The spin-spin correlation function in the context of the Ising model (and statistical mechanics in general) provides a measure of how the orientation of a spin at one lattice site is related to the orientation of a spin at another lattice site. It quantifies the degree to which two spins tend to be aligned (or anti-aligned) with each other.

Definition: The spin-spin correlation function $G(i, j)$ between spins $S_i$ at site $i$ and $S_j$ at site $j$ is typically defined as:

$$ G(i, j)=\left\langle S_i S_j\right\rangle-\left\langle S_i\right\rangle\left\langle S_j\right\rangle $$

where:

• $S_i$ and $S_j$ are the spin variables at sites $i$ and $j$ (in the Ising model, $S=+1$ or -1 ).
• $\langle\ldots\rangle$ denotes the statistical average (usually the thermal average in the canonical ensemble).

🧠Example

A custom example of a Monte Carlo simulation for the Spin-Spin Correlation Function.

https://gist.github.com/viadean/a30b52fd33c5a27ed02505972538523c

Explanation:

1. Initialization:
   • A 2D lattice of size  is initialized with random spins ((+1) or (-1)).
2. Monte Carlo Simulation:
   • The Metropolis algorithm is used to simulate the system's evolution. Spins are flipped based on the energy change ($\Delta E$) and the Metropolis acceptance criteria.
3. Spin-Spin Correlation:
   • For a given distance $r$, the correlation between spins is computed by averaging the product of spin pairs separated by $r$.
4. Visualization:
   • The correlation function $G(r)$ is plotted as a function of distance $r$ .

This code provides a basic framework for studying spin systems with Monte Carlo methods.