On a RPA method with angular momentum projection

In view of projecting the angular momentum eigenstates out of the intrinsic RPA states, we first study linear relations between deformed and spherical phonons. By use of these relations, the deformed RPA Hamiltonian can be expressed in terms of the spherical phonons in the form of a displaced harmonic oscillator. The angular momentum projection can be applied to an intrinsic RPA state determined by the reduced Hamiltonian, and the correspondence with the macroscopic deformed phonon model of Lipaset al. can be done at this stage. Two important parameters, the constant term in the linear relations and the strength of the spherical phonon Hamiltonian, are evaluated and compared with the best-fit values obtained by Lipaset al.     

Surface plasmon dispersion for very small metallic spheres: A quantum mechanical formulation

We have derived a pair of matrix equations describing the surface plasmon dispersion for a very small spherical metal particle, using the quantum mechanical random-phase approximation (RPA). The surface of the sphere is represented by an infinite spherical potential well for the conduction electrons and the positive ions are assumed to be uniformly distributed into a “jellium”. The dispersion relation is obtained as the condition that a solution of the RPA integral equation for charge fluctuation would exist for the particle.     

Random phase approximation and extensions applied to a bosonic field theory

An application of a self-consistent version of RPA to quantum field theory with broken symmetry is presented. Although our approach can be applied to any bosonic field theory, we specifically study the ϕ⁴ theory in 1 + 1 dimensions. We show that the standard RPA approach leads to an instability which can be removed when going to a superior version, i.e. the renormalized RPA. We present a method based on the so-called charging formula of the many-electron problem to calculate the correlation energy and the RPA effective potential. Received: 18 February 2002 / Accepted: 8 May 2002     

Second-order Green's function theory of a Heisenberg paramagnet

The Heisenberg paramagnet in one, two, and three dimensions is analyzed by a second-order Green's function theory similar to that used by Knapp and ter Haar. This theory, which incorporates the exact values for the zero, first, and second moments of the relaxation function as boundary conditions, yields results satisfying the rotational symmetry of the paramagnetic region as well as the principle of detailed balance. We find that our predictions for equal time properties in the classical limit are identical with the RPA Green's function theory of Liu as well as the spherical model results of Lax. The quantum limit is analyzed, and our predictions for the 1/T series coefficients for both internal energy and susceptibility are compared with exact results.Supported by National Science Foundation.     

From sum rules to RPA: 1. Nuclei

We take up the flexible formulation of RPA in subspaces of coordinate-like and momentum-like 1 ph operators which was developed in the preceeding paper. We draw the connection to many of the known collective approximations to RPA by proper choice of the subspace of operators. We finally apply the newly developed scheme to comparative studies of a hierarchy of approaches for the RPA spectra of doubly magic nuclei described with Skyrme forces.     

Is there a ground state deformation in nuclei near closed shells?

It is shown that the stability of spherical HFB states against deformation may be completely altered if angular momentum conservation is taken into account exactly, leading to a smooth transition from spherical to deformed nuclei. Furthermore the close connection between the stability matrix in HFB theory and the RPA matrix is destroyed by angular momentum projection.     

Entropy of a two-component fluid in a square-well model

An analytical expression valid for two approximations, notably, the random phase approximation (RPA) and the mean spherical approximation (MSA), is derived for entropy of a two-component system with the pair potential of a square well (SW). This expression is applied to calculate the excess entropy of mixing of the Na-K and Na-Cs equiatomic compositions. It is shown that the use of the SW model leads to better results than the application of the hard-sphere model. The MSA gives better convergence with the experiment compared with the RPA.     

On some properties of the Surface Delta Interaction

A method is discussed for solving the RPA equations without discarding any term in the case of separable interactions. It is specialized to the SDI for both spherical and deformed nuclei and applied in two simple examples.     

Random-phase approximation for the grand-canonical potential of composite fermions in the half-filled lowest Landau level

We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current- and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the k^-2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-state energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations of interacting electrons in the lowest Landau level. Received: 19 February 1998 / Revised: 25 March 1998 / Accepted: 17 April 1998     

Shape coexistence in theN=59 isotone97Sr

Experimental evidence for shape coexistence in the N=59 isotone⁹⁷Sr is presented. The ground state and the lowest excited levels are confirmed to be spherical. At 585.1 keV, a K=3/2 rotational band built on the v[422 3/2] Nilsson configuration has been identified. Nilsson orbital assignments for three further levels of deformed origin are proposed. The results are compared to RPA shell-model predictions.     

Effects of the ground state correlations on the structure of vibrational states

A method to treat the ground state correlations beyond the RPA is presented. A set of nonlinear equations taking into account effects of the ground state correlations on the pairing and phonon-phonon coupling is derived. The influence of such correlations on the properties of the vibrational states in spherical nuclei is studied.     

Auto-tuning of a two-degree-of-freedom rotational pendulum absorber

A dynamic vibration absorber is effective in suppressing harmonic excitation by tuning its natural frequency to match the excitation frequency. The rotational pendulum absorber (RPA) has a wide-range of natural frequencies that are continuously tunable by setting a suitable rotational speed. In this paper it is shown how to automatically tune the rotational speed of a two-degree-of-freedom RPA by detecting the phase between the vibration of the primary structure and that of the RPA. For this purpose the speed response of the RPA is introduced in addition to the frequency response. It is seen that if the excitation frequency is above a critical value dependent on the parameters of the RPA, the second vibration mode of the RPA is effective, allowing a relatively low rotational speed for the pendulums. The speed tuning algorithm is tested on a flexible plate that is subject to excitations of around 80 Hz, which do not generate visible oscillations but emit audible noise instead. Experimental results confirm the noise-level reduction effect of the RPA.     

Random-phase approximation and its extension for the O(2) anharmonic oscillator

We apply the random-phase approximation (RPA) and its extension called renormalized RPA to the quantum anharmonic oscillator with an O(2) symmetry. We first obtain the equation for the RPA frequencies in the standard and in the renormalized RPAs using the equation-of-motion method. In the case where the ground state has a broken symmetry, we check the existence of a zero frequency in the standard and in the renormalized RPAs. Then we use a time-dependent approach where the standard-RPA frequencies are obtained as small oscillations arround the static solution in the time-dependent Hartree-Bogoliubov equation. We draw the parallel between the two approaches.Received: 8 July 2003, Published online: 5 January 2004PACS: 21.60.Jz Hartree-Fock and random-phase approximations - 24.10.Cn Many-body theory     

A dynamics-controlled truncation scheme for the hierarchy of density matrices in semiconductor optics

We discuss the interaction of coherent electromagnetic fields with the semiconductor band edge in a dynamic density matrix model. Due to the influence of the Coulomb-interaction then-point density matrices are coupled in an infinite hierarchy of equations of motion. We show how this hierarchy is related to an expansion of the density matrices in terms of powers of the exciting field. We make use of the above results to set up a closed set of equations of motion involving two-, four-and six-point correlation functions, from which all third order contributions to the polarization can be calculated exactly. Comparison of our treatment of the hierarchy with the widely used RPA decoupling on the two-point level, gives interesting insight into the validity of the RPA. In particular we find, that a RPA-like factorization for two of the relevant density-matrices yields a solution of their respective equations of motion to lowest order in the electric field.     

The smooth structural change in mesoscopic Peierls chains

We investigate the Peierls transition in finite chains by exact (Lanczos) diagonalization and within a seminumerical method based on the factorization of the electron-phonon wave function (Adiabatic Ansatz, AA). AA can be applied for mesoscopic chains up to micrometer sizes and its reliability can be checked self-consistently. Our study demonstrates the important role played for finite systems by the tunneling in the double well potential. The chains are dimerized only if their size N exceeds a critical value N_c which increases with increasing phonon frequency. Quantum phonon fluctuations yield a broad transition region. This smooth Peierls transition contrasts not only to the sharp mean field transition, but also with the sharp RPA soft mode instability, although RPA partially accounts for quantum phonon fluctuations. For weak coupling the dimerization disappears below micrometer sizes; therefore, this effect could be detected experimentally in mesoscopic systems. Received: 3 January 1998 / Revised: 13 March 1998 / Accepted: 3 April 1998     

The compression modes in atomic nuclei and their relevance for the nuclear equation of state

Accurate assessment of the value of the incompressibility coefficient, K _∞, of symmetric nuclear matter, which is directly related to the curvature of the equation of state (EOS), is needed to extend our knowledge of the EOS in the vicinity of the saturation point. We review the current status of K _∞ as determined from experimental data on isoscalar giant monopole and dipole resonances (compression modes) in nuclei by employing the microscopic theory based on the Random Phase Approximation (RPA). The importance of full self-consistent calculations is emphasized. In recent years, a comparision between RPA calculations based on either non-relativistic effective interactions or relativistic Lagrangians has been pursued in great detail. It has been pointed out that these two types of models embed different ansatz for the density dependence of the symmetry energy. This fact has consequences on the extraction of the nuclear incompressibility, as it is discussed. The comparison with other ways of extracting K _∞ from experimental data is highlighted. The text was submitted by the author in English.     

Damping of zero sound in liquid3He

The dynamic structure factor in liquid³He is calculated within the framework of Dissipative Linear Response (DLR) equations. The effective interaction is represented by the Landau parameters up tol=2 and the Random Phase Approximation (RPA)-scattering amplitudes are evaluated in thes–p–d approximation. The dissipative linear response operator is calculated on the basis of RPA solutions of undamped zero sound using integral equation techniques.The dynamical structure factor has a strong resonance structure whose width agrees with measurements of the attenuation coefficient for small momentum transfers. At larger values of momentum transfer, where the dynamical structure factor has been directly measured, the Landau parameters do not give more than a qualitative agreement with the data.     

Random-matrix approach to RPA equations. I

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitary or an orthogonally invariant random-matrix model that does not appear in the Altland–Zirnbauer classification. The average spectrum of the model is studied with the help of a generalized Pastur equation. Two independent parameters govern the behaviour of the system: the strength α² of the coupling between positive- and negative-energy states and the distance between the origin and the centers of the two semicircles that describe the average spectrum for α² = 0, the latter measured in units of the equal radii of the two semicircles. With increasing α², positive- and negative-energy states become mixed and ever more of the spectral strength of the positive-energy states is transferred to those at negative energy, and vice versa. The two semicircles are deformed and pulled toward each other. As they begin to overlap, the RPA equations yield non-real eigenvalues: The system becomes unstable. We determine analytically the critical value of the strength for the instability to occur. Several features of the model are illustrated numerically.