The Ising model is a deceptively simple model for magnets which exhibits a phase transition in 2 dimensions. The model is made up of "spins" which are sitting on a regular lattice. Each spin ($\sigma$) on the lattice can be pointing "up" ($\sigma = \uparrow \mathrm{~or~} +1$) or "down" ($\sigma = \downarrow \mathrm{~or~} -1$).
Temperature ($T$) =
Field ($H$) =
Coupling ($J$) =
This is a Monte Carlo simulation of a moderate size Ising model which allows you to play with the temperature ($T$), the field strength ($H$), and the coupling constant ($J$). The two strip charts show instantaneous readouts of the energy and the magnetization as the simulation progresses.
The initial temperature is set just above the critical temperature ($T_c \approx 2.2692$) in 2 dimensions. Above this temperature, there's no spontaneous magnetization into a bulk "up" or "down" state. But dial the temperature down below $T_c$, and one of those two states will probably win out if you wait long enough.
We represent the energy of the arrangement of spins with the Ising model Hamiltonian: \[ H_{Ising} = -H \sum_{i=1}^{N} \sigma_i -\frac{J}{2} \sum_{i = 1}^{N} \sum_{j \in NN_i} \sigma_i \sigma_j \]
The first summation in the Ising Hamiltonian describes the interactions of individual spins with the external field ($H$). The second set of summations describes the interactions of spins with their nearest neighbors. $J$ is the coupling constant that we use to specify the strength of the interactions. In 2 dimensions, each spin has four nearest neighbors.
$J$ is a very important parameter — if this coupling strength is positive, the spins prefer to be aligned to their nearest neighbor, and if it is negative, the spins prefer to be anti-aligned (e.g. forming a checkerboard pattern).
The version here was originally written by Daniel V. Schroeder in the Physics Department at Weber State University and has been modified by Dan Gezelter to allow interaction with the field, coupling, energy, and magnetization.