Simulation Results
Results appear here after calculation and remain above the form.
Your phase interpretation will appear here.
Order Parameter Trace
Energy Trace
| Metric | Value | Unit / Meaning |
|---|
Calculator Inputs
This simulator uses a 2D square lattice with periodic boundaries and Metropolis updates.
Example Data Table
Illustrative outputs for a 24 × 24 ferromagnetic lattice with J = 1.00, h = 0.00, random initial spins, 300 thermalization sweeps, and 500 measurement sweeps.
| Temperature | Energy / Spin | Magnetization / Spin | Susceptibility | Heat Capacity | Interpretation |
|---|---|---|---|---|---|
| 1.50 | -1.94 | 0.97 | 0.08 | 0.41 | Strongly ordered regime |
| 2.25 | -1.48 | 0.61 | 1.68 | 1.37 | Near critical crossover |
| 3.40 | -0.68 | 0.08 | 0.19 | 0.33 | Disordered regime |
Formula Used
The simulator uses the standard Ising Hamiltonian for a 2D square lattice:
For a proposed spin flip, the local energy change is:
The Metropolis acceptance rule is:
The calculator reports these observables from measurement sweeps:
- Energy per spin: e = ⟨E⟩ / N
- Magnetization per spin: m = ⟨Σsᵢ⟩ / N
- Staggered magnetization: ms = ⟨Σ(-1)x+ysᵢ⟩ / N
- Susceptibility: χ = N(⟨q²⟩ - ⟨q⟩²) / T, using q = m or ms
- Heat capacity: C = (⟨E²⟩ - ⟨E⟩²) / (N T²)
How to Use This Calculator
- Enter lattice width and height. Larger lattices give smoother averages but need more computation.
- Set temperature, coupling, and external field. Positive coupling favors alignment; negative coupling favors checkerboard ordering.
- Choose the initial state. Random starts are common for unbiased experiments.
- Select thermalization sweeps to relax the system before measurements begin.
- Choose measurement sweeps and a sampling interval to control averaging depth.
- Press Run Simulation. The results appear above the form with summary cards, traces, and an export table.
- Use CSV for spreadsheet analysis or PDF for quick reporting and documentation.
Frequently Asked Questions
1) What does this calculator simulate?
It simulates a two-dimensional Ising lattice using nearest-neighbor interactions and Metropolis updates. It estimates energy, magnetization, susceptibility, heat capacity, and ordering behavior from repeated Monte Carlo sweeps.
2) Why do results vary between runs?
The simulation is stochastic. Different random spin-flip sequences produce slightly different trajectories, especially near the critical region where fluctuations become large and convergence is slower.
3) What is the critical temperature here?
For the zero-field two-dimensional square lattice with positive coupling and kB = 1, the reference critical temperature is about 2.269 × J in magnitude.
4) When should I increase sweeps?
Increase thermalization and measurement sweeps when the lattice is larger, the temperature is near the transition region, or the traces still show strong drift rather than steady fluctuations.
5) Why include staggered magnetization?
Negative coupling prefers alternating spin patterns. Ordinary magnetization can stay near zero even when strong antiferromagnetic order exists, so staggered magnetization helps reveal that structure.
6) Does this represent real materials exactly?
No. It is a simplified teaching and exploration model. Real materials may include longer-range interactions, quantum effects, anisotropy, impurities, or dimensional differences.
7) What does the acceptance ratio mean?
It shows the fraction of proposed spin flips that were accepted. Very low ratios can indicate a frozen state, while very high ratios often appear at hotter, more disordered settings.
8) Which export format should I use?
Use CSV when you want to sort, chart, or compare runs in a spreadsheet. Use PDF when you want a compact report for class notes, documentation, or sharing.