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Lecture notes in physics

Applications of generalized holstein-primakoff transformations to problems of nuclear collective motion

Abraham Klein, Yue Li, Mathilde Vallières

Published Jan 1, 1981• 0 citations
Publisher
DOI
10.1007/bfb0018144

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hep-phCross-listedarXiv:2610.36026•Oct 28, 2026•0 citations
A comparison between the finite Dyson-type and the infinite Holstein-Primakoff-type boson expansion is presented, considering both particle-hole and pairing type excitations. The significance of truncating to the truly collective branches is emphasized, and the convergence of the Holstein-Primakoff expansion is discussed. It is shown that in both expansions there is no need to construct physical basis states. Complete equivalence between the two types of expansion is exhibited by treating these excitations within models which are schematic but not totally unrealistic.NUCLEAR STRUCTURE Boson expansion theory, convergence, comparison of different methods, finite Dyson expansion, infinite Holstein-Primakoff expansion, pairing and particle-hole collective motions.
Spin-wave theory provides a quasiparticle description of excitation spectra in quantum spin systems. In this theory, the spin operators are usually bosonized by the Holstein-Primakoff transformation or the Dyson-Maleev transformation. In practical calculations, the bosonized Hamiltonian has to be truncated, and thus the resulting excitation spectra depend on the choice of bosonic representation. Here, we introduce a generalized Holstein-Primakoff transformation that continuously interpolates between the conventional Holstein-Primakoff and Dyson-Maleev transformations through a single parameter. The formulation naturally extends to the SU(N) algebra and provides a flexible framework for optimizing bosonic truncations beyond harmonic order. As an application, we investigate the spin-1 bilinear-biquadratic model on the square lattice. By combining the generalized Holstein-Primakoff transformation with continuous similarity transformations, we obtain excitation spectra in excellent agreement with tensor-network calculations. Our results demonstrate that optimizing the bosonic representation substantially reduces truncation errors and provides quantitatively reliable descriptions of both quasiparticle dispersions and multi-boson continua.
The generalised Holstein-Primakoff boson representation of the bifermion operators is used to construct a boson image of the microscopic nuclear Hamiltonian. The microscopic analogue of the interacting boson model (IBM) Hamiltonian is obtained. It is shown how the square-root factor in the IBM Hamiltonian can be generalised so as to take into account its dependence on the angular momentum and seniority of the collective state. The method is applicable to systems where the assumption that the projection operator P is approximately equal to unity can be used. The method is proposed to include, apart from the collective quadrupole, other collective and non-collective degrees of freedom in the Hamiltonian.
Recent years have seen a surge of interest in adapting non-classical states to search for physics beyond the Standard Model, including both qubit- and boson-based systems. These two classes of systems are closely related through the Holstein--Primakoff transformation (HPT). In this work, we show that the HPT offers novel insights into inherently quantum-mechanical detection of new physics. We begin by discussing how the HPT can be used to unify certain existing qubit- and boson-based proposals in a single framework. We then show how it naturally reveals a novel generalization of GHZ states which offer superior performance for the detection of rare interactions. We demonstrate that these novel states can be readily prepared, and combine the coherent advantage of GHZ states with superior error rejection. Finally, we show that the HPT also reveals a natural extension of the Hamiltonian-Not-in-Lindblad-Span theorem for collections of bosonic modes.