Calculator inputs
Example data table
| Case | Method | T | J | h | r | Expected behavior |
|---|---|---|---|---|---|---|
| High temperature chain | 1D exact | 5.00 | 1.00 | 0.00 | 6 | Fast decay and small C(r) |
| Low temperature chain | 1D exact | 0.80 | 1.00 | 0.00 | 6 | Slow decay and stronger alignment |
| Square lattice estimate | 2D approximation | 2.25 | 1.00 | 0.00 | 8 | Near-critical long range trend |
| Field biased lattice | Mean-field | 3.00 | 1.00 | 0.20 | 10 | Nonzero magnetization shifts C(r) |
Formula used
For a one dimensional chain, the calculator uses the transfer matrix eigenvalue ratio. The reduced coupling is K = J / kT. The reduced field is H = h / kT.
The two eigenvalues are λ± = eᴷ cosh(H) ± √(e²ᴷ sinh²(H) + e⁻²ᴷ). The correlation is estimated by C(r) = m² + (1 - m²)(λ-/λ+)ʳ. The connected part is C(r) - m².
For the two dimensional option, the zero-field critical temperature is Tc = 2J / [k ln(1 + √2)] for a ferromagnetic square lattice. The displayed decay uses an exponential factor and a critical power factor. Mean-field mode solves m = tanh[(zJm + h) / kT].
How to use this calculator
- Select a calculation method. Use 1D exact for a transfer matrix chain.
- Enter temperature, coupling strength, external field, and Boltzmann scale.
- Set the target distance r and maximum plotted distance.
- Use lattice size and spacing to control approximate finite-scale interpretation.
- Press Calculate. The result appears above the form.
- Use CSV for spreadsheet work. Use PDF for a report snapshot.
Spin correlations in the Ising model
What the value means
Spin correlation measures how two lattice spins move together. A value near one means strong alignment. A value near zero means weak memory between sites. A negative value can appear in antiferromagnetic settings. It means alternating order is favored. The distance r controls how far apart the two spins are placed.
Why temperature matters
Temperature competes with coupling. High temperature adds disorder. Correlations then fall quickly. Low temperature supports order. Neighboring spins tend to agree when the coupling is positive. The decay can become slow near a critical point. That slow decay is the reason correlation length is useful.
Exact and approximate modes
The one dimensional option uses a transfer matrix expression. It is reliable for an infinite chain. It also handles a nonzero external field. The two dimensional option is a practical approximation. It uses the square lattice critical temperature and a decay form. Mean-field mode is broader. It works with a selected coordination number.
Reading connected correlation
The raw correlation includes the effect of average magnetization. Connected correlation removes that background. It better shows the extra relationship caused by distance. This is helpful when an external field makes most spins point the same way. The connected value usually decays toward zero.
Using exported results
The table gives values from r equals zero to your maximum distance. The chart shows the decay pattern. Export the CSV when you need further analysis. Export the PDF when you need a clean summary. Always match the method to the physical model before using the numbers in a report.
FAQs
What is a spin correlation function?
It measures how strongly two spins are related at a chosen separation. Positive values show alignment. Negative values show alternating behavior. Values near zero show weak dependence between the sites.
Which method should I choose?
Use the 1D exact method for a chain. Use the 2D approximation for square lattice insight. Use mean-field mode when you want a simple estimate for a chosen coordination number.
What does coupling J mean?
Coupling controls how neighboring spins interact. Positive J favors aligned spins. Negative J favors alternating spins. Larger absolute values usually make correlation decay more slowly at the same temperature.
What does external field h do?
The field biases spins toward one direction. It can create nonzero magnetization. That background raises raw correlation, so connected correlation becomes useful for measuring distance-based dependence.
Why is C(0) equal to one?
A spin compared with itself has perfect agreement because s squared equals one for Ising variables. That is why the table starts with C(0) equal to one.
What is correlation length?
Correlation length estimates the distance scale over which spin memory remains important. A larger value means slower decay. It often grows near a critical temperature.
Is the two dimensional result exact?
No. It uses the known square lattice critical temperature with a practical decay approximation. It is useful for exploration, but detailed research may require simulation or exact methods.
Can I export the results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button after calculation to save the summary, chart, and table in a report-friendly format.