Heisenberg and Ising spin Hamiltonians

Let each unordered pair occur once in $H=\sum_{\langle ij\rangle}J_{ij}\mathbf s_i\cdot\mathbf s_j$. The operators are dimensionless, so J is an energy and a physical angular momentum is ħ times s. Positive J favors antiferromagnetic correlations. An ordered sum over both i,j and j,i needs a factor one half. A paper that instead puts a minus sign before the entire sum reverses which sign of its coupling is called ferromagnetic.

For spin one half, $\mathbf s=\boldsymbol\sigma/2$, not the full vector of Pauli matrices. Consequently a coefficient multiplying a Pauli-matrix product differs by a factor of four from the same coefficient multiplying spin operators. A classical Ising variable taking values plus and minus one is another normalization. Its coupling must be translated before comparing a transition temperature or bond energy with a quantum spin-half Hamiltonian. Energy differences provide a quick invariant check on those translations.

Resolve the scalar product into a longitudinal term and two ladder-operator terms. The longitudinal part measures the projections already present. A transverse term raises one spin while lowering its neighbor and conserves total z projection. It therefore connects the product states with opposite projections. The symmetric and antisymmetric combinations of those states become the triplet-zero and singlet eigenstates of the isotropic bond; a lone antiparallel product state is not an energy eigenstate.

For a longitudinal Ising bond, both antiparallel product states have the same energy and no such mixing occurs. The spin-only Ising Hamiltonian is diagonal in the common z basis. Adding a transverse field makes an Ising model quantum mechanically noncommuting again, so describing every Ising model as classical would be too broad. Exchange anisotropy is a physical selection of couplings, whereas mean-field factorization is a later approximation to the chosen interacting model.

The isotropic Heisenberg model has continuous global spin-rotation symmetry, whereas a longitudinal easy-axis interaction distinguishes an axis and can retain a discrete reversal symmetry. Those differences affect collective modes and thermal ordering. In an infinite one- or two-dimensional isotropic short-range Heisenberg system at positive temperature, the Mermin–Wagner result forbids spontaneous breaking of continuous spin rotation. A finite sample may have strong correlations without defining a thermodynamic singularity or evading the theorem.

A simple uniform molecular-field treatment gives the high-temperature Curie–Weiss intercept $\Theta=-zJS(S+1)/(3k_B)$ for equal nearest-neighbor couplings and coordination z in our convention. Negative J produces a positive intercept; positive J produces a negative one. This formula is an approximation, and its associated ordering scale can remain nonzero in dimensions where exact thermal order is forbidden. Frustration, anisotropy, longer-range interactions, and quantum fluctuations require further analysis; an intercept alone does not identify a Néel temperature.