model using:
- Metropolis Monte-Carlo algorithm
- Periodic boundary conditions
- Finite-size scaling
- Nematic correlation function
- Thermal averages over long Monte-Carlo runs
The 2D XY model exhibits a Berezinskii–Kosterlitz–Thouless (BKT) topological phase transition at a critical temperature around:
[ T_c(\infty) \approx 0.655 \pm 0.015 ]
Features of the BKT transition:
- No spontaneous magnetization for (T>0)
- Quasi-long-range order below (T_c)
- Vortex–antivortex unbinding at (T_c)
| Quantity | Symbol | Purpose |
|---|---|---|
| Order parameter | ⟨S⟩ | Detect degree of alignment |
| Susceptibility | χ(T) | Locate transition peak |
| Heat capacity | Cᵥ(T) | Check thermodynamic anomaly |
| Nematic correlation | g₂(r) | Confirm algebraic vs exponential decay |
| Auto-correlation | ACF | Check statistical independence |
| Snapshots | θ(x,y) | Identify vortex behavior |
Simulated lattice sizes:
[ L = 20, 30, 40, 50, 60, 80 ]
Temperature range:
[ T = 0.60 \rightarrow 0.90 \quad (\Delta T = 0.01) ]