An Introduction to the Holstein-Primakoff Transformation, with Applications in Magnetic Resonance

Abstract

We have witnessed an impressive advancement in computer performance in the last couple of decades. One would therefore expect a trickling down of the benefits of this technological advancement to the borough of computational simulation of multispin magnetic resonance spectra, but that has not been quite the case. Though some significant progress has been made, chiefly by Kuprov and collaborators, one cannot help but observe that there is still much to be done. In our view, the difficulties are not to be entirely ascribed to technology, but, rather, may mostly stem from the inadequacy of the conventional theoretical tools commonly used. We introduce in this paper a set of theoretical tools which can be employed in the description and efficient simulation of multispin magnetic resonance spectra. The so-called Holstein-Primakoff transformation lies at the heart of these, and provides a very close connection to discrete mathematics (from graph theory to number theory). The aim of this paper is to provide a reasonably comprehensive and easy-to-understand introduction to the Holstein-Primakoff (HP) transformation (and related bosons) to researchers and students working in the field of magnetic resonance. We also focus on how through the use of the HP transformation, we can reformulate many challenging computing problems encountered in multispin systems as enumerative combinatoric problems. This, one could say, is the HP transformation's primary forte. As a matter of illustration, our main concern here will be on the use of the HP bosons to characterize and eigendecompose a class of multispin Hamiltonians often employed in high-resolution magnetic resonance.