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:

“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)

Related Phrases