Ising Models

An Ising model is an array of spins (atoms that can take states $\pm 1$ ) that are magnetically coupled to each other. If one spin is, say $+1, is energetically favourable for its immediate neighbours to be in the same state.

Let the state $x$ of an Ising model with $N$ spins be a vector in which each component $x_{n}$ takes values $+ 1$ and $- 1$ . If two spins $m$ and $n$ are neighbours, we say $(m, n) \in N$ . The coupling between neighbouring spins is $J$ . We define $J_{m n} = J$ if $m$ and $n$ are neighbours, and $J_{m n} = 0$ otherwise. The energy of a state $x$ is:

$E (x | J, H) = - [\frac{1}{2} \sum_{m, n} J_{m n} x_{m} x_{n} + \sum_{n} H x_{n}]$

where $H$ is the applied field. The probability that the state is $x$ is:

$P (x | J, H) = \frac{1}{Z (β, J, H)} exp [- - β E (x; J, H)]$

where $β = \frac{1}{k_{B} T}$ , $k_{B}$ is the Boltzmann’s constant, and:

$Z (β, J, H) = \sum_{x} exp [- β E (x; J, H)]$

The Ising model is also an example of a Markov Random Field (MRF). We create a graph in the form of a 2D or 3D lattice, and connect neighbouring variables. We can define the following clique potential:

$ϕ_{s t} (y_{s}, y_{t}) = (\begin{matrix} e^{w_{s t}} & e^{- w_{s t}} \\ e^{- w_{s t}} & e^{w_{s t}} \end{matrix})$

Here $w_{s t}$ is the coupling strength between nodes $s$ and $t$ . If two nodes are not connected, $w_{s t} = 0$ . The weight matrix $W$ is symmetric: $w_{s t} = w_{t s}$ . All edges have the same strength $J$ , where $w_{s t} \neq 0$ .

The Ising model is analogous to the Gaussian graphical models. First, assuming $y_{t} \in {- 1, + 1}$ , we can write the unnormalized log probability of an Ising model as follows:

$\log \tilde{p} (y) = - \sum_{s \sim t} y_{s} w_{s t} y_{t} + \sum_{s} b_{s} y_{s} = - \frac{1}{2} y^{T} W y + b^{T} y$

where $θ = (W, b)$ , and $b$ is the bias term, corresponding to the local fields. If we define:

$μ = - \frac{1}{2} Σ^{- 1} b, Σ^{- 1} = - W, c = \frac{1}{2} μ^{T} Σ^{- 1} μ$

we can rewrite this in a form that looks similar to a Gaussian:

$\log \tilde{p} (y) = - \frac{1}{2} (y - μ)^{T} Σ^{- 1} (y - μ) + c$

Jethro's Braindump

Ising Models

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