HolsteinPrimakoff

The Holstein–Primakoff transformation is a method in quantum magnetism that maps spin operators onto bosonic creation and annihilation operators, enabling the treatment of spin systems with techniques developed for bosons. Introduced by T. Holstein and H. Primakoff in 1940, it provides a bosonic representation of the SU(2) spin algebra for spins on a lattice.

For a spin S at site i, the standard representation uses a bosonic operator a_i and its adjoint a_i†:

S_i^+ = sqrt(2S − a_i† a_i) a_i

S_i^− = a_i† sqrt(2S − a_i† a_i)

S_i^z = S − a_i† a_i

This representation preserves the spin commutation relations exactly, with the constraint that a_i† a_i ≤ 2S to keep the Hilbert space dimension correct.

In the dilute-magnon or large-S limit, the square-root factors are expanded, yielding the linear spin-wave theory (LSWT):

S_i^+ ≈ sqrt(2S) a_i

S_i^− ≈ sqrt(2S) a_i†

S_i^z ≈ S − a_i† a_i

This approximation describes noninteracting magnons and provides tractable expressions for the magnon spectrum in ordered magnets.

Applications include analysis of the Heisenberg model for ferromagnets and antiferromagnets, calculation of ground-state energy corrections, and the magnon dispersion relations. For antiferromagnets, one typically employs a sublattice decomposition and then applies Bogoliubov transformations to diagonalize the quadratic bosonic Hamiltonian.

The Holstein–Primakoff method is one among several bosonic representations of spins, alongside Dyson–Maleev and Schwinger bosons. Its main limitations arise from truncation of the square-root expansion, which introduces magnon interactions and is most reliable at low temperatures and low magnon densities.