The Holstein-Primakoff boson representation of quantum spins and associated large- $S$ expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite $S$ . We will present the exact normal-ordered forms of the finite- $S$ projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.The Holstein-Primakoff boson representation of quantum spins and associated large-$S$ expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite $S$. We will present the exact normal-ordered forms of the finite-$S$ projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.

arXiv:2509.26607Physical review. B./Physical review. BDec 22, 20250 citations

Quantum mechanical operator equivalents used in the theory of magnetism

Oluf Danielsen, Per-Anker Lindgård

Jan 1, 197210 citations

Representation of group generators by Boson or Fermion operators application to spin perturbation theory

Charles Paul Enz

Aug 15, 19660 citations

On the phase space representation for spin systems

J. Kutzner

Physics Letters AOct 1, 197211 citations

Bond-operator representation of quantum spins: Mean-field theory of frustrated quantum Heisenberg antiferromagnets

Subir Sachdev A, R. N. Bhatt

We introduce a new representation of S=1/2 quantum spins in terms of bond operators. The bond operators create and annihilate singlet and triplet bonds between a pair of spins. The representation is useful in describing the transition between dimerized and magnetically ordered phases of quantum antiferromagnets. It is used to obtain a mean-field theory of the two-dimensional frustrated quantum Heisenberg antiferromagnets considered recently by Gelfand, Singh, and Huse. The method should also be useful in the analysis of random quantum antiferromagnets.We introduce a new representation of S=1/2 quantum spins in terms of bond operators. The bond operators create and annihilate singlet and triplet bonds between a pair of spins. The representation is useful in describing the transition between dimerized and magnetically ordered phases of quantum antiferromagnets. It is used to obtain a mean-field theory of the two-dimensional frustrated quantum Heisenberg antiferromagnets considered recently by Gelfand, Singh, and Huse. The method should also be useful in the analysis of random quantum antiferromagnets.

Physical review. B, Condensed matterMay 1, 1990486 citations

The most general representation of spin operators via boson operators

Vladimir Chernyak

Physics Letters AMar 1, 19925 citations

The Holstein-Primakoff boson representation of quantum spins and associated large-

S

expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite

S

. We will present the exact normal-ordered forms of the finite-

S

projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.