Holstein-Primakoff Transformation Research Articles

Quantum phase transition for the Dicke model with the dipole–dipole interactions

In this paper, we reveal a second-order phase transition for the Dicke model with the dipole–dipole interaction between the atoms. By means of Holstein–Primakoff transformation, the energy spectra for both the normal and superradiant phases are explicitly obtained and therefore the scaling behaviour near the critical transition point can be given clearly. It is also shown that the dipole–dipole interactions between the atoms deeply affect the critical transition point, the ground-state and excitation energies and the scaled atomic inversion in the ground state.

The Lipkin Model in Many-Fermion System as an Example of the su(1,1) ⊗ su(1,1)-Algebraic Model

Following the idea, recently, proposed by the present authors for the two-level pairing model, the Lipkin model is reexamined. It is a natural generalization of the method already developed by the present authors with Kuriyama. This model is a schematic model for many-fermion system and obeys the su(2)-algebra. It is shown that the use of the Schwinger boson representation, the model is expressed in terms of the su(1,1) times su(1,1)-algebra and with the aid of the MYT mapping method, it is disguised from the orginal form in terms of the Holstein-Primakoff representation. Further, under various coherent states, the classical counterparts are derived. It is concluded that the Lipkin model can be treated in the common ring as that of the two-level pairing model.

Quantum phase transition in the generalized single-mode superradiant model

In this paper we reveal a zero-temperature quantum phase transition for the single-mode superradiant model with the form A2 from the normal to superradiant phase by mean of the Holstein-Primakoff transformation. In the thermodynamic limit, in which the numbers of atoms becomes infinite, the ground state energy and corresponding wavefunctions of both the normal and superradiant phases are obtained and therefore the scaling behavior near the critical transition point is derived.

Boson Realization of the su(3)-Algebra. IV: Holstein-Primakoff Representation for the Elliott Model

On the basis of the Schwinger boson representation for the Elliott model developed in (III), the Holstein-Primakoff representation for the su(3)-algebra is presented. The basic idea is the same as that adopted in (II), and as an example, the case of the lowest approximation is described.

Molecular theory of nematic liquid crystals viewed as effect of collective excitation in ferromagnetic systems

We develop a microscopic theory of the nematic phase with consideration of the effect of the collective excitation on properties of nematic liquid crystals. The model is based on the Heisenberg's exchange model of the ferromagnetic materials. Since the orientation of the molecular long axis and the angular momentum of the molecule rotating around its long axis have the same direction, operators can be introduced to research the nematic liquid crystals. Using the lattice model and the Holstein–Primakoff transformation, the Hamiltonian of the system can be obtained, which has the same form as that of the ferromagnetic substance. The relation between the order parameter and reduced temperature can be gotten. It is in good agreement with the experimental results in the low temperature region, the accordance is better than that of the molecular field theory and the computer simulation. In high temperature region close to the transition point, by considering the effect of the higher-order terms in the Hamiltonian, theoretical prediction is in better agreement with the experiment. That indicates the many-body effect is important to nematic liquid crystals.

Boson Realization of the su(3)-Algebra. II: -- Holstein-Primakoff Representation for the Lipkin Model --

On the basis of the Schwinger boson representation for the Lipkin model developed in (I), the Holstein-Primakoff representation for the su(3)-algebra is presented. Including the symmetric case, the representation obtained in this paper contains all the cases.

Periodic spin domains of spinor Bose–Einstein condensates in an optical lattice

The periodic spin domains of spinor Bose–Einstein condensates confined in a one-dimensional optical lattice are studied in terms of the equation of motion of the spinor which is reduced to the nonlinear Schrödinger equation with the help of Holstein–Primakoff transformation. It is shown that the spin domains obtained analytically can be easily controlled by adjusting the light-induced dipole–dipole interaction, which is realizable in optical lattice created by red-detuned laser beams with modulating intensity. The dynamical stability of the spin domains is also demonstrated.

Fermionic versus bosonic descriptions of one-dimensional spin-gapped antiferromagnets

In terms of spinless fermions and spin waves, we describe the magnetic properties of a spin-1/2 ferromagnetic-antiferromagnetic bond-alternating chain which behaves as a Haldane-gap antiferromagnet. On the one hand, we employ the Jordan–Wigner transformation and treat the fermionic Hamiltonian within the Hartree–Fock approximation. On the other hand, we employ the Holstein–Primakoff transformation and modify the conventional spin-wave theory so as to restore the sublattice symmetry. We calculate the excitation gap, the specific heat, the magnetic susceptibility, magnetization curves, and the nuclear spin-lattice relaxation rate with varying bond alternation. These schemes are further applied to a bond-alternating tetramerized chain which behaves as a ferrimagnet. The fermionic language is particularly stressed as a useful tool for investigating one-dimensional spin-gapped antiferromagnets, while the bosonic one works better for ferrimagnets.

Continuous unitary transformations and finite-size scaling exponents in the Lipkin-Meshkov-Glick model

We analyze the finite-size scaling exponents in the Lipkin-Meshkov-Glick model by means of the Holstein-Primakoff representation of the spin operators and the continuous unitary transformations method. This combination allows us to exactly compute the leading corrections to the ground-state energy, the gap, the magnetization, and the two-spin correlation functions. We also present numerical calculations for large system size which confirm the validity of this approach. Finally, we use these results to discuss the entanglement properties of the ground state focusing on the (rescaled) concurrence that we compute in the thermodynamical limit.

On modulational instability of nonlinear waves in 1D ferromagnetic spin chains

We report a theoretical study of modulational instability of extended nonlinearspin waves in a one-dimensional ferromagnetic chain. The investigation is madeboth analytically within the framework of the linear stability analysis and alsonumerically by means of molecular dynamics simulations. Using a Holstein–Primakofftransformation for the spin operators, the Hamiltonian, which is constituted by aHeisenberg exchange term, a biquadratic exchange energy, an anisotropic energy and aZeeman term, is bosonized. Then we derive a discrete nonlinear Schrödinger-likeequation for the spin-wave motion. Using a linear stability analysis, we establish thestability criteria of the spin waves in such a ferromagnetic chain. From our numericalsimulations of the discrete spin chain for the onset of instability, it emerges that theanalytical predictions are correctly verified. For a long timescale, depending on thestrength of the biquadratic exchange interaction relative to the exchange energy andthe anisotropy energy, on the one hand an intrinsic localized wave train can becreated displaying properties of the breather motion. On the other hand, due to theincreasing size of the instability domain, with increase of the biquadratic parameter,the instability can fully develop and the linear stability fails; consequently, thetime evolution of the modulated spin waves can show both regular and chaoticbehaviour.

Two-mode entanglement in two-component Bose-Einstein condensates

We study the generation of two-mode entanglement in a two-component Bose-Einstein condensate trapped in a double-well potential. By applying the Holstein-Primakoff transformation, we show that the problem is exactly solvable as long as the number of excitations due to atom-atom interactions remains low. In particular, the condensate constitutes a symmetric Gaussian system, thereby enabling its entanglement of formation to be measured directly by the fluctuations in the quadratures of the two constituent components [Giedke {\it et al.}, Phys. Rev. Lett. {\bf 91}, 107901 (2003)]. We discover that significant two-mode squeezing occurs in the condensate if the interspecies interaction is sufficiently strong, which leads to strong entanglement between the two components.

Spin wave contribution to entanglement in Heisenberg models

Employing the Holstein–Primakoff transformation, we obtain an approximate description of both the ground state entanglement and the entanglement dynamics in a spin system in terms of spin waves. As an example, we apply the method to a two-dimensional spin system.

Magnetic solitons of spinor Bose-Einstein condensates in an optical lattice

Magnetic solitons in spinor Bose-Einstein condensates confined in a one-dimensional optical lattice are studied by the Holstein-Primakoff transformation method. It is shown that due to the long-range light-induced and static magnetic dipole-dipole interactions, there exist different types of magnetic solitary excitations in different parameter regions. Compared to conventional solid-state materials, the parameters of this type of magnetic solitons in an optical lattice can be easily tuned by the above dipole-dipole interactions, which are highly controllable in experiments.

1/ N-Expansion for the Dicke model and the decoherence program

An analysis of the Dicke model, N two-level atoms interacting with a single radiation mode, is done using the Holstein–Primakoff transformation. The main aim of the paper is to show that, changing the quantization axis with respect to the common usage, it is possible to prove a general result either for N or the coupling constant going to infinity for the exact solution of the model. This completes the analysis, known in the current literature, with respect to the same model in the limit of N and volume going to infinity, keeping the density constant. For the latter the proper axis of quantization is given by the Hamiltonian of the two-level atoms and for the former the proper axis of quantization is defined by the interaction. The relevance of this result relies on the observation that a general measurement apparatus acts using electromagnetic interaction and so, one can state that the thermodynamic limit is enough to grant the appearance of classical effects. Indeed, recent experimental results give first evidence that superposition states disappear interacting with an electromagnetic field having a large number of photons.

On the extra phase correction to the semiclassical spin coherent-state propagator

The problem of an origin of the Solari–Kochetov extra-phase contribution to the naive semiclassical form of a generalized phase-space propagator is addressed with the special reference to the su(2) spin case which is the most important in applications. While the extra-phase correction to a flat phase-space propagator can straightforwardly be shown to appear as a difference between the principal and the Weyl symbols of a Hamiltonian in the next-to-leading order expansion in the semiclassical parameter, the same statement for the semiclassical spin coherent-state propagator holds provided the Holstein–Primakoff representation of the su(2) algebra generators is employed.

Dynamical decoherence in a cavity with a large number of two-level atoms

We consider a large number of two-level atoms interacting with the mode of a cavity in the rotating-wave approximation (Tavis–Cummings model). We apply the Holstein–Primakoff transformation to study the model in the limit of the number of two-level atoms, all in their ground state, becoming very large. The unitary evolution that we obtain in this approximation is applied to a macroscopic superposition state showing that, when the coherent states forming the superposition are distant enough, then the state collapses on a single coherent state describing a classical radiation mode. This appears as a true dynamical effect that could be observed in experiments with cavities.

Order from disorder in lattice QCD at high density

We investigate the properties of the ground state of strong coupling lattice QCD at finite density. Our starting point is the effective Hamiltonian for color singlet objects, which looks at lowest order as an antiferromagnet, and describes meson physics with a fixed baryon number background. We concentrate on uniform baryon number backgrounds (with the same baryon number on all sites), for which the ground state was extracted in an earlier work, and calculate the dispersion relations of the excitations. Two types of Goldstone boson emerge. The first, antiferromagnetic spin waves, obey a linear dispersion relation. The others, ferromagnetic magnons, have energies that are quadratic in their momentum. These energies emerge only when fluctuations around the large-N_c ground state are taken into account, along the lines of ``order from disorder'' in frustrated magnetic systems. Unlike other spectrum calculations in order from disorder, we employ the Euclidean path integral. For comparison, we also present a Hamiltonian calculation using a generalized Holstein-Primakoff transformation. The latter can only be constructed for a subset of the cases we consider.

Saturation of dephasing time in mesoscopic devices produced by a ferromagnetic state

We consider an exchange model of itinerant electrons in a Heisenberg ferromagnet and we assume that the ferromagnet is in a fully polarized state. Using the Holstein-Primakoff transformation we are able to obtain a boson-fermion Hamiltonian that is well-known in the interaction between light and matter. This model describes the spontaneous emission in two-level atoms that is the proper decoherence mechanism when the number of modes of the radiation field is taken increasingly large, the vacuum acting as a reservoir. In the same way one can see that the interaction between the bosonic modes of spin waves and an itinerant electron produces decoherence by spin flipping with a rate proportional to the size of the system. In this way we are able to show that the experiments on quantum dots, described in D. K. Ferry et al. [Phys. Rev. Lett. {\bf 82}, 4687 (1999)], and nanowires, described in D. Natelson et al. [Phys. Rev. Lett. {\bf 86}, 1821 (2001)], can be understood as the interaction of itinerant electrons and an electron gas in a fully polarized state.

Boson-fermion Holstein-Primakoff mapping at nonzero temperatures for the example of the Lipkin model

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.

Excited atomic coherent states

Abstract The excited atomic coherent (EAC) states are introduced and some of their statistical properties are discussed. When the Holstein-Primakoff transformations are used then these states become intermediate states interpolating between Fock and displaced Fock states. A generation scheme is discussed. The quasi-probability distribution functions (QDF) for these states are investigated. The Pegg-Barnett phase distribution of these states is studied.