Classical Heisenberg model ($E=-J{\sum }_{⟨ij⟩}{\mathbf{s}}_i\cdot {\mathbf{s}}_j,|{\mathbf{s}}_i|=1$)

The classical Heisenberg model assigns a three-component unit vector to each site and couples pairs by their dot product. Positive $J$ favors parallel alignment. With no applied magnetic field or anisotropy, simultaneous rotations of all vectors preserve the energy, so an ordered state chooses a direction and exhibits spontaneous symmetry breaking. The vector order parameter is the spatially averaged orientation. A quartic Landau free energy describes its mean-field onset near a continuous transition; fluctuations can change the critical exponents.^[ref]

For unit three-component classical spins and energy $-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ on a lattice of coordination $z_c$, mean-field decoupling gives $E_{\mathrm{MF}}=-z_{c}JS{\sum }_i{S}_{iz}+N{z}_{c}J{S}^2/2$. Integrating each spin over solid angle yieldsThe constant term corrects double counting. A normalized angular measure removes $(4\pi {)}^N$ but does not change the self-consistency equation. The lattice dimension and spin-space dimension are different data.

The single-site average in molecular field $z_{c}JS$ is $S=\coth x-1/x$, where $x=\beta z_{c}JS$. Thus $\tau x=\coth x-1/x$ with $\tau =k_{B}T/(z_{c}J)$. The small-$x$ slope $1/3$ gives $k_B{T}_c^{\mathrm{MF}}=z_{c}J/3$. On a square lattice $z_c=4$. This finite mean-field transition does not imply actual positive-temperature magnetization for the ideal two-dimensional short-range isotropic model, where the Mermin-Wagner theorem applies.