© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2006.06871 · Physical Review Research

Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots

Michael Rudolf Vogl, Pontus Laurell, Hao Zhang, Satoshi Okamoto, Gregory A. Fiete

Published Nov 17, 2020• 35 citations

Abstract

Operator square roots are ubiquitous in theoretical physics.They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation.Often the use of a perturbative expansion is the only recourse when dealing with them.In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots.Specifically, for the number operator n = a † a we show that the square root √ n near n = 0 can be approximated by a polynomial in n.This result is unexpected because a Taylor expansion fails.A polynomial expression in n is possible because n is an operator, and its constituents a and a † have a non trivial commutator [a, a † ] = 1 and do not behave as scalars.We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators.We prove that these new expressions exactly reproduce spin operators.Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.

DOI: 10.1103/physrevresearch.2.043243

Related papers

Newton series expansion of bosonic operator functions

Jürgen König

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

•Jan 1, 2021•31 citations

Newton series expansion of bosonic operator functions

Jürgen König, Alfred Hucht

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

arXiv:2008.11139•SciPost Physics•Jan 13, 2021•3 citations

Asymptotic Equation for Zeros of Hermite Polynomials from the Holstein-Primakoff Representation

Lucas Kocia

The Holstein-Primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of Hermite polynomials $H_n(x)$ as $n \rightarrow \infty$. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.

arXiv:1506.00541•Jun 1, 2015•0 citations

Convergence of Boson Expansion Theory

Motoyuki Sato, Kosuke Takada

When the Holstein-Primakoff (Hermitian)-type boson expansion theory is applied for nuclear structure calculations, truncation at the fourth-order expansion of the boson Hamiltonian is often used. The Dyson-type boson mapping has no difficulty concerning boson expansion, since the Dyson boson Hamiltonian is written in up to sixth-order boson operators, and we do not have to make any truncation of the boson expansion. In the present work, we propose a new clear-cut method to obtain the normal-ordered Holstein-Primakoff boson expansion. Using this method, we test whether or not the truncation of the boson expansion is good approximation in various cases. We show that truncation at the fourth-order expansion of the normal-ordered Holstein-Primakoff boson Hamiltonian is a rather good approximation, but it is not sufficient for detailed analyses.

Progress of Theoretical Physics•Sep 1, 1998•1 citation

Projected Holstein-Primakoff boson representation of quantum spins for spin wave theory

Ke Liu, Fangyu Xiong, Fa Wang

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:2509.26607•Physical review. B./Physical review. B•Dec 22, 2025•0 citations

Generalized Holstein-Primakoff transformation with optimizable bosonic truncation

Yang Liu, Jia‐Wei Mei, Rong Yu +3 more

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.

arXiv:2607.28983•Jul 31, 2026•0 citations

Operator square roots are ubiquitous in theoretical physics.They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation.Often the use of a perturbative expansion is the only recourse when dealing with them.In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots.Specifically, for the number operator n = a † a we show that the square root √ n near n = 0 can be approximated by a polynomial in n.This result is unexpected because a Taylor expansion fails.A polynomial expression in n is possible because n is an operator, and its constituents a and a † have a non trivial commutator [a, a † ] = 1 and do not behave as scalars.We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators.We prove that these new expressions exactly reproduce spin operators.Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

The Holstein-Primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of Hermite polynomials

H_n(x)

as

n \rightarrow \infty

. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.

When the Holstein-Primakoff (Hermitian)-type boson expansion theory is applied for nuclear structure calculations, truncation at the fourth-order expansion of the boson Hamiltonian is often used. The Dyson-type boson mapping has no difficulty concerning boson expansion, since the Dyson boson Hamiltonian is written in up to sixth-order boson operators, and we do not have to make any truncation of the boson expansion. In the present work, we propose a new clear-cut method to obtain the normal-ordered Holstein-Primakoff boson expansion. Using this method, we test whether or not the truncation of the boson expansion is good approximation in various cases. We show that truncation at the fourth-order expansion of the normal-ordered Holstein-Primakoff boson Hamiltonian is a rather good approximation, but it is not sufficient for detailed analyses.

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.

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.