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

TL;DR

This paper introduces a differential equation method to derive exact polynomial approximations of operator square-roots, notably improving the Holstein-Primakoff representation of spin operators with Hermitian expressions.

Contribution

The authors develop a novel differential equation approach to obtain exact polynomial expressions for operator square-roots, surpassing traditional perturbative methods.

Findings

Derived differential equations for operator square-roots.

Obtained polynomial approximations near zero for the number operator.

Produced Hermitian polynomial expressions for spin operators.

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 work 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 $\hat n=\hat a^\dag a$ we show that the square-root $\sqrt{\hat n}$ near $\hat n=0$ can be approximated by a polynomial in $\hat n$. This result is unexpected because a Taylor expansion fails. A polynomial expression in $\hat n$ is possible because $\hat n$ is an operator, and its constituents $a$ and $a^\dag$ have a non-trivial commutator $[a,a^\dag]=1$ 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 offer an advantage over other methods, such as the Dyson-Maleev representation.