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 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
L = |Λ|: the total number of sites on the lattice, σj ∈ {−1, … 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 turns to steam. His idea was that small changes in atomic-scale properties would … 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 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