Ising model

The Ising model (/ˈaɪsɪŋ/; German: ), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model (/ˈaɪsɪŋ/; German: ), named after the physicist Ernst Ising, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. The model allows the identification of phase transitions, as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model has no phase transition and was solved by Ising (1925) himself in his 1924 thesis. The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by Lars Onsager (1944). It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory. In dimensions greater than four, the phase transition of the Ising model is described by mean field theory. Consider a set Λ of lattice sites, each with a set of adjacent sites (e.g. a graph) forming a d-dimensional lattice. For each lattice site k ∈ Λ there is a discrete variable σk such that σk  ∈ {+1, −1}, representing the site's spin. A spin configuration, σ = (σk)k ∈ Λ is an assignment of spin value to each lattice site. For any two adjacent sites i, j ∈ Λ there is an interaction Jij. Also a site j ∈ Λ has an external magnetic field hj interacting with it. The energy of a configuration σ is given by the Hamiltonian function where the first sum is over pairs of adjacent spins (every pair is counted once). The notation ⟨ij⟩ indicates that sites i and j are nearest neighbors. The magnetic moment is given by µ. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally. The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥ 0: where β = (kBT)−1 and the normalization constant is the partition function. For a function f of the spins ('observable'), one denotes by the expectation (mean) value of f.