Let us explain the Ising model, which is one of the simple models that can mathematically express the paramagnetism and ferromagnetic phase transition of ferromagnetic materials. In the Ising model, the spins which handle two different numbers of ±1 are arranged spatially. When aligned spins interact with each other, the interaction is determined depending on the value of each spin.
For example, assume that only two different adjacent spins interact. If an interaction (ferromagnetic interaction) is given in which the energy decreases when the two spins take the same value, then a disordered phase (ferromagnetic state) arise in which all spins have the same value at absolute zero. However, at high temperatures, it becomes a disordered phase (paramagnetic state) in which each of the all spins takes independently different values.
The phenomenon of switching between an ordered phase and a disordered phase at a certain temperature is called a finite temperature phase transition. Furthermore, if the sign and magnitude of the interaction are randomized for each spin pair, it becomes difficult to even find the absolute zero state (the ground state). Due to its property, there may appear a spin glass state in which the thermal relaxation time is extremely long. It becomes difficult to solve the spin states even by numerical calculation.
強磁性体の常磁性や強磁性相転移を数学的に表現することができる簡易的なモデルの1つであるイジングモデルについて説明しよう．イジングモデルでは，±1の2つの異なる数値を取り扱うスピンを空間的に並べる．並べられたスピン同士が相互作用するとき，その相互作用は各々のスピンの取った値に依存して決定される．
例えば隣り合う異なる2つのスピン間だけ相互作用すると仮定すると，その2つのスピンが互いに同じ値を取るとエネルギーが下がるような相互作用（強磁性相互作用）が与えられるとき，絶対零度ではすべてのスピンが同じ値を取る秩序相（強磁性状態）になる．しかし，高温ではすべてのスピンが互いに独立にバラバラな値を取るような無秩序相（常磁性状態）となる．
ある特定の温度で，秩序相と無秩序相が切り替わる現象は有限温度相転移と呼ばれている．また，相互作用の符号や大きさをスピン対ごとにランダム化すると絶対零度の状態（基底状態）を求めることでさえ困難になり，さらに熱緩和時間が非常に長くなるスピングラス状態が現れたりと，たとえ数値計算であっても解くことは困難になる．
