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 the case of ferromagnetic interaction
• In … 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-neighbor coupling in the zero magnetic … 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 … 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 … 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 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 See more