Questions: The Berezinskii-Kosterlitz-Thouless Transition
4 questions to test your understanding
Score: 0 / 4
Question 1 Multiple Choice
For the XY model with coupling J, a single vortex in a system of size L with lattice spacing a has energy about πJ ln(L/a) and entropy about 2k_B ln(L/a). At what temperature does this simple argument predict free vortices appear?
AT = J/k_B
BT = πJ/(2k_B)
CT = 2πJ/k_B
DT = 0
The free energy is F = E − TS = (πJ − 2k_BT) ln(L/a). It becomes negative, favouring free vortices, when T > πJ/(2k_B).
Question 2 True / False
Below T_KT, the two-dimensional XY model has a nonzero average magnetization in the limit of infinite system size.
TTrue
FFalse
Answer: False
The Mermin-Wagner theorem forbids spontaneous breaking of a continuous symmetry in two dimensions at any T > 0. The low-temperature phase has power-law correlations but zero average magnetization.
Question 3 Multiple Choice
How does the spin-spin correlation function behave at large distance r in the low-temperature phase?
AIt approaches a nonzero constant
BIt decays as a power law, r^(−η)
CIt decays exponentially, e^(−r/ξ)
DIt grows with r
Spin-wave fluctuations give quasi-long-range order, with an exponent η that depends on temperature and reaches 1/4 at T_KT. Above T_KT, free vortices make the decay exponential.
Question 4 Short Answer
Explain how the BKT transition is consistent with the Mermin-Wagner theorem.
Think about your answer, then reveal below.
Model answer: Mermin-Wagner forbids a nonzero order parameter, not a phase transition. The BKT transition separates two phases that both have zero magnetization; they differ in how correlations decay and in whether vortices are bound or free.
The low-temperature phase has a finite spin stiffness (or superfluid density) without long-range order. That stiffness drops to zero at T_KT.