Antiferromagnetic order and magnons

Consider the nearest-neighbor square-lattice Hamiltonian $H=J\sum_{\langle ij\rangle}\mathbf s_i\cdot\mathbf s_j$ with J positive and each bond once. The lattice is bipartite, with four neighbors per site. A classical Néel state has angular momenta opposite on its two sublattices. Its uniform average vanishes, but its staggered average is finite. For electron-like moments the directions of all moment arrows reverse relative to angular momentum; the alternating pattern remains antiferromagnetic.

On an antiparallel bond, a ladder-operator term can lower the up spin and raise the down spin. The resulting state differs from the Néel product, proving that product is not an exact eigenstate. Quantum zero-point fluctuations reduce the sublattice moment from its classical value. A finite balanced antiferromagnet can have a rotationally symmetric singlet ground state while correlations reveal the pattern that becomes ordered after suitable thermodynamic and selecting-field limits. A zero uniform magnetization alone distinguishes neither case from a disordered state.

Expand fluctuations around opposite local quantization axes. One boson counts deviations on sublattice A and another on B. To quadratic order the exchange contains both number terms and pair creation or annihilation terms, because a transverse process changes both local deviations together. A bosonic Bogoliubov transformation preserves commutators while diagonalizing this quadratic Hamiltonian. It yields a spectrum involving a square root, rather than the simple difference between a constant and a cosine found for a ferromagnet.

At zero field and without anisotropy, the two positive-frequency branches are degenerate in the reduced magnetic Brillouin zone, corresponding to the two transverse polarizations, which can be chosen circular with opposite angular momenta. Many plots show just one energy curve over the larger structural zone instead. Its zeros at zero and the antiferromagnetic wavevector are related by magnetic-zone folding; counting two polarizations and also independently counting the unfolded momenta would count the same states twice. Fields or anisotropies can lift the degeneracy.

The square-root band is the leading large-S spin-wave result. Interactions among the fluctuation bosons produce energy renormalizations and finite lifetimes, particularly where decays are allowed. For the spin-half nearest-neighbor square model, the zero-temperature ground state does support Néel order, but the leading dispersion is not numerically exact. Successful identification of a collective mode does not license replacing all spectral weight by a single sharp noninteracting magnon in every frustrated or low-dimensional magnet.

For a strictly two-dimensional isotropic short-range system, continuous spin-rotation symmetry cannot break spontaneously at positive temperature. The spin-wave construction at zero temperature does not contradict that restriction. Interlayer coupling, symmetry-reducing anisotropy, or interactions outside the short-range assumptions can permit real quasi-two-dimensional materials to order. Finite samples can display large staggered correlations and an apparent crossover scale. A positive molecular-field estimate remains an estimate of local exchange strength, not proof of a true nonzero Néel temperature.