ANNNI model

Example

Let $\Lambda$ be a finite box of $\Z^d$ and pick one coordinate axis, the axial direction. Let $J_0, J_1 > 0$ and $J_2 > 0$ be constants and let $\beta \ge 0$ be the inverse temperature. A configuration is $\sigma \colon \Lambda \to \set{-1, +1}$, and $e$ denotes the unit vector of the axial direction. Its energy is $$H(\sigma) = -J_0 \sum_{\text{in-layer}} \sigma_x \sigma_y - J_1 \sum_{x} \sigma_x \sigma_{x + e} + J_2 \sum_{x} \sigma_x \sigma_{x + 2e},$$ where the first sum runs over nearest-neighbour pairs $\set{x, y}$ with $y - x$ orthogonal to $e$. The ANNNI model (axial next-nearest-neighbour Ising model) is the probability measure on $\set{-1, +1}^\Lambda$ proportional to $e^{-\beta H(\sigma)}$.