Heisenberg model

In the context of classical lattice spin systems, the Heisenberg model (often called the O(3) model) is a continuous-spin model in which each site carries a three-dimensional unit vector.

Let $\Lambda\subset\mathbb{Z}^d$ be finite. A spin configuration is a map $S:\Lambda\to\mathbb{S}^2\subset\mathbb{R}^3$, where the spin space is the unit sphere $\mathbb{S}^2$. A standard nearest-neighbor Hamiltonian is

with external field $\mathbf{h}\in\mathbb{R}^3$ (an external-field coupling ). Boundary effects can be incorporated via a boundary condition .

At inverse temperature $\beta$, the finite-volume Gibbs measure is proportional to $\exp(-\beta H_\Lambda(S))$ with respect to product surface measure on $(\mathbb{S}^2)^\Lambda$, normalized by the partition function .

(Remark: “Heisenberg model” is also used for a quantum spin system; this knowl refers to the classical O(3) lattice model.)

Key properties

• Continuous O(3) symmetry. For $\mathbf{h}=0$, the model is invariant under global rotations $S_x\mapsto R S_x$ with $R\in\mathrm{O}(3)$. This makes it a central example for studying spontaneous symmetry breaking and Goldstone-mode fluctuations.

• Ferromagnetic vs antiferromagnetic.

  • $J>0$ favors alignment ($S_x\approx S_y$), producing ferromagnetic order in sufficiently high dimension/low temperature.
  • $J<0$ favors antiparallel neighbors, and the nature of ordering depends strongly on lattice geometry (bipartite vs frustrated).

• Dimensional effects (short-range interactions). For finite-range, translation-invariant interactions (see finite-range interactions ), continuous symmetry suppresses conventional long-range order in low dimensions, while in higher dimensions one can have multiple pure phases and nontrivial phase transitions .

• Correlations and response. The behavior of two-point correlations , the correlation length , and the susceptibility are central diagnostics of ordering and criticality.

• Relation to other O(n) models. The XY model is the O(2) analogue with spins on $\mathbb{S}^1$, and many structural arguments (symmetries, spin waves, low-dimensional constraints) parallel between O(2) and O(3).

Physical interpretation

The Heisenberg model is a basic model for isotropic classical magnetism, where local magnetic moments can point in any direction in $\mathbb{R}^3$ and energetically prefer to align (ferromagnet) or anti-align (antiferromagnet). Its continuous symmetry makes it a standard laboratory for:

• how long-range order depends on dimension,
• how low-energy collective excitations affect correlations,
• how ordered phases are described by infinite-volume Gibbs measures (often extremal ones; see extremal Gibbs measures ).