Antiferromagnetic Ising model

Nearest-neighbor Ising spin system with couplings that favor antiparallel alignment, leading to Néel (staggered) order on bipartite lattices.

Antiferromagnetic Ising model

The antiferromagnetic Ising model is a special case of the Ising model in which the interaction energetically favors opposite spins on neighboring sites.

Let $\Lambda$ be a finite subset of $\mathbb{Z}^d$ with the usual nearest-neighbor adjacency (or more generally a finite graph). A spin configuration is a map $\sigma:\Lambda\to\{-1,+1\}$ (so the spin space is $\{-1,+1\}$ ). With a boundary condition $\eta$ on $\Lambda^c$ , the standard nearest-neighbor antiferromagnetic lattice Hamiltonian is

H_\Lambda(\sigma\mid \eta) ={} J\sum_{\langle x,y\rangle:\, x,y\in\Lambda}\sigma_x\sigma_y + J\sum_{\substack{\langle x,y\rangle:\\ x\in\Lambda,\,y\notin\Lambda}}\sigma_x\eta_y -{} h\sum_{x\in\Lambda}\sigma_x, \qquad J>0,

where $h$ is an external field and $\langle x,y\rangle$ denotes a nearest-neighbor edge.

At inverse temperature $\beta$ , the corresponding finite-volume Gibbs measure is

\mu_{\Lambda,\beta}^{\eta}(\sigma) ={} \frac{1}{Z_{\Lambda,\beta}(\eta)}\exp\!\big(-\beta H_\Lambda(\sigma\mid\eta)\big),

where $Z_{\Lambda,\beta}(\eta)$ is the partition function .

Key properties

Local energetic preference (antialignment). Since $J>0$ multiplies $\sigma_x\sigma_y$ , an edge contributes lower energy when $\sigma_x\sigma_y=-1$ , i.e. neighboring spins are opposite.
Bipartite lattices and Néel order. On a bipartite lattice (e.g. $\mathbb{Z}^d$ ), the ground states are the two “checkerboard” configurations (Néel states) obtained by fixing opposite spins on the two sublattices. The natural order parameter is the staggered magnetization
$m_{\mathrm{stag}}(\sigma)=\frac{1}{|\Lambda|}\sum_{x\in\Lambda}\varepsilon_x\,\sigma_x,$
where $\varepsilon_x=\pm 1$ records the sublattice (for $\mathbb{Z}^d$ , one choice is $\varepsilon_x=(-1)^{x_1+\cdots+x_d}$ ).
Gauge transform at zero field (bipartite case). When $h=0$ and the underlying graph is bipartite, flipping all spins on one sublattice maps the antiferromagnetic model to a ferromagnetic Ising model (up to an additive constant in the energy). This equivalence generally fails when $h\neq 0$ .
Frustration on non-bipartite graphs. On non-bipartite lattices (e.g. triangular lattice), not all edges can be simultaneously satisfied; this produces frustration and can lead to highly degenerate ground states and altered (or suppressed) ordering behavior.
Phase transitions and symmetry breaking. In dimensions $d\ge 2$ (and for sufficiently short-range interactions), the antiferromagnetic model on a bipartite lattice typically exhibits a low-temperature ordered phase characterized by nonzero staggered magnetization and spontaneous symmetry breaking in the infinite-volume limit, formulated via infinite-volume Gibbs measures and phase transitions .

Physical interpretation

The antiferromagnetic Ising model is a minimal model for antiferromagnets, where neighboring magnetic moments prefer to point in opposite directions due to exchange interactions. On bipartite lattices this produces Néel order (alternating sublattice magnetization). Unlike the ferromagnetic case, the usual uniform spontaneous magnetization can vanish even in an ordered phase; the physically meaningful “magnetization” is staggered.