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The operators of localized spins within a magnetic material commute atdifferent sites of its lattice and anticommute on the same site, so they areneither fermionic nor bosonic operators. Thus, to construct diagrammaticmany-body perturbation theory, the spin operators are usually mapped to thebosonic ones with Holstein-Primakoff (HP) transformation being the most widelyused in magnonics and spintronics literature. However, to make calculationstractable, the square root of operators in the HP transformation is expandedinto a Taylor series truncated to some low order. This poses a question on therange of validity of truncated HP transformation when describing nonequilibriumdynamics of localized spins interacting with each other or with conductionelectron spins. Here we apply exact diagonalization techniques to Hamiltonianof fermions (i.e., electrons) interacting with HP bosons vs. Hamiltonian offermions interacting with the original localized spin operators in order tocompare their many-body states and one-particle equilibrium or nonequilibriumGreen functions. The Hamiltonian of fermions interacting with HP bosons givesincorrect ground state and electronic spectral function, unless large number ofterms are retained in truncated HP transformation. Furthermore, trackingnonequilibrium dynamics of localized spins over longer time intervals requiresprogressively larger number of terms in truncated HP transformation. Finally,we show that recently proposed [M. Vogl et al., Phys. Rev. Research 2, 043243(2020); J. K\"{o}nig et al., SciPost Phys. 10, 007 (2021)] resummed HPtransformation resolves the trouble with truncated HP transformation, whileallowing us to derive an exact (manifestly Hermitian) Hamiltonian consisting offinite and fixed number of boson-boson and electron-boson interacting terms.

physical review. b./physical review. b

Quantum many-body states and Green functions of nonequilibrium  electron-magnon systems: Localized spin operators vs. their mapping to  Holstein-Primakoff bosons