Lipkin Model Research Articles

Note on the Deformed Boson Scheme in Time-Dependent Variational Method

The Holstein-Primakoff representation for the su(2)-algebra is derived in the deformed boson scheme. The following two points are discussed: (i) the connection between a simple Hamiltonian and the Hamiltonian obeying the su(2)-algebra, such as the Hamiltonian of the Lipkin model and (ii) derivation of the Hamiltonian for describing the damped and amplified motion for the su(2)-boson model.

Importance of single-boson and single-fermion mappings in the thermal boson expansion

Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna RU-141980 Russia~Received 18 January 2001; published 4 June 2001!In the context of the boson expansion theory, it is usually the case that the bosonization of single-boson~fermion! states is ignored. Although this is tolerable to some extent in cold systems, it causes serious diffi-culties in the treatment of thermal ensembles where the single-boson ~fermion! density of states plays animportant role. In the framework of the thermo-field dynamics it is shown how extended forms of the Holstein-Primakoff mapping for both bosons and fermions can lead to consistent thermal-boson expansions. Applica-tions to the O(N) anharmonic oscillator and the Lipkin model are presented.DOI: 10.1103/PhysRevC.64.015201 PACS number~s!: 11.15.Pg, 12.39.Fe, 13.75.Lb, 21.60.2nI. INTRODUCTION

TFD extension of a self-consistent RPA to finite temperatures

The self-consistent RPA (SCRPA) developed by Schuck and coauthors is extended to finite temperatures. The corresponding equations are derived by using the formalism of thermofield dynamics. The intrinsic energy of a system is calculated as the expectation value of the Hamiltonian with respect to a T-dependent thermal vacuum state for a thermal-phonon operator. A nonvanishing number of thermal quasiparticles in the vacuum state are assumed. By virtue of the assumption, the thermal Hartree-Fock (HF) equations appear to be coupled to the equations of motion for phonon variables. The thermal occupation numbers are also calculated in a consistent way with the energies of the HF quasiparticles. The approximation is applied to the two-level Lipkin model. Advantages of the thermal SCRPA (TSCRPA) are most obvious at temperatures near the phase-transition point. In the TSCRPA, the phase transition occurs at lower T than in other approximations. Moreover, within the TSCRPA, a statistical behavior of the Lipkin model is described with an appropriate accuracy at any T even if the HF transformation parameter is kept fixed at a value corresponding to the “spherical” phase of the HF field.

A Possible Boson Realization of Generalized Lipkin Model for Many-Fermion System: The su(M + 1)-Algebraic Model in Non-Symmetric Boson Representation

On the basis of the formalism proposed by three of the present authors (A. K., J. P. & M. Y.), the generalized Lipkin model consisting of (M +1) single-particle levels is investigated. This model is essentially a kind of the su(M + 1)-algebraic model and, in contrast to the conventional treatment, the case in which fermions are partially occupied in each level is discussed. The scheme for obtaining the orthogonal set for the irreducible representation is presented.

Extension of random-phase approximation preserving energy weighted sum rules: An application to a 3-level Lipkin model

A limitation common to all extensions of random-phase approximation including only particle-hole configurations is that they violate to some extent the energy weighted sum rules. Considering one such extension, the improved RPA (IRPA), already used to study the electronic properties of metallic clusters, we show how it can be generalized in order to eliminate this drawback. This is achieved by enlarging the configuration space, including also elementary excitations corresponding to the annihilation of a particle (hole) and the creation of another particle (hole) on the correlated ground state. The approach is tested within a solvable 3-level model.

Spatio-Temporal Evolution of Initial Coherent State Under the Action of Quantum Integrable and Nonintegrable Systems**The project supported by National Natural Science Foundation of China and the National Basic Research Project “Nonlinear Science” of China

According to the exactly formulated concepts of quantum integrability and nonintegrability,an initial coherent state evolving under the action of an integrable Hamiltonian having the same dynamical symmetry is expressed as an analytical function of and . The expectation values of various dynamical variables should be expressed as correlated analytical functions of and . Thus, the general characters of initial coherent state would be maintained during the spatio-temporal evolution under the action of such a system even if the wavepacket is slightly distorted. In contrast, when the integrability condition is destroyed, these features will be lost during the evolution. This theoretical prediction is illustrated with numerical results computed for the two-level Lipkin model with a dou ble-well potential and dynamical explanations are given. And the Eherenfest time is verified with this model.

A Note on a Boson Realization in Many-Boson System

As preparation for extending a new boson realization in the Lipkin model, recently proposed by the present authors, a possible boson realization for a many-boson system is developed. The basic idea comes from the MYT boson mapping method. In order to obtain the classical counterpart to the quantal system in this new boson realization, a certain wave packet is used.

On the Coupling of Two su(1,1) Spins in the Holstein-Primakoff Type Boson Representation

The su(1, 1) algebra is quite interesting, because with the help of this algebra, schematic aspects of thermal effects in many-boson and oscillating systems can be described in a conservative form. 1), 2) In addition to the above, we have another example in many-body systems. The classical and semi-classical time evolution of many-boson systems are treated in terms of squeezed boson coherent states. Such systems have been investigated by the present authors, together with Tsue and Fujiwara. 3), 4) In this case, the motion under investigation can be classified into two groups. One is classical motion and the other quantal fluctuations around the classical motion. Both can be treated formally in terms of two su(1, 1) spins. This will be demonstrated in a subsequent paper. 5) Further, in a new boson realization of the Lipkin model, 6) we can find two su(1, 1) spins. As mentioned above, we know that there exist a few systems composed of two su(1, 1) spins. Therefore, it may be interesting to investigate the coupling rule of two su(1,1) spins, because the coupling rule of su(2) spins is quite well known, but the case of su(1, 1) spins is not so familiar. The main aim of this paper is to give a general scheme for two coupled su(1, 1) spins and, further, to investigate the classical aspects. A certain part of the rule investigated in this paper will be used in a subsequent paper, 5) in which a possible interpretation is given for the decomposition of the classical and quantal fluctuation parts in the system described in terms of the squeezed state. For this purpose, in this section we breifly describe the su(1, 1) algebra as preparation for later sections. The su(1, 1) algebra consists of three operators, which we denote T±,0. These operators obey the relations

The Lipkin Model in a New Boson Realization: Basic Idea

With the use of the basic idea of the boson mapping method, a new boson realization, which is suitable for the Lipkin model, is proposed. This realization is different from the forms which are obtained with the Holstein-Primakoff and the Schwinger boson representations for the su(2) spin system. Further, with the aid of the basic idea of the TDHF method in canonical form, a method analysing the Lipkin model in the framework of classical mechanics is developed. The condition for the solution of the Hamilton equation of motion is discussed.

Time evolution of the Wigner function in discrete quantum phase space for a soluble quasi-spin model

The discrete phase space approach to quantum mechanics of degrees of freedom without classical counterparts is applied to the many-fermions/quasi-spin Lipkin model. The Wigner function is written for some chosen states associated to discrete angle and angular momentum variables, and the time evolution is numerically calculated using the discrete von Neumann-Liouville equation. Direct evidences in the time evolution of the Wigner function are extracted that identify a tunnelling effect. A connection with a SU (2)-based semiclassical continuous approach to the Lipkin model is also presented.

Unitary flow of the bosonized large-N Lipkin model

The flow equations describing the continuous unitary transformation which brings the Hamiltonian closer to diagonality are derived and solved exactly for the Lipkin model in the Holstein-Primakoff boson representation and for a large particle number N. The transformed Hamiltonian is diagonal in order 1/N3, extending known linear transformations to next-higher orders in the inverse particle number. This approach quite naturally allows one to preserve the tridiagonal structure of the original Lipkin Hamiltonian in the course of the transformation. Exact analytical results for the coupling functions and explicit expressions for the ground state energy and for the energy gap to the first excited state in order 1/N2 are presented and are compared with the accurate numerical values.

Random-phase approximation approach to rotational symmetry restoration in a three-level Lipkin model

We study an extended Lipkin-Meshkov-Glick model that permits a transition to a deformed phase with a broken continuous symmetry. Unlike simpler models, one sees a persistent zero-frequency Goldstone mode past the transition point into the deformed phase. We found that the RPA formula for the correlation energy provides a useful correction to the Hartree-Fock energy when the number of particle N satisfies N > 3, and becomes accurate for large N. We conclude that the RPA correlation energy formula offers a promising way to improve the Hartree-Fock energy in a systematic theory of nuclear binding energies.

Quasiparticle random phase approximation with the Pauli exclusion principle fully fulfilled

The violation of the Pauli Exclusion Principle (PEP) in the framework of the Quasi-particle Random Phase Approximation is examined within the proton-neutron monopole Lipkin model. For the first time, the solution of the QRPA equation including an exact description of the PEP is presented. Restoring the PEP, the QRPA solution is considerably stabilized and a better agreement with the exact solution is achieved for the energy of the first excited state, the ground state expectation value of the quasiparticle number operator and beta transition amplitudes.

Shape-phase transitions in mixed parity systems and the onset of octupole deformation

Shape-phase transitions in mixed parity boson systems are studied using mean field theory. It is shown that without parity projection, shape transitions in mixed parity systems are very similar to those in positive parity systems, both requiring a finite interaction strength for the onset of deformation. The shape-phase diagram in mixed parity systems, however, changes significantly after parity projection, with the critical point for the onset of octupole deformation moving to zero strength. Shape transitions in the Lipkin model are shown to exhibit the same features, indicating that this is a direct consequence of parity projection in finite, mixed parity systems independent of the nature of interacting particles.

PARITY-PROJECTED RESONATING HARTREE-FOCK APPROXIMATION TO THE LIPKIN MODEL

An exact eigenstate of the Lipkin-model Hamiltonian has a parity-symmetry. The resonating Hartree-Fock (Res-HF) ground-state wave function however breaks the parity-symmetry. To restore the parity-symmetry, we develop a parity-projected (P-P) Res-HF approximation to the Lipkin model. The P-P Res-HF wave function is superposed by two P-P Slater determinants. Self-consistent calculation is carried out so as to optimize the P-P Res-HF energy functional. A Res-HF ground state with definite parity brings us the ground-state energy much nearer to the exact one than that the Res-HF and explains most of the ground-state correlation energy. We also show behaviour of P-P Res-HF orbital energies and mixing coefficients calculated by the P-P Res-HF approximation and compare them with those calculated by the parity-unprojected Res-HF approximation. From this comparison, we find very interesting relations between them which have never been seen in the parity-unprojected Res-HF calculation.

An extension of a two-shell solvable model to include two collective excitations

A simple extension of the two-shell Lipkin model includes two types of collective oscillations, isoscalar and isovector, described by the same Hamiltonian. This may offer additional opportunities to compare various approaches to the many-body problem with the exact solution of the model.

Anharmonic collective excitation in a solvable model

We apply the time-dependent variational principle, the nuclear field theory, and the boson expansion method to the Lipkin model to discuss anharmonicities of collective vibrational excitations. It is shown that all of these approaches lead to the same anharmonicity to leading order in the number of particles. Comparison with the exact solution of the Lipkin model shows that these theories reproduce quite well.

Model of a self-consistent theory of large amplitude collective motion at finite excitation energy

We illustrate some of the concepts introduced in a recent paper devoted to the formulation of a self-consistent theory of large amplitude collective motion at finite excitation energy. We apply these concepts to a Hamiltonian with the symmetry of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ (a doubled Lipkin model), in which the latter, exemplifying the slow, collective degrees of freedom, interacts with a set of harmonic oscillators representing the fast, noncollective variables. We pass to a mean-field approximation, which implies a collisionless regime, but show as well how the formalism may be extended to include the limit of collisions that establish instantaneous local equilibrium. In the representation of natural orbitals, the variables split into classical canonical sets and occupation numbers of the orbitals. Careful attention is paid to the problem of eliminating the intrinsic canonical coordinates from the equations of motion in such a way (white noise approximation) as to yield dissipative equations of motion for the collective variables. For the case of local equilibrium, the relaxation problem requires in addition an equation relating the slow change in temperature to the slow change of the collective variables. The two limiting relaxation scenarios are studied numerically.

Effect of thermal ground state correlations on the statistical properties of the Lipkin model

The renormalized random phase approximation for hot finite Fermi systems is evaluated with the use of the thermo field dynamics formalism. This approximation treats vibrations of a hot finite Fermi system as harmonic ones but takes into account the Pauli principle in a more proper way than the usual thermal RPA, thus incorporating a new type of correlations in a thermal ground state. To demonstrate advantages of the approximation and to analyze a range of its validity, it is applied to the exactly solvable Lipkin model. A comparison is made with the exact grand canonical ensemble calculations, results of the thermal Hartree – Fock approximation and the thermal random phase approximation. The intrinsic energy of the system, the heat capacity, the average value of the quasispin operator z-projection and the particle number variance are calculated as functions of temperature. On the whole, the thermal renormalized RPA appears to be a better approximation than the other two. Its advantage is especially evident in the vicinity of the phase transition point. It is found that within TRRPA the phase transition occurs at lower temperature than in THFA and TRPA.

Instability of Thermal Equilibrium State of the Lipkin Model

The Lipkin model is the simplest algebraic model in which we can investigate a ground state transition in a fermion system. This model is also applied to the description of the thermal equilibrium state. In this paper, we investigate its stability with respect to changes in the particle number and the volume. In order to enable us to take the thermal limit, an appropriate number dependence is introduced into the force strength. We show with the use of a mean field approximation that the thermal equilibrium state in the condensed phase of particle-hole pairs becomes unstable.