Virtues and limitations of the truncated Holstein-Primakoff description of quantum rotors

TL;DR

This paper analyzes the truncated Holstein-Primakoff approximation for quantum rotors, especially near phase transitions, highlighting its virtues, limitations, and how renormalization can recover exact behaviors in the Lipkin model.

Contribution

It provides a detailed analysis of the truncated Holstein-Primakoff method's virtues and limitations, and introduces renormalization techniques to correct finite-size divergences in the Lipkin model.

Findings

Truncated solutions predict singularities near phase transitions.

Renormalization recovers exact observable behavior.

Analysis focused on the Lipkin model.

Abstract

A Hamiltonian describing the collective behaviour of N interacting spins can be mapped to a bosonic one employing the Holstein-Primakoff realisation, at the expense of having an infinite series in powers of the boson creation and annihilation operators. Truncating this series up to quadratic terms allows for the obtention of analytic solutions through a Bogoliubov transformation, which becomes exact in the limit N -> infinit. The Hamiltonian exhibits a phase transition from single spin excitations to a collective mode. In a vicinity of this phase transition the truncated solutions predict the existence of singularities for finite number of spins, which have no counterpart in the exact diagonalization. Renormalisation allows to extract from these divergences the exact behaviour of relevant observables with the number of spins around the phase transition, and relate it with the class of universality to which the model belongs. In the present work a detailed analysis of these aspects is presented for the Lipkin model.