Consider a set $${\displaystyle \Lambda }$$ of lattice sites, each with a set of adjacent sites (e.g. a graph) forming a $${\displaystyle d}$$-dimensional lattice. For each lattice site $${\displaystyle k\in \Lambda }$$ there … See more

The thermodynamic limit exists as long as the interaction decay is $${\displaystyle J_{ij}\sim |i-j|^{-\alpha }}$$ with α > 1.
• In … See more

• In the ferromagnetic case there is a phase transition. At low temperature, the Peierls argument proves positive magnetization for the nearest neighbor case and then, by the Griffiths inequality, also when longer range interactions are added. Meanwhile, … See more

The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a d-dimensional lattice, namely, Λ = Z , Jij = 1, h = 0.
No phase transition in one dimension
In his 1924 PhD thesis, Ising solved the model for the d = 1 … See more

One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in materials , as when ice melts to water or water turns to … See more

Definitions
The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with See more

In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest … See more