### Ising Model

One of the most fundamental models in statistical physics that is first introduced as an effective model of the ferromagnetic-paramagnetic phase transition. This model consists of “spins” that take value of +/- 1 on a lattice. When each spin tends to take the same value as that of neighbor spins (ferromagnetic interaction), all spins take the same value at absolute zero (ferromagnetic phase or ordered phase). At the high temperature limit, on the other hand, all spins takes independent values from each other (paramagnetic phase or disordered phase). Finite-temperature phase transition is a phenomena that the system changes from one phase to the other phase at some temperature. When randomness of interactions is introduced, this model becomes difficult to solve even numerically. For example, the ground state cannot be found in polynomial time and the thermal-relaxation time becomes very long (spin glass). Ising model is a classical model originally, but one can obtain quantum version of Ising model by introducing transverse magnetic field flipping a spin as a source of quantum fluctuation. In this model (transverse field Ising model,) ferromagnetic-paramagnetic phase transition manifests oneself even at absolute zero by controlling the strength of a field (quantum phase transition). Since Ising model itself is very simple but supports many phenomena, it has been well studied and used in not only classical / quantum statistical physics but also classical / quantum information theory and machine learning.