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arXiv:1206.1847 · Journal of Physics A Mathematical and Theoretical

A nontrivial bosonic representation of large spin systems at high temperatures

Yamen Hamdouni

Published Oct 8, 2012• 4 citations
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Abstract

We report on a nontrivial bosonization scheme for spin operators. It is shown that in the large N limit, at an infinite temperature, the operators behave like the creation and annihilation operators, i.e. a † and a , corresponding to a harmonic oscillator in thermal equilibrium, whose temperature and frequency are related by ℏω/( k B T ) = ln 3. The z -component is found to be equivalent to the position variable of another harmonic oscillator occupying its ground Gaussian state at zero temperature. The obtained results are applied to the Heisenberg XY Hamiltonian at a finite temperature.

DOI
10.1088/1751-8113/45/42/425301

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We report on a nontrivial bosonization scheme for spin operators. It is shown that in the large N limit, at an infinite temperature, the operators behave like the creation and annihilation operators, i.e. a † and a , corresponding to a harmonic oscillator in thermal equilibrium, whose temperature and frequency are related by ℏω/( k B T ) = ln 3. The z -component is found to be equivalent to the position variable of another harmonic oscillator occupying its ground Gaussian state at zero temperature. The obtained results are applied to the Heisenberg XY Hamiltonian at a finite temperature.
Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U-type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.
In this paper, infinite translation-invariant and periodic systems of classical particles are treated directly—not as limits of finite systems. A formalism of classical creation and annihilation operators that create and destroy classical identical particles at points of phase space is employed. By use of this formalism, classical dynamical variables are expressed without reference to the canonical coordinates and momenta of individual particles, or to the number of particles. Different physical situations are described by different representations of the algebra of creation and annihilation operators. The concept of thermal equilibrium is generalized so as to be meaningful in infinite systems. Stationary states and thermal-equilibrium states for an infinite system of noninteracting particles (possibly in a periodic external potential) are exhibited explicitly.
A harmonic oscillator is an indefinite-frequency one if the parameter ωωω is replaced by an operator. An ensemble of NNN such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and NNN can be finite. Due to the operator character of ωωω the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large NNN one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail.
We outline a method for rigorously calculating the time-dependent as well as the static thermal properties of localized spin systems from those of an ordinary many-boson system. Our method retains the advantages of both the Holstein-Primakoff and the Dyson-Maleev transformations without having their main disadvantages. The spin operators and the boson Hamiltonian are all finite series in the boson creation and annihilation operators. Our boson Hamiltonian is Hermitian. This method establishes a rigorous correspondence between the thermal properties of the spin-\textonehalf{} isotropic Heisenberg model and those of a hardcore boson system with only two-body interactions.
We show that spin systems with infinite-range interactions can violate at thermal equilibrium a multipartite Bell inequality, up to a finite critical temperature Tc . Our framework can be applied to a wide class of spin systems and Bell inequalities, to study whether nonlocality occurs naturally in quantum many-body systems close to the ground state. Moreover, we also show that the low-energy spectrum of the Bell operator associated to such systems can be well approximated by the one of a quantum harmonic oscillator, and that spin-squeezed states are optimal in displaying Bell correlations for such Bell inequalities.
In this paper, we treat a quantum harmonic oscillator in thermal equilibrium with any systems in certain classes of bosons with infinitely many degrees of freedom. We describe the following results: (i) when a canonical correlation function is given, we so reconstmct a Hamiltonian by the rotating wave approximation from it that the Hamiltonian restores it. Namely, we solve an inverse problem in the quantum field theory at finite temperature in a finite volume. (ii) Taking an infinite volume limit for the result in (i), we consider long-time behavior of the canonical correlation function in the finite volume limit.