Ising Model - Definition

Definition

Given a graph Λ (for example, a d-dimensional lattice), per each lattice site j ∈ Λ there is a discrete variable σ_j such that σ_j ∈{+1, −1}. A spin configuration, σ = (σ_j)_j∈Λ is an assignment of spin value to each lattice site.

For any two adjacent sites i, j ∈Λ one has an interaction J_ij, and a site i ∈ Λ has an external magnetic field h_i. The energy of a configuration σ is given by the Hamiltonian Function

$H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\sum_{j} h_j\sigma_j$

where the first sum is over pairs of adjacent spins (every pair is counted once). indicates that sites i and j are nearest neighbors. The configuration probability is given by the Boltzmann distribution with inverse temperature β ≥0:

$P_\beta(\sigma) ={e^{-\beta H(\sigma)} \over Z_\beta}, \,$

where β = (k_BT)-1

and the normalization constant

$Z_\beta = \sum_\sigma e^{-\beta H(\sigma)} \,$

is the partition function. For a function f of the spins ("observable"), one denotes by

the expectation (mean value) of f.

The configuration probabilities represent the probability of being in a state with configuration σ in equilibrium.

Famous quotes containing the word definition:

“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)

“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)

“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)