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.

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