Lipkin Model Research Articles

A Possible Extension of a Trial State in the TDHF Theory with a Canonical Form to the Lipkin Model: -- Canonicity Conditions and Dynamical Treatment of Classical and Quantal Variables in an Extended State of the su(2)-Coherent State --

With the aim of the extension of the TDHF theory in the canonical form in the Lipkin model, the trial state for the variation is constructed, which is an extension of the Slater determinant. The canonicity condition is imposed to formulate the variational approach in the canonical form. A possible solution of the canonicity condition is given and the zero-point fluctuation induced by the uncertainty principle is investigated. As an application, the ground state energy is evaluated.

Non-perturbative flow equations from continuous unitary transformations

The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner's flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow equation to be rewritten as a nonlinear partial differential equation. The implementation is non-perturbative in that the derivation of the PDE is based on an expansion controlled by the size of the system rather than the coupling constant. We apply this method to the Lipkin model and obtain very accurate results for the spectrum, expectation values and eigenstates for all values of the coupling and in the thermodynamic limit. New aspects of the phase structure, made apparent by this non-perturbative treatment, are also investigated. The Dicke model is treated using a two-step diagonalization procedure which illustrates how an effective Hamiltonian may be constructed and subsequently solved within this framework.

Extended RPA with ground-state correlations in a solvable model

The ground states and excited states of the Lipkin model hamiltonian are calculated using a new theoretical approach which has been derived from an extended time-dependent Hartree-Fock theory known as the time-dependent density-matrix theory (TDDM). TDDM enables us to calculate correlated ground states, and its small amplitude limit (STDDM), which is a version of extended RPA theories based on a correlated ground state, can be used to calculate excited states. It is found that this TDDM plus STDDM approach gives much better results for both the ground states and the excited states than the Hartree-Fock ground state plus RPA approach.

Effective operator treatment of the Lipkin model

We analyze the Lipkin Model using effective operator techniques. We present both analytical and numerical results for effective Hamiltonians. The accuracy of the cluster approximation is investigated.

The two-level Lipkin model solution within the RPA approach for deformed fermions

We use a deformed algebra formalism, the so-called quon algebra, to investigate the role of deformation in an approximate treatment of a many-body system in a simple schematic model. A previously developed method is used in order to find the Hartree–Fock ground state energy and the RPA frequency as a function of the deformation parameter. We show that this dependence is very sensitive to small deviations of the algebra from the pure fermionic case and that the frequency is renormalized by those deviations. Also, we conclude that the well-known RPA collapse still occurs, although the critical value for the interaction constant now depends on the deformation parameter. Finally, we show that it is possible to make our results converge to the exact solution, allowing for a dependence of the deformation parameter on the two-body interaction strength of the model.

The two-level Lipkin model solution within the RPA approach for deformed fermions

The two-level Lipkin model solution within the RPA approach for deformed fermions

A FINITE-TEMPERATURE GENERALISATION OF THE COUPLED CLUSTER METHOD: A NON-PERTURBATIVE ACCESS TO GRAND PARTITION FUNCTIONS

We present here a succinct account of the finite-temperature generalisation of the coupled cluster (CC) method. It requires the concept of thermal normal ordered products whose Boltzmann trace is zero. The basic idea is to express the Boltzmann operator as a normal ordered exponential containing cluster operators and a number, and express the free energy as a logarithm of a suitable Boltzmann trace, where only the number part in the ordered exponential survives. The free energy is manifestly extensive. Free energies of Lipkin model and anharmonic oscillators obtained from the grand partition function are discussed as example applications.

The Lipkin model. Beyond mean field with generalized coherent states

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The ground-state energy is obtained within several variant types of coherent states. It has been found that two of the kinds considered describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone. The time evolution predicted by generalized coherent states has been investigated numerically by comparing its dynamics with the exact one.

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.

Nonperturbative Flow Equations from Running Expectation Values

We show that Wegner's flow equations, as recently discussed in the Lipkin model, can be solved self-consistently. This leads to a nonlinear differential equation which fully determines the order parameter as a function of the dimensionless coupling constant, even across the phase transition. Since we consider an expansion in the fluctuations, rather than the conventional expansion in the coupling constant, convergence to the exact results is found in both phases when taking the thermodynamic limit.

Extended random phase approximation in a solvable model

We propose an extended Lipkin model that can describe decay processes of particle-hole states and is still solvable. We examine several RPA-type theories using the model. We show explicitly the roles played by the self-energies of particle-hole propagators, and clarify how the extended RPA theory describes collective states better than other RPA-type theories.

Quasiparticle random phase approximation with a nonlinear phonon operator

We present model calculations of the quasiparticle random phase approximation (QRPA) with a new form of the phonon operator. This modification of the QRPA is applied to the proton-neutron Lipkin model and we shall review it briefly. The present calculations show that the inclusion of nonlinear terms in the phonon operator leads to a much better agreement with the exact results obtained by the diagonalization of the nuclear Hamiltonian. It is further found that if all relevant nonlinear terms of the phonon operator are taken into account, all odd excited states are also exactly reproduced.

Schematic model for QCD. II. Finite temperature regime

A schematic model for QCD, developed in a previous paper, is applied to calculate meson properties in the high temperature (up to 0.5 GeV) regime. It is a Lipkin model for quark-antiquark pairs coupled to gluon pairs of spin zero. The partition function is constructed with the obtained meson spectrum and several thermodynamical observables are calculated, such as the energy density, heat capacity, as well as relative production rates of mesons and absolute production rates for pions and kaons. The model predictions show a qualitative agreement with data. Based on these results, we advocate the use of the model as a toy model for QCD.

Boson-mapping-based extension of the random-phase approximation in a three-level Lipkin model

Boson-mapping-based extension of the random-phase approximation in a three-level Lipkin model

RPA-like calculations within limited particle–hole spaces

Working in a boson formalism, we define a new class of phonon operators for RPA-like calculations. These operators remind the standard RPA ones, namely they have “upward” and “backward” amplitudes and an overall one-particle one-hole nature. However, their structure is more complex. As a basic feature, the vacuum of these phonon operators is characterized by only a limited (and variable at will) set of particle–hole excitations. In addition, multiphonon states constructed by repeated actions of these phonon operators on the vacuum state have a particle–hole structure never exceeding in complexity that of the vacuum itself. We discuss advantages and disadvantages deriving from the use of this new class of phonon operators and show an application of the formalism within the Lipkin model. Diagonalizations in multiphonon spaces built in terms of new and standard RPA phonon operators exhibit a faster convergence toward the exact results in the former case.

The partition function of an interacting many body system

Based on the path integral representation of the partition function of a many body system with separable two body interaction we propose a systematic extension of the perturbed static path approximation (PSPA) to lower temperatures. Thereby, special attention must be paid to instabilities of the classical mean field solution in functional space that cause divergencies within the conventional PSPA. As a result we develop an approximation applicable from high to very low temperatures. These findings are tested against exact results for the archetypical cases of a particle moving in a one dimensional double well and the exactly solvable Lipkin model. In particular, we obtain a very good approximation to the level density of the Lipkin model even at low thermal excitations. Our results may have potential applications in low temperature nuclear physics and mesoscopic systems, e.g. for gap fluctuations in nanoscale superconducting devices previously studied within a PSPA type of approximation. PACS: 5.30.-d, 24.60.-k, 21.10.Ma, 74.25.Bt

Uniqueness of the solution of the contracted Schrödinger equation

In this paper two fundamental questions in the contracted Schr\"odinger equation (CSE) approach are considered by using Lipkin's quasispin model: 1-1 mapping between the second-order reduced density matrix (2-RDM) and the wave function of an excited state, and the uniqueness of the solution of CSE under incomplete N-representability conditions. We present some examples of the wave functions that give the same 2-RDM as the excited state. Thus 2-RDM of an excited state does not determine the wave function uniquely, and it alone cannot be used as basic variable for excited states of the density-matrix theory. Under the incomplete representability constraints the solution of the second-order CSE contains all the exact 4-RDMs together with the spurious ones. We examined the distribution of the solutions as a function of energy, and found that the solutions are well separated from each other under the P- and G-representability conditions of 4-RDM in the low-energy region, but with moderate interaction, or in the higher-energy region, there exist spurious solutions for almost all energies. Thus the G condition of 4-RDM is not sufficient to solve the excited states, although it gives accurate results for the ground state of Lipkin's model.

Testing a New Class of Phonon Operators for RPA-Like Calculations Within an Exactly Solvable Model

Working in a boson formalism, we introduce a phonon operator different from that commonly used in RPA calculations. As a major feature, this new phonon operator has a vacuum carrying only a limited number of particle-hole excitations. Therefore, the form of this vacuum is quite different from the exponential one of the RPA. In addition, multiphonon excitations have a particle-hole structure never exceeding in complexity that of this vacuum, no matter how large is the number of phonons involved. We present the results of some preliminary tests performed within the exactly solvable two-level Lipkin model.

Looking for an Optimal Ground State Within Quasiparticle Random Phase Approximation

A new ansatz for the correlated ground state of the many-nucleon system is proposed which results in obtaining a modified Quasiparticle Random Phase Approximation (QRPA). An additional degree of freedom is introduced which allows to determine variationally the ground state simultaneously with solving the QRPA equations. This new approach, QRPA with an optimal ground state, is studied within the proton-neutron Lipkin model. New solutions have been found, in the range of the interaction strength where the standard QRPA formalism does not work. A relation between one of them and the solution obtained within a semi-classical approach is established. A detailed study of the expectation value of the quasiparticle number operator in the ground state and the transition amplitude for the two-neutrino double beta Fermi decay, is also presented.

A schematic calculation of the ββ-decay matrix element within the Green function approach

The time integral representation of the many-body Green function describing the two-neutrino double-beta-decay (2νββ-decay) matrix element is used in schematic calculations within the proton-neutron Lipkin model. The two-body Hamiltonian considered includes a quadratic polynomial in bosons to describe the motion of the selected degrees of freedom. The beta-transition operators also include higher-order terms in the boson expansion. They have been shown to be of crucial importance in the determination of the 2νββ-decay matrix element M F. We have found that in the standard QRPA approach, which exploits the form of M F with denominator, there is a dominant unphysical contribution to M F arising from the non-orthogonality of the initial and final quasiparticle ground states. The physical part of M F is negligible for small values of the particle-particle interaction strength, i.e., it exhibits a different behavior as that known from the QRPA approaches.