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Use dimensionless spin angular momentum operators, so their commutators have no additional $\hslash$. A fully polarized state $|F⟩={⨂}_n|S,S{⟩}_n$ is an exact ground state of the Heisenberg ferromagnet, with $E_0=-NJ{S}^2$. On each bond the largest eigenvalue of ${\mathbf{S}}_n\cdot {\mathbf{S}}_{n+1}$ is $S^2$, obtained in the two-spin total-spin-$2S$ sector. Hence the bondwise lower bound on the Hamiltonian is attained by this state.

The ferromagnetic ground-state multiplet has total spin $NS$. Applying the global lowering operator $(S_{\mathrm{tot}}^{-}{)}^r$, $r=0,\dots ,2NS$, generates its $2NS+1$ magnetic states; their linear combinations are also ground states. Global spin rotations generate polarized ground states pointing in arbitrary directions and span the same multiplet. Uniform reorientation therefore costs no energy. Slowly varying orientations give gapless ferromagnetic magnons, with quadratic rather than linear long-wavelength dispersion. In the ordered thermodynamic ground state, the two broken spin generators are canonically paired because $⟨[S_{\mathrm{tot}}^x,S_{\mathrm{tot}}^y]⟩=i⟨S_{\mathrm{tot}}^z⟩\ne 0$, producing one type-B Goldstone boson. For a finite chain the zero-wavevector motion remains inside the exact ground multiplet; nonzero wavevectors have a finite-size spacing that vanishes as $N^{-2}$. This is a ground-state statement, not a claim of finite-temperature long-range order in one dimension.

Letwhere the two signs describe the Heisenberg antiferromagnet and Heisenberg ferromagnet. A standard Lieb-Robinson bound is obtained by iterating the Heisenberg picture equation and bounding nested commutators by operator norms. Its constants depend on the interaction only through quantities such asChanging $J$ to $-J$ changes neither the supports nor the norms $\Vert \Phi (X)\Vert$. Every term in the nested-commutator estimate acquires at most an irrelevant sign before its absolute value is taken. Therefore both chains obey exactly the same estimatewith the same $C,\mu$, and Lieb-Robinson velocity $v_{\mathrm{LR}}$. No unitary equivalence of the two Hamiltonians is required.

The large effective field primarily represents exchange interactions, which arise from Coulomb interactions, electronic hopping and the Pauli exclusion principle. It is not usually the much smaller classical magnetic dipole field of neighbouring ions. Direct exchange involves overlapping magnetic-ion orbitals. Superexchange instead uses virtual charge transfer through a ligand while real charge transport remains gapped, allowing substantial magnetic order in an electrical insulator.

Use dimensionless spin operators ${\mathbf{S}}_i$ with eigenvalue $S=1/2$, and count each bond once. One consistent Heisenberg model convention isThen $J_{ij}<0$ favours a Heisenberg ferromagnet, while $J_{ij}>0$ favours a Heisenberg antiferromagnet. Replacing neighbouring spins by their mean values gives the effective fieldFor $z$ equivalent ferromagnetic neighbours with $J_{ij}=-J_F<0$ and uniform order, $M=n{g}_J{\mu }_B⟨S_z⟩$, soThe last equality is the same molecular-field result derived above, not an exact transition temperature for every lattice.

Microscopically, a half-filled orbital pair with virtual hopping $t_{ij}$ and a large on-site repulsion $U$ permits an energy lowering of the singlet but not the parallel-spin triplet, yielding antiferromagnetic $J\simeq 4|t_{ij}{|}^2/U$ in this convention. Ligand-mediated hopping is the analogous mechanism in many ionic insulators. Different orbital occupancies and nearly orthogonal ligand paths can suppress this antiferromagnetic term and permit ferromagnetic exchange to dominate. Thus the sign depends on electronic structure and bonding geometry, not simply whether the sample conducts.

For antiferromagnetic coupling on a bipartite lattice, the field on each sublattice is proportional to the opposite sublattice magnetization, with opposite alignment preferred. Two coupled self-consistency equations then permit $M_A=-M_B\ne 0$ and vanishing total magnetization: the order parameter is staggered. The single positive-$\lambda$ uniform equation describes ferromagnetism and must be extended to these sublattices to describe an antiferromagnet.