Theory
dlc = 1 (Ising, Z2)
dlc = 2 (XY, Heisenberg)
Mermin-Wagner (1966): No spontaneous breaking of continuous symmetry in d ≤ 2 at T > 0. Goldstone mode fluctuations diverge in low d.
⟨φ(x)φ(0)⟩ ~ exp(−r/ξ) d=3
⟨φ(x)φ(0)⟩ ~ r−η d=2 (BKT)
⟨φ(x)φ(0)⟩ ~ const? d=1 NO!
In d=1, domain walls have zero energy cost → order is destroyed at any T > 0 (Peierls argument). In d=2 XY model: BKT transition via vortex unbinding.