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arXiv:1210.0028 · Physica Scripta

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

Jorge G Hirsch, Octavio Castaños, Ramón López-Peña, Eduardo Nahmad-Achar

Published Feb 11, 2013• 23 citations
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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.

DOI
10.1088/0031-8949/87/03/038106

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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.
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 → ∞. 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.
We discuss the behaviour of the entanglement entropy of the ground state in various collective systems. Results for general quadratic two-mode boson models are given, yielding the relation between quantum phase transitions of the system (signalled by a divergence of the entanglement entropy) and the excitation energies. Such systems naturally arise when expanding collective spin Hamiltonians at leading order via the Holstein–Primakoff mapping. In a second step, we analyse several such models (the Dicke model, the two-level Bardeen–Cooper–Schrieffer model, the Lieb–Mattis model and the Lipkin–Meshkov–Glick model) and investigate the properties of the entanglement entropy over the whole parameter range. We show that when the system contains gapless excitations the entanglement entropy of the ground state diverges with increasing system size. We derive and classify the scaling behaviours that can be met.
Spin-wave theory provides a quasiparticle description of excitation spectra in quantum spin systems. In this theory, the spin operators are usually bosonized by the Holstein-Primakoff transformation or the Dyson-Maleev transformation. In practical calculations, the bosonized Hamiltonian has to be truncated, and thus the resulting excitation spectra depend on the choice of bosonic representation. Here, we introduce a generalized Holstein-Primakoff transformation that continuously interpolates between the conventional Holstein-Primakoff and Dyson-Maleev transformations through a single parameter. The formulation naturally extends to the SU(N) algebra and provides a flexible framework for optimizing bosonic truncations beyond harmonic order. As an application, we investigate the spin-1 bilinear-biquadratic model on the square lattice. By combining the generalized Holstein-Primakoff transformation with continuous similarity transformations, we obtain excitation spectra in excellent agreement with tensor-network calculations. Our results demonstrate that optimizing the bosonic representation substantially reduces truncation errors and provides quantitatively reliable descriptions of both quasiparticle dispersions and multi-boson continua.
A unified approach to the analysis of quantum phase transitions in some different Curie-Weiss models is proposed such that they are treated and analyzed under the same general scheme. This approach takes three steps: balancing the quantum Hamiltonian by an appropriate factor, rewriting the Hamiltonian in terms of SU(2)SU(2)SU(2) operators only, and obtention of a classical Hamiltonian. SU(2)SU(2)SU(2) operators are obtained from creation and annihilation operators as linear combinations in the case of fermions and as an inverse Holstein-Primakoff transformation in the case of bosons. This scheme is successfully applied to Lipkin, pairing, Jaynes-Cummings, bilayer, and Heisenberg models.
Within the formalism of thermo field dynamics, the boson-fermion Holstein-Primakoff transformation is constructed for the case of nonzero temperatures. For the example of the Lipkin model, the transformation in question is used to construct a thermal Hamiltonian in the form of an expansion in the parameter $${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}$$ , where N is the number of particles in the system. The temperature dependence of quasiparticle (fermion) and collective-excitation (boson) energies is calculated to terms of order 1/N.
The Holstein-Primakoff boson representation of quantum spins and associated large-SSS expansion have been the standard framework for describing the spin wave excitations in magnetically order phases of quantum spin systems. However, we will show that the omission of projection operators and normal-ordering in this representation can produce incorrect magnon hamiltonians for finite SSS. We will present the exact normal-ordered forms of the finite-SSS projection operators and projected Holstein-Primakoff boson representations of spin and quadrupole operators, which can produce exact two-magnon interaction terms under ferromagnetic or fully polarized states. We will also discuss the difficulties of applying this projected representation to antiferromagnetic spin wave theory.