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

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.

11 pages, 5 figures, match published version. Use of the discontinuity in the critical exponents of the fidelity susceptibility to obtain analytically the class of universality associated with the Lipkin model. The mention to spurious divergences has been replaced with "have no counterpart in the exact diagonalization"

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• Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model
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• Comparative quantum and semi-classical analysis of Atom-Field Systems II: Chaos and regularity
• Classical chaos in atom-field systems
• Delocalization and quantum chaos in atom-field systems
• Squeezing by Critical Speeding-up: Applications in Quantum Metrology
• Quantum geometric tensor and quantum phase transitions in the Lipkin-Meshkov-Glick model
• Adiabatic invariants for the regular region of the Dicke model
• Quantum many-body states and Green functions of nonequilibrium electron-magnon systems: Localized spin operators vs. their mapping to Holstein-Primakoff bosons
• Perfect Intrinsic Squeezing at the Superradiant Phase Transition Critical Point
• Tricritical Dicke model with and without dissipation
• Critical steady states of all-to-all squeezed and driven superradiance: An analytic approach
• Fidelity, susceptibility and critical exponents in the Dicke model
• Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models
• Goldilocks Probes for Noisy Interferometry via Quantum Annealing to Criticality
• Quantum theory of the Josephson junction between finite islands
• Investigating Lipkin-Meshkov-Glick Model and Criticality-Enhanced Metrology in a Coherent Ising Machine