Mathematical Treatment of Symmetry Breaking: MCQ Questions
with Mathematical Problems and Answers
1. In the Landau theory of phase transitions, the free energy F is expanded in powers of the
order parameter φ. Which of the following expressions represents the simplest form of F for a
second-order phase transition?
a) F = a φ² + b φ⁴
b) F = a φ + b φ³
c) F = a φ² + b φ³
d) F = a φ + b φ⁴
Answer: a) F = a φ² + b φ⁴
2. For a system with Z symmetry, what is the form of the potential V(φ) that exhibits ₂
spontaneous symmetry breaking?
a) V(φ) = μ²φ² + λφ⁴, μ² > 0, λ > 0
b) V(φ) = μ²φ² + λφ⁴, μ² < 0, λ > 0
c) V(φ) = μ²φ² - λφ⁴, μ² > 0, λ > 0
d) V(φ) = -μ²φ² + λφ , μ² > 0, λ > 0⁶
Answer: b) V(φ) = μ²φ² + λφ⁴, μ² < 0, λ > 0
3. In the context of spontaneous symmetry breaking, what is the value of the vacuum
expectation value (VEV) φ for the potential V(φ) = μ²φ² + λφ⁴, when μ² < 0 and λ > 0?⟨ ⟩
a) φ = ±√(-μ²/λ)⟨ ⟩
b) φ = ±√(μ²/λ)⟨ ⟩
c) φ = ±√(-μ²/2λ)⟨ ⟩
d) φ = ±√(μ²/2λ)⟨ ⟩
Answer: c) φ = ±√(-μ²/2λ)⟨ ⟩
4. For a complex scalar field φ = (φ + iφ )/√2, what is the form of the U(1) symmetric potential ₁ ₂
V(φ) that exhibits spontaneous symmetry breaking?
a) V(φ) = μ²|φ|² + λ|φ|⁴, μ² > 0, λ > 0
b) V(φ) = μ²|φ|² + λ|φ|⁴, μ² < 0, λ > 0
c) V(φ) = μ²|φ|² - λ|φ|⁴, μ² > 0, λ > 0
d) V(φ) = -μ²|φ|² + λ|φ| , μ² > 0, λ > 0⁶
Answer: b) V(φ) = μ²|φ|² + λ|φ|⁴, μ² < 0, λ > 0
5. In the context of the Higgs mechanism, what is the mass of the Higgs boson for the potential
V(φ) = μ²|φ|² + λ|φ|⁴, where μ² < 0 and λ > 0?
a) m_H = √(-2μ²)
b) m_H = √(2λv²)
c) m_H = √(-μ²)
d) m_H = √(λv²)
Answer: b) m_H = √(2λv²), where v is the VEV
6. For a system with O(N) symmetry, what is the dimension of the order parameter φ?
a) 1
b) 2
c) N
d) N-1
Answer: c) N
7. In the Ginzburg-Landau theory of superconductivity, what is the form of the free energy
functional F[ψ] for a complex order parameter ψ?
a) F[ψ] = ∫d³r [α|ψ|² + β|ψ|⁴ + γ| ψ|²]∇
b) F[ψ] = ∫d³r [α|ψ|² - β|ψ|⁴ + γ| ψ|²]∇
c) F[ψ] = ∫d³r [α|ψ|² + β|ψ|⁴ - γ| ψ|²]∇
d) F[ψ] = ∫d³r [α|ψ|² - β|ψ|⁴ - γ| ψ|²]∇
Answer: a) F[ψ] = ∫d³r [α|ψ|² + β|ψ|⁴ + γ| ψ|²]∇
8. What is the Goldstone theorem for a system with a continuous symmetry group G broken to a
subgroup H?
a) There are dim(G) - dim(H) massless bosons
b) There are dim(G) + dim(H) massless bosons
c) There are dim(G) * dim(H) massless bosons
d) There are dim(G) / dim(H) massless bosons
Answer: a) There are dim(G) - dim(H) massless bosons
9. In the context of the Standard Model electroweak symmetry breaking, what is the dimension
of the Higgs field φ?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2 (it's a complex doublet)
10. For a system with SU(2) symmetry, what is the form of the potential V(φ) that exhibits
spontaneous symmetry breaking?
a) V(φ) = μ²Tr(φ†φ) + λ[Tr(φ†φ)]², μ² > 0, λ > 0
b) V(φ) = μ²Tr(φ†φ) + λ[Tr(φ†φ)]², μ² < 0, λ > 0
c) V(φ) = μ²Tr(φ†φ) - λ[Tr(φ†φ)]², μ² > 0, λ > 0
d) V(φ) = -μ²Tr(φ†φ) + λ[Tr(φ†φ)]³, μ² > 0, λ > 0
Answer: b) V(φ) = μ²Tr(φ†φ) + λ[Tr(φ†φ)]², μ² < 0, λ > 0
11. In the context of chiral symmetry breaking in QCD, what is the dimension of the quark
condensate ψXψ ?⟨ ⟩
a) 1
b) 2
c) 3
d) 4
Answer: c) 3 (mass dimension)
12. What is the form of the effective potential V_eff(φ) in the one-loop approximation for a scalar
field theory?
a) V_eff(φ) = V(φ) + ( /2) ∫ d⁴k/(2π)⁴ ln[k² + V''(φ)]ℏ
b) V_eff(φ) = V(φ) - ( /2) ∫ d⁴k/(2π)⁴ ln[k² + V''(φ)]ℏ
c) V_eff(φ) = V(φ) + ( /2) ∫ d⁴k/(2π)⁴ ln[k² - V''(φ)]ℏ
d) V_eff(φ) = V(φ) - ( /2) ∫ d⁴k/(2π)⁴ ln[k² - V''(φ)]ℏ
Answer: b) V_eff(φ) = V(φ) - ( /2) ∫ d⁴k/(2π)⁴ ln[k² + V''(φ)]ℏ
13. In the context of the Coleman-Weinberg mechanism, what is the condition for radiative
symmetry breaking?
a) λ > e⁴/(32π²)
b) λ < e⁴/(32π²)
c) λ = e⁴/(32π²)
d) λ > e⁴/(64π²)
Answer: b) λ < e⁴/(32π²)
14. What is the form of the Nambu-Goldstone boson field π(x) in terms of the symmetry
generator Q and the order parameter φ?
a) π(x) = [Q, φ(x)]
b) π(x) = {Q, φ(x)}
c) π(x) = Q φ(x)
d) π(x) = φ(x) Q
Answer: a) π(x) = [Q, φ(x)]
15. In the context of the Higgs mechanism, what is the gauge-invariant combination of the Higgs
field φ and the gauge field A_μ?
a) D_μφ = ∂_μφ + ieA_μφ
b) D_μφ = ∂_μφ - ieA_μφ
c) D_μφ = ∂_μφ + e²A_μφ
d) D_μφ = ∂_μφ - e²A_μφ
Answer: b) D_μφ = ∂_μφ - ieA_μφ
16. What is the dimension of the coupling constant λ in the potential V(φ) = λφ⁴ for a scalar field
in 4D spacetime?
a) 0
b) 1
c) 2
d) -1
Answer: a) 0 (dimensionless)
17. In the context of the Gross-Neveu model, what is the form of the potential V(σ) that exhibits
dynamical symmetry breaking?
a) V(σ) = σ²/2g - N/(4π) σ²ln(σ²/μ²)
b) V(σ) = σ²/2g + N/(4π) σ²ln(σ²/μ²)
c) V(σ) = -σ²/2g + N/(4π) σ²ln(σ²/μ²)
d) V(σ) = -σ²/2g - N/(4π) σ²ln(σ²/μ²)
Answer: b) V(σ) = σ²/2g + N/(4π) σ²ln(σ²/μ²)
18. What is the form of the Gell-Mann-Lévy sigma model Lagrangian?
a) L = (∂_μπ)² + (∂_μσ)² - λ(π² + σ² - v²)²
b) L = (∂_μπ)² + (∂_μσ)² + λ(π² + σ² - v²)²
c) L = (∂_μπ)² - (∂_μσ)² - λ(π² + σ² - v²)²
d) L = -(∂_μπ)² - (∂_μσ)² - λ(π² + σ² - v²)²
Answer: a) L = (∂_μπ)² + (∂_μσ)² - λ(π² + σ² - v²)²
19. In the context of the BCS theory of superconductivity, what is the form of the gap equation?
a) Δ = V ∑_k tanh(E_k/2kT) / 2E_k
b) Δ = V ∑_k tanh(E_k/2kT) / E_k
c) Δ = V ∑_k coth(E_k/2kT) / 2E_k
d) Δ = V ∑_k coth(E_k/2kT) / E_k
Answer: a) Δ = V ∑_k tanh(E_k/2kT) / 2E_k
20. What is the form of the Higgs potential in the Standard Model?
a) V(H) = -μ²H†H + λ(H†H)²
b) V(H) = μ²H†H + λ(H†H)²
c) V(H) = -μ²H†H - λ(H†H)²
d) V(H) = μ²H†H - λ(H†H)²
Answer: b) V(H) = μ²H†H + λ(H†H)²
21. In the context of the Abelian Higgs model, what is the mass of the gauge boson after
symmetry breaking?
a) m_A = ev
b) m_A = e²v
c) m_A = √(ev)
d) m_A = √(e²v)
Answer: a) m_A = ev, where v is the VEV
22. What is the form of the effective potential for a fermion field ψ in the large-N limit?
a) V_eff(σ) = N/(4π²) [σ⁴ln(σ²/μ²) - σ⁴/2]
b) V_eff(σ) = N/(4π²) [σ⁴ln(σ²/μ²) + σ⁴/2]
c) V_eff(σ) = N/(4π²) [-σ⁴ln(σ²/μ²) + σ⁴/2]
d) V_eff(σ) = N/(4π²) [-σ⁴ln(σ²/μ²) - σ⁴/2]
Answer: a) V_eff(σ) = N/(4π²) [σ⁴ln(σ²/μ²) - σ⁴/2]
23. In the context of the Nambu-Jona-Lasinio model, what is the form of the gap equation?
a) 1 = 2G ∫ d³p/(2π)³ 1/√(p² + m²)
b) 1 = G ∫ d³p/(2π)³ 1/√(p² + m²)
c) 1 = 2G ∫ d³p/(2π)³ 1/(p² + m²)
d) 1 = G ∫ d³p/(2π)³ 1/(p² + m²)
Answer: a) 1 = 2G ∫ d³p/(2π)³ 1/√(p² + m²)
24. What is the form of the Landau-Ginzburg free energy functional for a superconductor in a
magnetic field?
a) F = ∫d³r [α|ψ|² + β|ψ|⁴ + γ|( - ieA)ψ|² + B²/2μ ]∇ ₀
b) F = ∫d³r [α|ψ|² - β|ψ|⁴ + γ|( - ieA)ψ|² + B²/2μ ]∇ ₀
c) F :
25. In the context of the 2D Ising model, what is the form of the free energy near the critical
point?
a) F ~ |t|²ln|t|
b) F ~ |t|²
c) F ~ |t|³
d) F ~ |t|ln|t|
Answer: a) F ~ |t|²ln|t|, where t = (T-Tc)/Tc
26. What is the scaling dimension of the order parameter φ at the Wilson-Fisher fixed point in 4-
ε dimensions?
a) d_φ = 1 - ε/2 + O(ε²)
b) d_φ = 1 + ε/2 + O(ε²)
c) d_φ = 1 - ε/3 + O(ε²)
d) d_φ = 1 + ε/3 + O(ε²)
Answer: a) d_φ = 1 - ε/2 + O(ε²)
27. In the context of the O(N) model, what is the β-function for the coupling constant g to one-
loop order?
a) β(g) = εg - (N+8)g²/8π² + O(g³)
b) β(g) = -εg + (N+8)g²/8π² + O(g³)
c) β(g) = εg + (N-8)g²/8π² + O(g³)
d) β(g) = -εg - (N-8)g²/8π² + O(g³)
Answer: b) β(g) = -εg + (N+8)g²/8π² + O(g³)
28. What is the form of the Callan-Symanzik equation for the n-point correlation function?
a) [μ∂/∂μ + β(g)∂/∂g - nγ_φ]G^(n) = 0
b) [μ∂/∂μ + β(g)∂/∂g + nγ_φ]G^(n) = 0
c) [μ∂/∂μ - β(g)∂/∂g - nγ_φ]G^(n) = 0
d) [μ∂/∂μ - β(g)∂/∂g + nγ_φ]G^(n) = 0
Answer: a) [μ∂/∂μ + β(g)∂/∂g - nγ_φ]G^(n) = 0
29. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation
function?
a) G(r) ~ r^(-η(T))
b) G(r) ~ exp(-r/ξ)
c) G(r) ~ r^(-d+2-η)
d) G(r) ~ ln(r)
Answer: a) G(r) ~ r^(-η(T))
30. What is the form of the Landau-Lifshitz equation for the magnetization M?
a) ∂M/∂t = -γM × H_eff + αM × ∂M/∂t
b) ∂M/∂t = γM × H_eff + αM × ∂M/∂t
c) ∂M/∂t = -γM × H_eff - αM × ∂M/∂t
d) ∂M/∂t = γM × H_eff - αM × ∂M/∂t
Answer: a) ∂M/∂t = -γM × H_eff + αM × ∂M/∂t
31. In the context of the Ginzburg-Landau theory, what is the expression for the coherence
length ξ?
a) ξ = √( ²/2m|α|)ℏ
b) ξ = √( ²/4m|α|)ℏ
c) ξ = √( ²/m|α|)ℏ
d) ξ = √( ²/8m|α|)ℏ
Answer: a) ξ = √( ²/2m|α|)ℏ
32. What is the form of the Gross-Pitaevskii equation for a Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|²]ψℏ ℏ ∇
b) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|²]ψℏ ℏ ∇
c) i ∂ψ/∂t = [- ² ²/2m + V(r) - g|ψ|²]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) - g|ψ|²]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|²]ψℏ ℏ ∇
33. In the context of the XY model, what is the form of the spin-wave Hamiltonian?
a) H_SW = (J/2)∫d²r( θ)²∇
b) H_SW = J∫d²r( θ)²∇
c) H_SW = (J/2)∫d²r( ²θ)²∇
d) H_SW = J∫d²r( ²θ)²∇
Answer: a) H_SW = (J/2)∫d²r( θ)²∇
34. What is the form of the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature T_BKT?
a) T_BKT = πJ/2
b) T_BKT = πJ
c) T_BKT = 2πJ
d) T_BKT = πJ/4
Answer: a) T_BKT = πJ/2
35. In the context of the Hubbard model, what is the form of the interaction term?
a) H_int = U∑_i n_i↑n_i↓
b) H_int = U∑_i (n_i↑ + n_i↓)
c) H_int = U∑_i (n_i↑ - n_i↓)²
d) H_int = U∑_i n_i↑n_i↓²
Answer: a) H_int = U∑_i n_i↑n_i↓
36. What is the form of the Landau free energy for a system with Z symmetry near the critical ₂
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
37. In the context of the Ising model, what is the form of the critical exponent β?
a) β = 1/8 (2D), 5/16 (3D)
b) β = 1/4 (2D), 1/3 (3D)
c) β = 1/2 (2D), 3/4 (3D)
d) β = 3/8 (2D), 7/16 (3D)
Answer: a) β = 1/8 (2D), 5/16 (3D)
38. What is the form of the Landau-de Gennes free energy for nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴
39. In the context of the Heisenberg model, what is the form of the spin-wave dispersion
relation?
a) ω_k = 2JS|k|
b) ω_k = JS|k|²
c) ω_k = √(2JS|k|)
d) ω_k = √(JS|k|²)
Answer: a) ω_k = 2JS|k|
40. What is the form of the Mermin-Wagner theorem for continuous symmetries in low
dimensions?
a) No spontaneous symmetry breaking in d ≤ 2
b) No spontaneous symmetry breaking in d < 2
c) No spontaneous symmetry breaking in d ≤ 3
d) No spontaneous symmetry breaking in d < 3
Answer: a) No spontaneous symmetry breaking in d ≤ 2
41. In the context of the Gross-Neveu model, what is the form of the dynamically generated
mass?
a) m = μexp(-π/g²N)
b) m = μexp(-2π/g²N)
c) m = μexp(-π/gN)
d) m = μexp(-2π/gN)
Answer: b) m = μexp(-2π/g²N)
42. What is the form of the Ginzburg criterion for the validity of mean-field theory?
a) |T-Tc|/Tc >> (a³/V )(kTc/ΔC)²₀
b) |T-Tc|/Tc << (a³/V )(kTc/ΔC)²₀
c) |T-Tc|/Tc >> (V /a³)(kTc/ΔC)²₀
d) |T-Tc|/Tc << (V /a³)(kTc/ΔC)²₀
Answer: b) |T-Tc|/Tc << (a³/V )(kTc/ΔC)²₀
43. In the context of the Nambu-Goto action for strings, what is the form of the action?
a) S = -T∫dτdσ√(-det(g_αβ))
b) S = T∫dτdσ√(-det(g_αβ))
c) S = -T∫dτdσ√(det(g_αβ))
d) S = T∫dτdσ√(det(g_αβ))
Answer: a) S = -T∫dτdσ√(-det(g_αβ))
44. What is the form of the Polyakov action for strings?
a) S = -(T/2)∫d²σ√(-γ)γ^αβ∂_αX^μ∂_βX_μ
b) S = (T/2)∫d²σ√(-γ)γ^αβ∂_αX^μ∂_βX_μ
c) S = -(T/2)∫d²σ√(γ)γ^αβ∂_αX^μ∂_βX_μ
d) S = (T/2)∫d²σ√(γ)γ^αβ∂_αX^μ∂_βX_μ
Answer: a) S = -(T/2)∫d²σ√(-γ)γ^αβ∂_αX^μ∂_βX_μ
45. In the context of the Coleman-Weinberg mechanism, what is the form of the one-loop
effective potential?
a) V_eff(φ) = (λφ⁴/4!) + ( /64π²)φ⁴[ln(φ²/μ²) - 25/6]ℏ
b) V_eff(φ) = (λφ⁴/4!) - ( /64π²)φ⁴[ln(φ²/μ²) - 25/6]ℏ
c) V_eff(φ) = (λφ⁴/4!) + ( /32π²)φ⁴[ln(φ²/μ²) - 25/6]ℏ
d) V_eff(φ) = (λφ⁴/4!) - ( /32π²)φ⁴[ln(φ²/μ²) - 25/6]ℏ
Answer: a) V_eff(φ) = (λφ⁴/4!) + ( /64π²)φ⁴[ln(φ²/μ²) - 25/6]ℏ
46. What is the form of the Bogomolny bound for monopoles?
a) M ≥ v|n|/g
b) M ≥ v|n|g
c) M ≤ v|n|/g
d) M ≤ v|n|g
Answer: a) M ≥ v|n|/g
47. In the context of the sine-Gordon model, what is the form of the soliton solution?
a) φ(x) = 4arctan[exp(mx)]
b) φ(x) = 4arctan[exp(-mx)]
c) φ(x) = 2arctan[exp(mx)]
d) φ(x) = 2arctan[exp(-mx)]
Answer: b) φ(x) = 4arctan[exp(-mx)]
48. What is the form of the Landau-Zener formula for the transition probability?
a) P = exp(-πΔ²/2 v)ℏ
b) P = 1 - exp(-πΔ²/2 v)ℏ
c) P = exp(-2πΔ²/ v)ℏ
d) P = 1 - exp(-2πΔ²/ v)ℏ
Answer: b) P = 1 - exp(-πΔ²/2 v)ℏ
49. In the context of the Kibble-Zurek mechanism, what is the form of the correlation length ξ
near the critical point?
a) ξ ~ |ε|^(-ν)
b) ξ ~ |ε|^(ν)
c) ξ ~ |ε|^(-1/ν)
d) ξ ~ |ε|^(1/ν)
Answer: a) ξ ~ |ε|^(-ν)
50. What is the form of the Brazovskii free energy functional?
a) F[ψ] = ∫d³r [r|ψ|² + ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
b) F[ψ] = ∫d³r [r|ψ|² - ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
c) F[ψ] = ∫d³r [-r|ψ|² + ( ²+q ²)²|ψ|² + u|ψ∇ ₀
50. What is the form of the Brazovskii free energy functional?
a) F[ψ] = ∫d³r [r|ψ|² + ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
b) F[ψ] = ∫d³r [r|ψ|² - ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
c) F[ψ] = ∫d³r [-r|ψ|² + ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
d) F[ψ] = ∫d³r [-r|ψ|² - ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
Answer: a) F[ψ] = ∫d³r [r|ψ|² + ( ²+q ²)²|ψ|² + u|ψ|⁴]∇ ₀
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
51. In the context of the Higgs mechanism, what is the mass of the Higgs boson in terms of the
vacuum expectation value v and the coupling λ?
a) m_H = √(2λ)v
b) m_H = √(λ)v
c) m_H = 2√(λ)v
d) m_H = λv
Answer: a) m_H = √(2λ)v
52. What is the form of the BCS gap equation at zero temperature?
a) 1 = V∑_k 1/(2E_k)
b) 1 = V∑_k 1/E_k
c) 1 = V∑_k tanh(E_k/2kT)/2E_k
d) 1 = V∑_k tanh(E_k/2kT)/E_k
Answer: a) 1 = V∑_k 1/(2E_k)
53. In the context of the Landau theory of phase transitions, what is the order parameter for the
superfluid transition in liquid helium?
a) Complex scalar field ψ
b) Real scalar field φ
c) Vector field A_μ
d) Tensor field g_μν
Answer: a) Complex scalar field ψ
54. What is the form of the Josephson current in a superconducting junction?
a) I = I_c sin(Δφ)
b) I = I_c cos(Δφ)
c) I = I_c tan(Δφ)
d) I = I_c cot(Δφ)
Answer: a) I = I_c sin(Δφ)
55. In the context of the Ising model, what is the form of the critical exponent γ for the
susceptibility?
a) γ = 7/4 (2D), 5/4 (3D)
b) γ = 5/4 (2D), 7/4 (3D)
c) γ = 3/2 (2D), 5/4 (3D)
d) γ = 5/4 (2D), 3/2 (3D)
Answer: a) γ = 7/4 (2D), 5/4 (3D)
56. What is the form of the Ginzburg-Landau coherence length ξ in terms of the critical
temperature Tc and the reduced temperature t = (T-Tc)/Tc?
a) ξ(T) = ξ / √|t|₀
b) ξ(T) = ξ √|t|₀
c) ξ(T) = ξ / |t|₀
d) ξ(T) = ξ |t|₀
Answer: a) ξ(T) = ξ / √|t|₀
57. In the context of the XY model, what is the form of the spin-spin correlation function above
the BKT transition?
a) G(r) ~ exp(-r/ξ)
b) G(r) ~ r^(-η)
c) G(r) ~ r^(-1/4)
d) G(r) ~ ln(r)
Answer: b) G(r) ~ r^(-η)
58. What is the form of the Landau-Lifshitz-Gilbert equation for magnetization dynamics?
a) dM/dt = -γM×H_eff + αM×dM/dt
b) dM/dt = γM×H_eff + αM×dM/dt
c) dM/dt = -γM×H_eff - αM×dM/dt
d) dM/dt = γM×H_eff - αM×dM/dt
Answer: a) dM/dt = -γM×H_eff + αM×dM/dt
59. In the context of the Kosterlitz-Thouless transition, what is the form of the correlation length
ξ near the transition?
a) ξ ~ exp(b/√(T-T_KT))
b) ξ ~ exp(b/√(T_KT-T))
c) ξ ~ exp(b/(T-T_KT))
d) ξ ~ exp(b/(T_KT-T))
Answer: b) ξ ~ exp(b/√(T_KT-T))
60. What is the form of the Gross-Pitaevskii equation for a rotating Bose-Einstein condensate?
a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
b) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
c) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
d) i ∂ψ/∂t = [ ² ²/2m + V(r) + g|ψ|² + ΩL_z]ψℏ ℏ ∇
Answer: a) i ∂ψ/∂t = [- ² ²/2m + V(r) + g|ψ|² - ΩL_z]ψℏ ℏ ∇
61. In the context of the Hubbard model, what is the form of the antiferromagnetic exchange
coupling J in the large-U limit?
a) J = 4t²/U
b) J = 2t²/U
c) J = t²/U
d) J = t²/2U
Answer: a) J = 4t²/U
62. What is the form of the Landau free energy for a system with O(3) symmetry near the critical
point?
a) F = at² + bt⁴ + c( t)²∇
b) F = at² - bt⁴ + c( t)²∇
c) F = -at² + bt⁴ + c( t)²∇
d) F = -at² - bt⁴ + c( t)²∇
Answer: c) F = -at² + bt⁴ + c( t)²∇
63. In the context of the Ising model, what is the form of the critical exponent ν for the
correlation length?
a) ν = 1 (2D), 5/8 (3D)
b) ν = 5/8 (2D), 1 (3D)
c) ν = 1 (2D), 1/2 (3D)
d) ν = 1/2 (2D), 1 (3D)
Answer: a) ν = 1 (2D), 5/8 (3D)
64. What is the form of the Landau-de Gennes free energy for biaxial nematic liquid crystals?
a) F = aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
c) F = -aQ_ij² + bQ_ij³ + cQ_ij⁴ + dQ_ij⁵ + eQ_ij⁶
d) F = -aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
Answer: b) F = aQ_ij² - bQ_ij³ + cQ_ij⁴ - dQ_ij⁵ + eQ_ij⁶
65. In the context of the Heisenberg model, what is the form of the spin-wave dispersion relation
for antiferromagnets?
a) ω_k = √(2JSΔK)
b) ω_k = 2JS√(ΔK)
c) ω_k = √(4J²S²ΔK)
d) ω_k = 4JS√(ΔK)
Answer: c) ω_k = √(4J²S²ΔK)
66. What is the form of the Mermin-Wagner-Hohenberg theorem for continuous symmetries in
low dimensions?
a) No long-range order in d ≤ 2 for short-range interactions
b) No long-range order in d < 2 for short-range interactions
c) No long-range order in d ≤ 3 for short-range interactions
d) No long-range order in d < 3 for short-range interactions
Answer: a) No long-range order in d ≤ 2 for short-range interactions
67. In the context of the Gross-Neveu model, what is the form of the beta function to leading
order?
a) β(g) = (N-2)g³/2π
b) β(g) = -(N-2)g³/2π
c) β(g) = (2-N)g³/2π
d) β(g) = -(2-N)g³/2π
Answer: b) β(g) = -(N-2)g³/2π
68. What is the form of the Ginzburg criterion for the validity of mean-field theory in terms of the
Ginzburg number Gi?
a) |t| >> Gi
b) |t| << Gi
c) |t| >> Gi^(1/2)
d) |t| << Gi^(1/2)
Answer: a) |t| >> Gi
69. In the context of the Nambu-Goto action for strings, what is the form of the equation of
motion?
a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
b) ∂_α(√(γ)γ^αβ∂_βX^μ) = 0
c) ∂_α(√(-γ)γ^αβ)∂_βX^μ = 0
d) ∂_α(√(γ)γ^αβ)∂_βX^μ = 0
Answer: a) ∂_α(√(-γ)γ^αβ∂_βX^μ) = 0
70. What is the form of the Polyakov path integral for string theory?
a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
b) Z = ∫DX^μDγ_αβ exp(iS_P[X,γ])
c) Z = ∫DX^μ exp(-S_P[X])
d) Z = ∫DX^μ exp(iS_P[X])
Answer: a) Z = ∫DX^μDγ_αβ exp(-S_P[X,γ])
71. In the context of the Coleman-Weinberg mechanism, what is the condition for the stability of
the vacuum?
a) λ > 0
b) λ < 0
c) λ = 0
d) λ > e⁴/32π²
Answer: d) λ > e⁴/32π²
72. What is the form of the Bogomolny equations for monopoles?
a) B_i = D_iφ
b) B_i = -D_iφ
c) E_i = D_iφ
d) E_i = -D_iφ
Answer: a) B_i = D_iφ
73. In the context of the sine-Gordon model, what is the form of the breather solution?
a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
b) φ(x,t) = 4arctan[β cos(ωt) / cosh(βx)]
c) φ(x,t) = 4arctan[β sin(ωt) / sinh(βx)]
d) φ(x,t) = 4arctan[β cos(ωt) / sinh(βx)]
Answer: a) φ(x,t) = 4arctan[β sin(ωt) / cosh(βx)]
74. What is the form of the Landau-Zener formula for the transition probability in terms of the
Landau-Zener parameter Λ?
a) P = exp(-πΛ)
b) P = 1 - exp(-πΛ)
c) P = exp(-2πΛ)
d) P = 1 - exp(-2πΛ)
Answer: b) P = 1 - exp(-πΛ)
75. In the context of the Kibble-Zurek mechanism, what is the form of the defect density n near
the critical point?
a) n ~ |ε|^(νd)
b) n ~ |ε|^(-νd)
c) n ~ |ε|^(ν(d-1))
d) n ~ |ε|^(-ν(d-1))
Answer: b) n ~ |ε|^(-νd)
76. What is the form of the Brazovskii transition temperature shift ΔT_c in terms of the Ginzburg
number Gi?
a) ΔT_c/T_c ~ Gi^(1/2)
b) ΔT_c/T_c ~ Gi^(-1/2)
c) ΔT_c/T_c ~ Gi
d) ΔT_c/T_c ~ Gi^(-1)
Answer: a) ΔT_c/T_c ~ Gi^(1/2)
77. In the context of the Higgs mechanism, what is the mass of the gauge boson in terms of the
gauge coupling g and the vacuum expectation value v?
a) m_A = gv
b) m_A = g²v
c) m_A = gv/2
d) m_A = g²v/2
Answer: a) m_A = gv
78. What is the form of the BCS coherence length ξ in terms of the Fermi velocity v_F and the ₀
superconducting gap Δ?
