms of the Berry
connection A(k)?
a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
b) x
Yn = (1/2π) dk An(k)∮
c) x
Yn = (1/2π) ∫ ^π dk An(k)₀
d) x
Yn = (1/π) ∫ ^π dk An(k)₀
Answer: a) x
Yn = (1/2π) ∫ ^2π dk An(k)₀
42. In the theory of topological superconductors, what is the mathematical form of the
Bogoliubov-de Gennes (BdG) Hamiltonian?
a) HBdG = (ε(k) - μ)τz + Δ(k)τx
b) HBdG = (ε(k) - μ)σz + Δ(k)σx
c) HBdG = ε(k)τz + Δ(k)τx
d) HBdG = ε(k)σz + Δ(k)σx
Where τi and σi are Pauli matrices in particle-hole and spin space, respectively.
Answer: a) HBdG = (ε(k) - μ)τz + Δ(k)τx
43. What is the formula for the winding number W in a 1D topological superconductor?
a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
b) W = (1/π) ∫ ^π dk ∂k arg(Δ(k))₀
c) W = (1/2π) dk k × A(k)∮ ∇
d) W = (1/4π) ∫ d²k Ω(k)
Answer: a) W = (1/2πi) ∫ ^2π dk ∂k log(Δ(k))₀
44. In the context of the quantum anomalous Hall effect, what is the expression for the Chern
number C in terms of the Berry curvature Ω(k)?
a) C = (1/2π) ∫BZ d²k Ω(k)
b) C = (1/4π) ∫BZ d²k Ω(k)
c) C = (1/2π) ∂BZ dk · A(k)∮
d) C = (1/4π) ∂BZ dk · A(k)∮
Where BZ is the Brillouin zone.
Answer: a) C = (1/2π) ∫BZ d²k Ω(k)
45. What is the mathematical condition for the existence of Majorana zero modes in a 1D
topological superconductor?
a) Δ² > μ² + t²
b) Δ² < μ² + t²
c) Δ² = μ² + t²
d) Δ > μ + t
Where Δ is the superconducting gap, μ is the chemical potential, and t is the hopping amplitude.
Answer: a) Δ² > μ² + t²
46. In the theory of higher-order topological insulators, what is the formula for the nested Wilson
loop?
a) W2[l] = P exp(-i l A2(k) · dk)∮
b) W2[l] = P exp(i l A2(k) · dk)∮
c) W2[l] = P exp(-i ∫l A2(k) · dk)
d) W2[l] = P exp(i ∫l A2(k) × dk)
Where P is the path-ordering operator and A2 is the second-order Berry connection.
Answer: b) W2[l] = P exp(i l A2(k) · dk)∮
47. What is the mathematical expression for the Fu-Kane formula for the Z2 invariant in 3D
topological insulators?
a) (-1)ν = Π_{i=1}^8 δi
b) ν = Σ_{i=1}^8 δi
c) (-1)ν = Π_{i=1}^8 sgn(δi)
d) ν = Π_{i=1}^8 |δi|
Where δi are the parity eigenvalues at time-reversal invariant momenta.
Answer: a) (-1)ν = Π_{i=1}^8 δi
48. In the context of topological crystalline insulators, what is the formula for the mirror Chern
number nM?
a) nM = (n+ - n-) / 2
b) nM = n+ + n-
c) nM = |n+ - n-|
d) nM = (n+ × n-)^(1/2)
Where n+ and n- are the Chern numbers for states with different mirror eigenvalues.
Answer: a) nM = (n+ - n-) / 2
49. What is the mathematical expression for the Berry phase γ in terms of the Chern number C
for a closed 2D manifold?
a) γ = 2πC
b) γ = πC
c) γ = C/2π
d) γ = C/π
Answer: a) γ = 2πC
50. In the theory of topological semimetals, what is the formula for the chiral anomaly in Weyl
semimetals?
a) ∂ρ/∂t + · j = (e²/4π² ²)E · B∇ ℏ
b) ∂ρ/∂t + · j = (e²/2π² ²)E · B∇ ℏ
c) ∂ρ/∂t + · j = (e/4π² ²)E · B∇ ℏ
d) ∂ρ/∂
