= Solution
The <Holstein–Primakoff transformation> realizes a <Spin commutation relations>[spin algebra] using <bosonic creation operators> and <bosonic annihilation operators>. A <boson> counts a unit of <spin> deviation from the local positive quantization axis: $\widetilde S^z=S-n$ with $n=a^\dagger a$. The square root in $\widetilde S^+$ supplies the correct <Spin ladder operator>[spin-ladder] <matrix elements>. In particular,
$$
\widetilde S^+|n\rangle=\sqrt{n(2S-n+1)}|n-1\rangle,\qquad
\widetilde S^-|n\rangle=\sqrt{(n+1)(2S-n)}|n+1\rangle.
$$
Their <commutator> is $2(S-n)$ on every physical <Fock state>[number state], reproducing the <Spin commutation relations>. Although the <Bosonic Fock space>[oscillator Fock space] is infinite, the <Holstein–Primakoff occupation constraint>[physical subspace] is only
$$
\boxed{n=0,1,\ldots,2S.}
$$
The order of the square root and oscillator <Linear operator>[operator] matters in obtaining these coefficients.
When the <Magnon>[spin deviations] are dilute, expand the square root in $n/(2S)$. The leading transverse <Linear operators>[operators] are linear in the <bosonic creation operators> and <bosonic annihilation operators> and turn <exchange interactions> into a <Quadratic bosonic Hamiltonian>[quadratic Hamiltonian], which can be diagonalized. Higher powers describe interactions between <spin waves>. The method therefore organizes corrections by inverse powers of $S$ about an ordered classical configuration. Large $S$ alone is insufficient: the average <One-dimensional planar spin-wave depletion>[depletion] must remain small compared with $S$. Possible problems include the <Bosonic occupation number>[occupation] constraint after truncation, noncollinear order requiring site-dependent axis <rotations>, <zero modes>, and <Infrared divergence>[infrared fluctuations] that destroy the assumed dilute expansion.
Back to article page
