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

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 n = â † 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 offer an advantage over other methods, such as the Dyson-Maleev representation. I. NOTICE OF COPYRIGHTThis manuscript has been authored in part by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http: //energy.gov/downloads/doe-public-access-plan).

Newton series expansion of bosonic operator functions

“…In contrast, the finite sum (15) leaves the physical and the unphysical parts of the Hilbert space unconnected. The observation that the truncation of the normal-ordered expansion of the Holstein-Primakoff representation after 2S +1 terms provides an exact spin representation was recently published in [10]. Instead of performing a Taylor expansion, the authors made the ansatz to write f HP (n) as a normal-ordered series in the form (11a).…”

“…What, to the best of our knowledge, has not been realized yet is that, in order to easily derive a compact formula for the coefficients F k , finite-difference calculus provides a natural and elegant tool that is well adapted to the discreteness of the domain of definition of f (n). It, furthermore, directly leads to closed and compact expressions instead of iterative equations [10] for the coefficients of the expansion.…”