Second Random-phase Approximation Research Articles

Simulating excited states of the Lipkin model on a quantum computer

We simulate the excited states of the Lipkin model using the recently proposed Quantum Equation of Motion (qEOM) method. The qEOM generalizes the EOM on classical computers and gives access to collective excitations based on quasi-boson operators $\hat{O}^\dagger_n(\alpha)$ of increasing configuration complexity $\alpha$. We show, in particular, that the accuracy strongly depends on the fermion to qubit encoding. Standard encoding leads to large errors, but the use of symmetries and the Gray code reduces the quantum resources and improves significantly the results on current noisy quantum devices. With this encoding scheme, we use IBM quantum machines to compute the energy spectrum for a system of $N=2, 3$ and $4$ particles and compare the accuracy against the exact solution. We found that the results of the approach with $\alpha = 2$, an analog of the second random phase approximation (SRPA), are, in principle, more accurate than with $\alpha = 1$, which corresponds to the random phase approximation (RPA), but the SRPA is more amenable to noise for large coupling strengths. Thus, the proposed scheme shows potential for achieving higher spectroscopic accuracy by implementations with higher configuration complexity, if a proper error mitigation method is applied.

Recent applications of the subtracted SRPA approximation

The Second Random Phase Approximation (SRPA) is a natural extension of the Random Phase Approximation obtained by introducing more general excitation operators where two particle-two hole configurations, in addition to the one particle-one hole ones, are considered. Only in the last years, large-scale SRPA calculations, without usually employed approximations have been performed. The SRPA model corrected by a subtraction procedure designed to cure double counting issues and the related instabilities has been recently implemented and applied in the study of different physical cases. We report here on some of the most recent results obtained by using this model. In particular, results on the dipole strength 48Ca and on a systematic study of the isoscalar giant quadrupole resonance in spherical nuclei will be shown and discussed.

Subtraction method in the Second Random Phase Approximation

We discuss the subtraction method applied to the Second Random Phase Approximation (SRPA). This method has been proposed to overcome double counting and stability issues appearing in beyond mean-field calculations. We show that the subtraction procedure leads to a considerable reduction of the SRPA downwards shift with respect to the random phase approximation (RPA) spectra and to results that are weakly cutoff dependent. Applications to the isoscalar monopole and quadrupole response in 16O and to the low-lying dipole response in 48Ca are shown and discussed.

Second RPA calculations with the Skyrme and Gogny interactions

The Second Random Phase Approximation (SRPA) is a natural extension of RPA where more general excitation operators are introduced. These operators contain, in addition to the one particle-one hole configurations already considered in RPA, also two particle-two hole excitations. Only in the last years, large-scale SRPA calculations have been performed, showing the merits and limits of this approach. In the first part of this paper, we present an overview of recent applications of the SRPA based on the Skyrme and Gogny interactions. Giant resonances in 16O will be studied and their properties discussed by using different models. In particular, we will present the first applications of the SRPA model with the finite-range Gogny interaction, discussing the advantages and drawbacks of using such an interaction in this type of calculations. After that, some more recent results, obtained by using a subtraction procedure to overcome double-counting in the SRPA, will be discussed. We will show that this procedure leads to results that are weakly cutoff dependent and that a strong reduction of the SRPA downwards shift with respect to the RPA spectra is found. Moreover, applying this procedure for the first time in the Gogny-SRPA framework, we will show that this method is able to reduce the anomalous shift found in previous calculations and related to some proton-neutron matrix elements of the residual interaction.

Beyond-mean-field corrections within the second random-phase approximation

A subtraction procedure, introduced to overcome double-counting problems in beyond-mean-field theories, is used in the second random-phase approximation (SRPA). Doublecounting problems arise in the energy-density functional framework in all cases where effective interactions tailored at leading order are used for higher-order calculations, such as those done in the SRPA model. It was recently shown that this subtraction procedure also guarantees that the stability condition related to the Thouless theorem is verified in extended RPA models. We discuss applications of the subtraction procedure, introduced within the SRPA model, to the nucleus 16O. The application of the subtraction procedure leads to: (i) stable results that are weakly cutoff dependent; (ii) a considerable upwards correction of the SRPA spectra (which were systematically shifted downwards by several MeV with respect to RPA spectra, in all previous calculations). With this important implementation of the model, many applications may be foreseen to analyze the genuine impact of 2 particle-2 hole configurations (without any cutoff dependences and anomalous shifts) on the excitation spectra of medium-mass and heavy nuclei.

Nuclear excitations as coupled one and two random-phase-approximation modes

We present an extension of the random--phase approximation (RPA) where the RPA phonons are used as building blocks to construct the excited states. In our model, that we call double RPA (DRPA), we include up to two RPA phonons. This is an approximate and simplified way, with respect to the full second random--phase approximation (SRPA), to extend the RPA by including two particle--two hole configurations. Some limitations of the standard SRPA model, related to the violation of the stability condition, are not encountered in the DRPA. We also verify in this work that the energy--weighted sum rules are satisfied. The DRPA is applied to low--energy modes and giant resonances in the nucleus $^{16}$O. We show that the model (i) produces a global downwards shift of the energies with respect to the RPA spectra; (ii) provides a shift that is however strongly reduced compared to that generated by the standard SRPA. This model represents an alternative way of correcting for the SRPA anomalous energy shift, compared to a recently developed extension of the SRPA, where a subtraction procedure is applied. The DRPA provides results in good agreeement with the experimental energies, with the exception of those low--lying states that have a dominant two particle--two hole nature. For describing such states, higher--order calculations are needed.

Subtraction method in the second random-phase approximation: First applications with a Skyrme energy functional

We make use of a subtraction procedure, introduced to overcome double-counting problems in beyond-mean-field theories, in the second random-phase-approximation (SRPA) for the first time. This procedure guarantees the stability of the SRPA (so that all excitation energies are real). We show that the method fits perfectly into nuclear density-functional theory. We illustrate applications to the monopole and quadrupole response and to low-lying ${0}^{+}$ and ${2}^{+}$ states in the nucleus $^{16}\mathrm{O}$. We show that the subtraction procedure leads to (i) results that are weakly cutoff dependent and (ii) a considerable reduction of the SRPA downwards shift with respect to the random-phase approximation (RPA) spectra (systematically found in all previous applications). This implementation of the SRPA model will allow a reliable analysis of the effects of two particle-two hole configurations (2p2h) on the excitation spectra of medium-mass and heavy nuclei.

Second random-phase approximation, Thouless' theorem, and the stability condition reexamined and clarified

It has been revealed through numerical calculations that the Second Random Phase Approximation (SRPA) with the Hartree-Fock solution as its reference state results in 1) spurious states at genuinely finite energy, contrary to common expectation, and 2) unstable solutions, which within the first-order Random Phase Approximation correspond to real low-energy collective vibrations. In the present work, these shortcomings of SRPA are shown to not contradict Thouless' theorem about the energy-weighted sum rule, and their origin is traced to the violation of the stability condition. A more general theorem is proven. Formal arguments are elucidated through numerical examples. Implications for the validity of SRPA are discussed.

Fragmentation of electric dipole strength inN=82isotones

Fragmentation of the dipole strength in the N=82 isotones 140Ce, 142Nd and 144Sm is calculated using the second random-phase approximation (SRPA). In comparison with the result of the random-phase approximation (RPA), the SRPA provides the additional damping of the giant dipole resonance and the redistribution of the low-energy dipole strength. Properties of the low-energy dipole states are significantly changed by the coupling to two-particle-two-hole (2p2h) states, which are also sensitive to the correlation among the 2p2h states. Comparison with available experimental data shows a reasonable agreement for the low-energy E1 strength distribution.

Extended second random phase approximation applied to metallic clusters

The coupling to two particle-two hole configurations is at the basis of the calculations of the spreading width of the collective excitations that in random phase approximation (RPA) are described as superpositions of one particle-one hole elementary modes. Second RPA (SRPA) is just the extension of RPA including both kinds of configurations. In SRPA use is made of the quasiboson approximation (QBA) as in RPA. It has been found that this is at the origin of the fact that the coupling among them strongly lowers the RPA collective excitations, both in metallic clusters and in nuclei. When the coupling of the two particle-two hole configurations among themselves is also included, as it has to be done in order to be consistent, this effect is even enhanced. Thus the reasonably good description of the collective vibrational states obtained at the RPA level (as expected on physical grounds, for example, for the dipole modes) is completely spoiled in SRPA. In the present paper we show quantitatively, for metallic clusters, that this disturbing behavior is eliminated when no use is made of QBA and a better description of ground-state correlations is introduced.

Collective excitations in metallic clusters within the second random phase approximation

The second random-phase approximation (SRPA) is a natural extension of the RPA. It essentially consists in enlarging the space of the elementary modes introduced to describe the collective states by adding 2 particle – 2 hole excitations to the 1 particle – 1 hole ones of the RPA. However SRPA calculations are really few and almost all are done by resorting to strong approximations. We apply the SRPA to the study of collective states in metallic clusters. All kinds of couplings among all 1 particle – 1 hole and 2 particle – 2 hole configurations are taken into account. The coupling of 1 particle – 1 hole configurations with the 2 particle – 2 hole ones strongly pushes down the multipole strength distribution, especially for the dipole case, with respect to RPA. The inclusion of the coupling of the 2 particle – 2 hole among themselves enhances this behaviour also for the collective states whose description within RPA is widely accepted as satisfactory. Some possible origins of these unpleasant results are discussed. In particular, the need for a better treatment of short range correlations is underlined.

Extension of the second random-phase approximation

The second random-phase approximation (SRPA) is the simplest and most natural extension of the RPA. It enlarges the space of the elementary modes introduced to describe the collective states by adding 2 particle -2 hole excitations to the 1 particle -1 hole ones of the RPA. In deriving the SRPA equations, use is made, as in the RPA, of the so-called quasi-boson approximation (QBA) where expectation values in the ground state of the system are approximated by their values in the uncorrelated reference state. This, however, has been shown to imply a degree of approximation worse than that in the RPA. It is, therefore, necessary to improve the QBA by considering a reference state which contains some correlations. Having in mind to perform such calculations for realistic systems, we consider a simple extension of the SRPA in which the reference state contains 2 particle - 2 hole correlations. The quality of such an extension is tested by applying it to a solvable three-level model and found to be good.

Transverse electron scattering form factors at low momentum transfer: Sensitivity to in-medium modifications of vector mesons?

Form factor measurements of $M2$ and $M4$ transitions to the lowest ${J}^{\ensuremath{\pi}}{=2}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states, respectively, in ${}^{48}\mathrm{Ca}$ with inelastic electron scattering at $180\ifmmode^\circ\else\textdegree\fi{}$ are reported for momentum transfers $q\ensuremath{\simeq}0.4\ensuremath{-}0.8{\mathrm{fm}}^{\ensuremath{-}1}.$ These form factors complement previous measurements at higher $q$ which have been treated by Lallena [Phys. Rev. C 48, 344 (1993)], as a test case to derive information on in-medium modifications of the $\ensuremath{\rho}$-meson mass. He deduced within the random-phase approximation (RPA) an effective mass ${m}_{\ensuremath{\rho}}^{*}\ensuremath{\simeq}0.9\ensuremath{-}0.95{m}_{\ensuremath{\rho}}$ assuming simultaneous scaling of the $\ensuremath{\pi}$ coupling constant (Brown-Rho scaling). The validity of the analysis is critically assessed by comparing the measured form factors to second random-phase approximation (SRPA) calculations with a $\ensuremath{\pi}+\ensuremath{\rho}$ exchange interaction. The dependence of the form factors on the choice of the interaction and corrections such as mesonic exchange currents (MEC) are found to be of comparable magnitude to effects from a dropping of ${m}_{\ensuremath{\rho}},$ and in-medium effects hence cannot be clearly inferred from the data. To quantify the latter would require first the construction of an interaction which is capable of describing simultaneously and optimally a large variety of low-energy nuclear properties in ${}^{48}\mathrm{Ca}.$ Remaining discrepancies in such a description could then be studied with the aim to single out in-medium effects of the type advocated by Lallena.

The higher random phase approximation and the stability of the energy spectrum of the nuclear shell model

Assuming only two-body nuclear forces, we study the dynamical two and many-nucleon correlations in spherical nuclei with the method of the Heisenberg equations of motion of the one-nucleon operators. After self-consistently solving the Hartree-Fock problem for the shell model we study the effect of the left-over non linear coupling terms using the method of the Higher Random Phase Approximation (HRPA). In particular, we formulate the secular problem and solve it in the Second Random Phase Approximation (SRPA) for spherical nuclei. The numerical work has been carried out for the example of the16O nucleus. The original Hartree-Fock shell model spectrum has been found rather stable against the inclusion of the « triple » correlations. The density fluctuation terms have been shown to be rather small, but, generally, capable of introducing a certain damping of some of the single-particle states as a consequence of the non-hermiticity of the extended SRPA secular matrix.