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The Heisenberg Hamiltonian H=J∑SiSj with antiferromagnetic interactions between spins (J>0) can lead to different kinds of ground state spin configurations, depending on the type of lattice.

Classical picture of an antiferromagnetic state.
Néel order : spins take on an alternating up-down arrangement . This state breaks the rotational symmetry. This is possible for structures that have even-sided loops such as the honeycomb or square lattices which are bipartite lattices . This is not the case for the triangular lattice, which is said to be frustrated. |

For large values of the spin quantum number S, and bipartite lattices, the ground state is close to the classical state , while in low dimensions and for small S, quantum fluctuations become more important and lead to new effects. Much is known, for example, about the properties of the square lattice or the honeycomb lattice spin S=1/2 antiferromagnets which are two-dimensional systems. Quantum fluctuations lead to a reduction of the local spin average value. At T=0 in the square lattice the order parameter is reduced by about 30 %. In the honeycomb lattice (fewer neighbors and therefore bigger fluctuations) by about 50 %.

The low energy excitations correspond to spin waves – magnons . In ferro- and antiferromagnets these give rise to characteristic power laws in thermodynamic properties at low temperatures.