Collective Coordinates Research Articles

Coriolis contribution to excited states of odd-mass nuclei with different deformation-dependent mass coefficients

Within the collective Bohr Hamiltonian, the adoption of a mass tensor as a function of collective coordinates has demonstrated its importance for describing the structure of nuclei. On the other hand, for odd-mass nuclei, the Coriolis interaction between the rotational and single-particle motions affects significantly the structure of nuclear excited states. In the framework of a recently developed extended Bohr Hamiltonian, by considering the deformation-dependent mass formalism whith different mass parameters for the rotation and the two β and γ vibrations and taking into account the Coriolis contribution, we investigate the bands structure of the 173Yb, 163Dy, 155Eu and 153Eu nuclei. Excited-state energies and B(E2) transition probabilities are calculated and compared with the available experimental data. Besides, we investigate the effect of DDMF and the Coriolis force on nuclear observables.

Curvature-induced skyrmion mass

We investigate the propagation of magnetic skyrmions on elastically deformable geometries by employing imaginary time quantum field theory methods. We demonstrate that the Euclidean action of the problem carries information of the elements of the surface space metric, and develop a description of the skyrmion dynamics in terms of a set of collective coordinates. We reveal that novel curvature-driven effects emerge in geometries with non-constant curvature, which explicitly break the translational invariance of flat space. In particular, for a skyrmion stabilized by a curvilinear defect, an inertia term and a pinning potential are generated by the varying curvature, while both of these terms vanish in the flat-space limit.

D-instanton perturbation theory

D-instanton world-volume theory has open string zero modes describing collective coordinates of the instanton. The usual perturbative amplitudes in the D-instanton background suffer from infra-red divergences due to the presence of these zero modes, and the usual approach of analytic continuation in momenta does not work since all open string states on a D-instanton carry strictly zero momentum. String field theory is well-suited for tackling these issues. However we find a new subtlety due to the existence of additional zero modes in the ghost sector. This causes a breakdown of the Siegel gauge, but a different gauge fixing consistent with the BV formalism renders the perturbation theory finite and unambiguous. At each order, this produces extra contribution to the amplitude besides what is obtained from integration over the moduli space of Riemann surfaces.

Empirical stability criteria for parametrically driven solitons of the nonlinear Schrödinger equation

For the parametrically driven nonlinear Schrödinger equation, two criteria for the dynamical stability of solitons are presented, using the driving force f(x, t) = a exp[2 i K(t) x]. Both criteria are based on the results of a collective coordinate theory. In this theory, the parameters of the 1-soliton solution are allowed to depend on time and a variational approach yields a set of coupled nonlinear ODEs for the collective coordinates. The solutions of these ODEs are used to compute the soliton’s normalized momentum p(t), velocity v(t), and a parametric plot p(v). The first criterion states that a sufficient condition for instability is fulfilled if the curve p(v), or a piece of it, has a negative slope. If the slope is positive then a necessary condition for stability is fulfilled. For constant K, a second criterion is presented. Here a complex wavefunction is calculated and a parametric plot of its imaginary part vs its real part is made. This yields a closed orbit in the complex plane. If this orbit has a negative sense of rotation with respect to a fixed point, then a sufficient condition for instability is fulfilled, whereas a positive sense is a necessary condition for stability. The ensemble of orbits and fixed points is a so-called phase portrait, which is very helpful for the classification of the different types of solutions of the ODEs. Finally, the validity of the two criteria is tested by comparing with simulations for the driven nonlinear Schrödinger equation. Both criteria make correct predictions about the stability and instability of the solitons, with the exception of a few cases at the border between stability and instability regions.

The Time-Dependent Generator Coordinate Method in Nuclear Physics

The emergence of collective behaviors and the existence of large amplitude motions are both central features in the fields of nuclear structure and reactions. From a theoretical point of view, describing such phenomena requires increasing the complexity of the many-body wavefunction of the system to account for long-range correlations. One of the challenges, when going in this direction, is to keep the approach tractable within our current computational resources while gaining a maximum of predictive power for the phenomenon under study. In the Generator Coordinate Method (GCM), the many-body wave function is a linear superposition of (generally non-orthogonal) many-body states (the generator states) labeled by a few collective coordinates. Such a method has been widely used in structure studies to restore the symmetries broken by single-reference approaches. In the domain of reactions, its time-dependent version (TDGCM) has been developed and applied to predict the dynamics of heavy-ion collisions or fission where the collective fluctuations play an essential role. In this review, we present the recent developments and applications of the TDGCM in nuclear reactions. We recall the formal derivations of the TDGCM and its most common approximate treatment, the Gaussian Overlap Approximation. We also emphasize the Schr\"odinger Collective-Intrinsic Model (SCIM) variant focused on the inclusion of quasiparticle excitations into the description. Finally, we highlight several exploratory studies related to a TDGCM built on time-dependent generator states.

Microscopic description of the self-conjugate Xe108 and Te104 α -decay chain

A microscopic calculation of half-lives for the recently observed $^{108}\mathrm{Xe}\ensuremath{\rightarrow}^{104}\mathrm{Te}\ensuremath{\rightarrow}^{100}\mathrm{Sn}\ensuremath{\alpha}$-decay chain is performed by using a self-consistent framework based on energy density functionals. The relativistic density functional DD-PC1 and a separable pairing interaction of finite range are used to compute axially symmetric deformation-energy surfaces of $^{104}\mathrm{Te}$ and $^{108}\mathrm{Xe}$ as functions of quadrupole, octupole, and hexadecupole collective coordinates. Dynamic least-action paths are determined that trace the $\ensuremath{\alpha}$-particle emission from the equilibrium deformation to the point of scission. The calculated half-lives, 197 ns for $^{104}\mathrm{Te}$ and $50\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{s}$ for $^{108}\mathrm{Xe}$, are compared with recent experimental values of the half-lives of superallowed $\ensuremath{\alpha}$ decay of $^{104}\mathrm{Te}$: $<18$ ns, and $^{108}\mathrm{Xe}$: ${58}_{\ensuremath{-}23}^{+106}\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{s}$.

Tree-like structures and Freak waves generation induced by quintic-nonlinearity and cubic-Raman effect in a nonlinear metamaterial

The goal of this investigation is based on electromagnetic wave behavior which is modeled by a nonlinear Schrodinger equation in a nonlinear metamaterial optical system. Two regimes are under consideration known as negative index regime and absorption regime. During the propagation, second-order dispersion, cubic-quintic nonlinearities and cubic-Raman effect are taken into account. The characterization of electromagnetic wave is assured by collective coordinates method and conventional ansatz function. The soliton light pulse is progressively modified by quintic-nonlinearity and cubic-Raman effect in order to provoke freak waves, tree structures and soliton self-frequency shift generation. Hence, the coefficients of the Schrodinger equation are plotted in order to improve the explanation of these strange phenomena. Moreover, all those above mentioned phenomena are strongly influenced by frequency and the regime under consideration. Thus, some specific freak waves such as Kuznetsov–Ma, Peregrine and Riemann waves are found. It is clearly pointed out the key role of internal and external disturbances due to cubic-Raman effects. That leads to freak waves, tree structures and soliton self frequency shift generation for both regimes. The so-called self-compensation process as technique of cancellation of these phenomena is also investigated.

Model reduction for the Kuramoto-Sakaguchi model: The importance of nonentrained rogue oscillators.

The Kuramoto-Sakaguchi model for coupled phase oscillators with phase frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite-size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model, where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at a given coupling strength, as well as the critical coupling strength for partial and for global synchronization.

Collective coordinate study of spin-wave emission from a dynamic domain wall

Radiation from a dynamic soliton is an issue of fundamental interest. Here, the authors study the emission from a domain wall in a ferromagnet taking into account the collective coordinate for the modulation of the wall thickness (breathing mode) in addition to the conventional coordinates, describing the sliding motion and the tilting angle of the wall plane. The breathing mode is shown to have a linear coupling to the fluctuation (spin wave), hence, it is the dominant source of spin-wave emission. The wave profile of the emitted spin wave is identified and the contribution to the Gilbert damping parameter due to the emission is estimated.

Efficient method for estimation of fission fragment yields of r -process nuclei

$\textbf{Background}$ More than half of all the elements heavier than iron are made by the rapid neutron capture process (or r process). For very neutron-rich astrophysical conditions, such at those found in the tidal ejecta of neutron stars, nuclear fission determines the r-process endpoint, and the fission fragment yields shape the final abundances of $110\le A \le 170$ nuclei. The knowledge of fission fragment yields of hundreds of nuclei inhabiting very neutron-rich regions of the nuclear landscape is thus crucial for the modeling of heavy-element nucleosynthesis. $\textbf{Purpose}$ In this study, we propose a model for the fast calculation of fission fragment yields based on the concept of shell-stabilized prefragments defined with help of the nucleonic localization functions. $\textbf{Methods}$ To generate realistic potential energy surfaces and nucleonic localizations, we apply Skyrme Density Functional Theory. The distribution of the neck nucleons among the two prefragments is obtained by means of a statistical model. $\textbf{Results}$ We benchmark the method by studying the fission yields of $^{178}$Pt, $^{240}$Pu, $^{254}$Cf, and $^{254,256,258}$Fm and show that it satisfactorily explains the experimental data. We then make predictions for $^{254}$Pu and $^{290}$Fm as two representative cases of fissioning nuclei that are expected to significantly contribute during the r-process nucleosynthesis occurring in neutron star mergers. $\textbf{Conclusions}$ The proposed framework provides an efficient alternative to microscopic approaches based on the evolution of the system in a space of collective coordinates all the way to scission. It can be used to carry out global calculations of fission fragment distributions across the r-process region.

Coupling of shape and pairing vibrations in a collective Hamiltonian based on nuclear energy density functionals

The quadrupole collective Hamiltonian, based on relativistic energy density functionals, is extended to include a pairing collective coordinate. In addition to quadrupole shape vibrations and rotations, the model describes pairing vibrations and the coupling between shape and pairing degrees of freedom. The parameters of the collective Hamiltonian are determined by constrained self-consistent relativistic mean-field plus Bardeen-Cooper-Schrieffer (RMF+BCS) calculations in the space of intrinsic shape and pairing deformations. The effect of coupling between shape and pairing degrees of freedom is analyzed in a study of low-energy spectra and transition rates of four axially symmetric $N=92$ rare-earth isotones. When compared to results obtained with the standard quadrupole collective Hamiltonian, the inclusion of dynamical pairing increases the moment of inertia, lowers the energies of excited $0^+$ states and reduces the E0-transition strengths, in better agreement with data.

Three-Dimensional Dynamics of a Magnetic Hopfion Driven by Spin Transfer Torque.

Magnetic hopfion is a three-dimensional (3D) topological soliton with novel spin structure that would enable exotic dynamics. Here, we study the current-driven 3D dynamics of a magnetic hopfion with a unit Hopf index in a frustrated magnet. Attributed to the spin Berry phase and symmetry of the hopfion, the phase space entangles multiple collective coordinates, thus the hopfion exhibits rich dynamics including longitudinal motion along the current direction, transverse motion perpendicular to the current direction, rotational motion, and dilation. Furthermore, the characteristics of hopfion dynamics is determined by the ratio between the nonadiabatic spin transfer torque parameter and the damping parameter. Such peculiar 3D dynamics of magnetic hopfion could shed light on understanding the universal physics of hopfions in different systems and boost the prosperous development of 3D spintronics.

Collective Inertial Masses in Nuclear Reactions

Towards the microscopic theoretical description for large amplitude collective dynamics, we calculate the coefficients of inertial masses for low-energy nuclear reactions. Under the scheme of energy density functional, we apply the adiabatic self-consistent collective coordinate (ASCC) method, as well as the Inglis' cranking formula to calculate the inertias for the translational and the relative motions, in addition to those for the rotational motion. Taking the scattering between two $\alpha$ particles as an example, we investigate the impact of the time-odd components of the mean-field potential on the collective inertial masses. The ASCC method asymptotically reproduces the exact masses for both the relative and translational motions. On the other hand, the cranking formula fails to do so when the time-odd components exist.

Excited states of odd-mass nuclei with different deformation-dependent mass coefficients

Experimental data indicate that the mass tensor of collective Bohr Hamiltonian cannot be considered as a constant but should be considered as a function of the collective coordinates. In this work our purpose is to investigate the properties of low-lying collective states of the odd nuclei $^{173}$Yb and $^{163}$Dy by using a new generalized version of the collective quadrupole Bohr Hamiltonian with deformation-dependent mass coefficients. The proposed new version of the Bohr Hamiltonian is solved for Davidson potential in $\beta$ shape variable, while the $\gamma$ potential is taken to be equal to the harmonic oscillator. The obtained results of the excitation energies and B(E2) reduced transition probabilities show an overall agreement with the experimental data. Moreover, we investigate the effect of the deformation dependent mass parameter on energy spectra and transition rates in both cases, namely: when the mass coefficients are different and when they are equal. Besides, we will show the positive effect of the present formalism on the moment of inertia.

Parametrically driven nonlinear Dirac equation with arbitrary nonlinearity

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase .

Triangular Rogue Waves and Multi- Wave Trains Generation in a Chameleon Electrical Transmission Line

In this work, we present a particular nonlinear transmission line called chameleon nonlinear transmission line. In fact, the chameleon’s behavior is related to the fact that without changing its appearance, the line can exhibit alternatively purely right - or left - handed behavior. This transmission line is different to the composite one. So, the goal of this work is to demonstrate that this special line can support the propagation of soliton light pulse subjected to quintic – phase modulation, improve the freak wave’s mechanism of generation and verify the chameleon’s behavior of the line. Consequently, we employ collective coordinate’s theory in order to give a great characterization of the light pulse. Then, we introduce an upgraded function called “type II Ansatz function” with eight collective coordinates compared to the conventional Gaussian Ansatz with six collective coordinates. We show that the two additional coordinates will allow us to improve the technique of measurement of internal excitation leading to the generation of rogue events. Moreover, these coordinates will give supplemental details on frequency shift and chirp distortions during the generation of specific rogue events such as “wall of waves”, tree structures, multi - wave trains, Kuznetsov - Ma breathers, Akhmediev breathers, Peregrine solitons and triangular rogue waves. The stability of the soliton light pulse will be also investigated at specific frequency ranges.

Statistical distributions of the tuning and coupling collective modes at a conical intersection using the hierarchical equations of motion.

We investigate the possibility of extracting the probability distribution of the effective environmental tuning and coupling modes during the nonadiabatic relaxation through a conical intersection. Dynamics are dealt with an open quantum system master equation by partitioning a multistate electronic subsystem out of all the nuclear vibrators. This is an alternative to the more usual partition retaining the tuning and coupling modes of a conical intersection in the active subsystem coupled to a residual bath. The minimal partition of the electronic system generally leads to highly structured spectral densities for both vibrational baths and requires a strongly nonperturbative non-Markovian master equation, treated here by the hierarchical equations of motion (HEOMs). We extend-for a two-bath situation-the procedure proposed by Shi et al. [J. Chem. Phys. 140, 134106 (2014)], whereby the information contained in the auxiliary HEOM matrices is exploited in order to derive the nuclear dissipative wave packet, i.e., the statistical distribution of the displacement of the two tuning and coupling collective coordinates in each electronic state and the coherence. This allows us to visualize the distribution, all along the nonadiabatic decay. We explore a large parameter space for a symmetrical conical intersection model and a symmetrical initial Franck-Condon preparation. Some parameters could be controlled by external fields, while others are molecule dependent and could be designed by molecular engineering. We illustrate the relation between the strongly coupled electronic and bath dynamics together with a geometric measure of non-Markovianity.

Effect of system–reservoir correlations on temperature estimation**Project supported by the National Natural Science Foundation of China (Grants Nos. 11704025, 11674139, and 11834005).

In many previous temperature estimation schemes, the temperature of a sample is directly read out from the final steady state of a quantum probe, which is coupled to the sample. However, in these studies, information of correlations between system (the probe) and reservoir (the sample) is usually eliminated, leading the steady state of the probe is a canonical equilibrium state with respect solely to system’s Hamiltonian. To explore the influence of system–reservoir correlations on the estimation precision, we investigate the equilibration dynamics of a spin interacting with a finite temperature bosonic reservoir. By incorporating an intermediate harmonic oscillator or a collective coordinate into the spin, the system–reservoir correlations can be correspondingly encoded in a Gibbs state of an effective Hamilton, which is size consistent with the original bare spin. Extracting information of temperature from this corrected steady state, we find the effect of the system-reservoir correlations on the estimation precision is highly sensitive to the details of the spectral density function of the measured reservoir.

Curvature as a Collective Coordinate in Enhanced Sampling Membrane Simulations.

The plasticity of membranes plays an important functional role in cells, cell components, and micelles, where bending, budding, and remodeling implement numerous recognition and communication processes. Comparatively, molecular simulation methods to induce, control, and quantitatively characterize such deformations remain scarce. This work defines a novel collective coordinate associated with membrane bending, which strives to combine realism (by preserving the notion of local atomic curvatures) and low computational cost (allowing its evaluation at every time step of a molecular dynamics simulation). Enhanced sampling simulations along this conformational coordinate provide convenient access to the underlying bending free energy landscape. To showcase its potential, the method is applied to three state-of-the-art problems: the determination of the bending free energy landscape of a 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) bilayer, the formation of a POPE liposome, and the study of the influence of the Pseudomonas quinolone signal on the budding of Gram-negative bacterial outer membranes.

Machine Learning for Vibrational Spectroscopic Maps.

Maps that relate spectroscopic properties of a vibrational mode and collective solvent coordinates have proven useful in theoretical vibrational spectroscopy of condensed-phase systems. It has been realized that the predictive power of such an approach is limited and there is no clear systematic way to improve its accuracy. Here, we propose an adaptation of Δ-machine-learning methodology that goes beyond the spectroscopic maps. The machine-learning part of our approach combines Gaussian process regression used to generate the data set with an artificial neural network used to predict spectroscopic properties of interest. A specific application to the OH-stretch frequencies and transition dipoles of water is presented. Our method approximates these properties about two times more accurately than the spectroscopic-maps-only-based approach. Our results suggest that the new approach may become useful in the study of vibrational spectroscopy of condensed-phase systems.