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M1 dipole strength from projected generator coordinate method calculations in the sd-shell valence space
Authors:
Stavros Bofos,
Jaime Martínez-Larraz,
Benjamin Bally,
Thomas Duguet,
Mikael Frosini,
Tomás R. Rodríguez,
Kamila Sieja
Abstract:
The low-energy enhancement observed in the deexcitation $γ$-ray strength functions, attributed to magnetic dipole (M1) radiations, has spurred theoretical efforts to improve on its description. Among the most widely used approaches are the quasiparticle random-phase approximation (QRPA) and its extensions. However, these methods often struggle to reproduce the correct behavior of the M1 strength a…
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The low-energy enhancement observed in the deexcitation $γ$-ray strength functions, attributed to magnetic dipole (M1) radiations, has spurred theoretical efforts to improve on its description. Among the most widely used approaches are the quasiparticle random-phase approximation (QRPA) and its extensions. However, these methods often struggle to reproduce the correct behavior of the M1 strength at the lowest $γ$ energies. An alternative framework, the projected generator coordinate method (PGCM), offers significant advantages over QRPA by restoring broken symmetries and incorporating both vibrational and rotational dynamics within a unified description. Due to these features, PGCM has been proposed as a promising tool to study the low-energy M1 strength function in atomic nuclei. However, comprehensive investigations employing this method are lacking. The PGCM is presently used within the frame of sd-shell valence space calculations based on the USDB shell-model interaction to benchmark its performance against the solutions obtained via exact diagonalization. The reliability of two different sets of generator coordinates in the PGCM calculations is gauged using ${}^{24}$Mg as a test case. The ability of the PGCM to reproduce results from exact diagonalization in the sd valence space is demonstrated for $1^{+}$ states and M1 transitions. Future work will need to assess whether the proposed method can be applied systematically and extended to large-scale calculations while maintaining a reasonable computational cost.
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Submitted 16 July, 2025;
originally announced July 2025.
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HFB3: an axial HFB solver with Gogny forces using a 2-center HO basis (C++/Python)
Authors:
N. Dubray,
J. P. Ebran,
P. Carpentier,
M. Frosini,
A. Zdeb,
N. Pillet,
J. Newsome,
M. Verrière,
G. Accorto,
D. Regnier
Abstract:
The HFB3 program solves the axial nuclear Hartree-Fock-Bogoliubov (HFB) equations using bases formed by either one or two sets of deformed Harmonic Oscillator (HO) solutions with D1-type and D2-type Gogny effective nucleon-nucleon interactions. Using two sets of HO solutions shifted along the z-axis (2-center basis) allows to accurately describe highly elongated nuclear systems while keeping a mod…
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The HFB3 program solves the axial nuclear Hartree-Fock-Bogoliubov (HFB) equations using bases formed by either one or two sets of deformed Harmonic Oscillator (HO) solutions with D1-type and D2-type Gogny effective nucleon-nucleon interactions. Using two sets of HO solutions shifted along the z-axis (2-center basis) allows to accurately describe highly elongated nuclear systems while keeping a moderate basis size, making this type of basis very convenient for the description of the nuclear fission process. For the description of odd-even and odd-odd systems, the equal-filling-approximation is used. Several observables can be calculated by the program, including the mean values of the multipole moments, nuclear radii, inertia tensors following Adiabatic Time-Dependent Hartree-Fock-Bogoliubov (ATDHFB) or Generator Coordinate Method (GCM) prescriptions, local and non-local one-body densities, local and non-local pairing densities, some fission fragment properties, etc. The program can ensure that the mean values associated with some specific operators take pre-defined values (constraints). Such constraints can be set on the usual multipole moments (for protons, neutrons or total mass). This program can be used as a monoprocess and monothreaded CLI executable, or through full-featured Python bindings (available through the Python Package Index PyPI).
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Submitted 8 August, 2025; v1 submitted 12 June, 2025;
originally announced June 2025.
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Extremely large oblate deformation of the first excited state in $^{12}$C: a new challenge to modern nuclear theory
Authors:
C. Ngwetsheni,
J. N. Orce,
P. Navrátil,
P. E. Garrett,
T. Faestermann,
A. Bergmaier,
M. Frosini,
V. Bildstein,
B. A. Brown,
C. Burbadge,
T. Duguet,
K. Hadyńska-Klȩk,
M. Mahgoub,
C. V. Mehl,
A. Pastore,
A. Radich,
S. Triambak
Abstract:
A Coulomb-excitation study of the high-lying first excited state at 4.439 MeV in the nucleus $^{12}$C has been carried out using the $^{208}$Pb($^{12}$C,$^{12}$C$^*$)$^{208}$Pb$^*$ reaction at 56 MeV and the {\sc Q3D} magnetic spectrograph at the Maier-Leibnitz Laboratorium in Munich. High-statistics achieved with an average beam intensity of approximately 10$^{11}$ ions/s together with state-of-t…
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A Coulomb-excitation study of the high-lying first excited state at 4.439 MeV in the nucleus $^{12}$C has been carried out using the $^{208}$Pb($^{12}$C,$^{12}$C$^*$)$^{208}$Pb$^*$ reaction at 56 MeV and the {\sc Q3D} magnetic spectrograph at the Maier-Leibnitz Laboratorium in Munich. High-statistics achieved with an average beam intensity of approximately 10$^{11}$ ions/s together with state-of-the-art {\it ab initio} calculations of the nuclear dipole polarizability permitted the accurate determination of the spectroscopic quadrupole moment, $Q_{_S}(2_{_1}^+) = +0.076(30)$~eb, in agreement with previous measurements. Combined with previous work, a weighted average of $Q_{_S}(2_{_1}^+) = +0.090(14)$ eb is determined, which includes the re-analysis of a similar experiment by Vermeer and collaborators, $Q_{_S}(2_{_1}^+) = +0.103(20)$~eb. Such a large oblate deformation challenges modern nuclear theory and emphasizes the need of $α$ clustering and associated triaxiality effects for full convergence of $E2$ collective properties.
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Submitted 3 June, 2025;
originally announced June 2025.
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Deformed natural orbitals for ab initio calculations
Authors:
Alberto Scalesi,
Thomas Duguet,
Mikael Frosini,
Vittorio Somà
Abstract:
The rapid development of ab initio nuclear structure methods towards doubly open-shell nuclei, heavy nuclei and greater accuracy occurs at the price of evermore increased computational costs, especially RAM and CPU time. While most of the numerical simulations are carried out by expanding relevant operators and wave functions on the spherical harmonic oscillator basis, alternative one-body bases o…
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The rapid development of ab initio nuclear structure methods towards doubly open-shell nuclei, heavy nuclei and greater accuracy occurs at the price of evermore increased computational costs, especially RAM and CPU time. While most of the numerical simulations are carried out by expanding relevant operators and wave functions on the spherical harmonic oscillator basis, alternative one-body bases offering advantages in terms of computational efficiency have recently been investigated. In particular, the so-called natural basis used in combination with symmetry-conserving methods applicable to doubly closed-shell nuclei has proven beneficial in this respect. The present work examines the performance of the natural basis in the context of symmetry-breaking many-body calculations enabling the description of superfluid and deformed open-shell nuclei at polynomial cost with system's size. First, it is demonstrated that the advantage observed for closed-shell nuclei carries over to open-shell ones. A detailed investigation of natural-orbital wave functions provides useful insight to support this finding and to explain the superiority of the natural basis over alternative ones. Second, it is shown that the use of natural orbitals combined with importance-truncation techniques leads to an even greater gain in terms of computational costs. The presents results pave the way for the systematic use of natural-orbital bases in future implementations of non-perturbative many-body methods.
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Submitted 24 December, 2024; v1 submitted 25 July, 2024;
originally announced July 2024.
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Dimensionality reduction through tensor factorization : application to \textit{ab initio} nuclear physics calculations
Authors:
Mikael Frosini,
Thomas Duguet,
Pierre Tamagno,
Lars Zurek
Abstract:
The construction of predictive models of atomic nuclei from first principles is a challenging (yet necessary) task towards the systematic generation of theoretical predictions (and associated uncertainties) to support nuclear data evaluation. The consistent description of the rich phenomenology of nuclear systems indeed requires the introduction of reductionist approaches that construct nuclei dir…
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The construction of predictive models of atomic nuclei from first principles is a challenging (yet necessary) task towards the systematic generation of theoretical predictions (and associated uncertainties) to support nuclear data evaluation. The consistent description of the rich phenomenology of nuclear systems indeed requires the introduction of reductionist approaches that construct nuclei directly from interacting nucleons by solving the associated quantum many-body problem. In this context, so-called \textit{ab initio} methods offer a promising route by deriving controlled (and systematically improvable) approximations both to the inter-nucleon interaction and to the solutions of the many-body problem. From a technical point of view, approximately solving the many-body Schrödinger equation in heavy open-shell systems typically requires the construction and contraction of large mode-4 (mode-6) tensors that need to be stored repeatedly. Recently, a new dimensionality reduction method based on randomized singular value decomposition has been introduced to reduce the numerical cost of many-body perturbation theory. This work applies this lightweight formalism to the study of the Germanium isotopic chain, where standard approaches would be too expansive to run. Inclusion of triaxiality is found to improve the overall agreement with experimental data on differential quantities.
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Submitted 25 July, 2024;
originally announced July 2024.
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Ab initio description of monopole resonances in light- and medium-mass nuclei: IV. Angular momentum projection and rotation-vibration coupling
Authors:
Andrea Porro,
Thomas Duguet,
Jean-Paul Ebran,
Mikael Frosini,
Robert Roth,
Vittorio Somà
Abstract:
Giant Resonances are, with nuclear rotations, the most evident expression of collectivity in finite nuclei. These two categories of excitations, however, are traditionally described within different formal schemes, such that vibrational and rotational degrees of freedom are separately treated and coupling effects between those are often neglected. The present work puts forward an approach aiming a…
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Giant Resonances are, with nuclear rotations, the most evident expression of collectivity in finite nuclei. These two categories of excitations, however, are traditionally described within different formal schemes, such that vibrational and rotational degrees of freedom are separately treated and coupling effects between those are often neglected. The present work puts forward an approach aiming at a consitent treatment of vibrations and rotations. Specifically, this paper is the last in a series of four dedicated to the investigation of the giant monopole resonance in doubly open-shell nuclei via the ab initio Projected Generator Coordinate Method (PGCM). The present focus is on the treatment and impact of angular momentum restoration within such calculations. The PGCM being based on the use of deformed mean-field states, the angular-momentum restoration is performed when solving the secular equation to extract vibrational excitations. In this context, it is shown that performing the angular momentum restoration only after solving the secular equation contaminates the monopole response with an unphysical coupling to the rotational motion, as was also shown recently for (quasi-particle) random phase approximation calculations based on a deformed reference state. Eventually, the present work based on the PGCM confirms that an a priori angular momentum restoration is necessary to handle consistently both collective motions at the same time. This further pleads in favor of implementing the full-fledged projected (quasi-particle) random phase approximation in the future.
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Submitted 1 July, 2024;
originally announced July 2024.
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Impact of correlations on nuclear binding energies
Authors:
Alberto Scalesi,
Thomas Duguet,
Pepijn Demol,
Mikael Frosini,
Vittorio Somà,
Alexander Tichai
Abstract:
A strong effort will be dedicated in the coming years to extend the reach of ab initio nuclear-structure calculations to heavy doubly open-shell nuclei. In order to do so, the most efficient strategies to incorporate dominant many-body correlations at play in such nuclei must be identified. With this motivation in mind, the present work pedagogically analyses the inclusion of many-body correlation…
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A strong effort will be dedicated in the coming years to extend the reach of ab initio nuclear-structure calculations to heavy doubly open-shell nuclei. In order to do so, the most efficient strategies to incorporate dominant many-body correlations at play in such nuclei must be identified. With this motivation in mind, the present work pedagogically analyses the inclusion of many-body correlations and their impact on binding energies of Calcium and Chromium isotopes. Employing an empirically-optimal Hamiltonian built from chiral effective field theory, binding energies along both isotopic chains are studied via a hierarchy of approximations based on polynomially-scaling expansion many-body methods. The corresponding results are compared to experimental data and to those obtained via valence-space in-medium similarity renormalization group calculations at the normal-ordered two-body level that act as a reference in the present study. The spherical mean-field approximation is shown to display specific shortcomings in Ca isotopes that can be understood analytically and that are efficiently corrected via the consistent addition of low-order dynamical correlations on top of it. While the same setting cannot appropriately reproduce binding energies in doubly open-shell Cr isotopes, allowing the unperturbed mean-field state to break rotational symmetry permits to efficiently capture the static correlations responsible for the phenomenological differences observed between the two isotopic chains. Eventually, the present work demonstrates in a pedagogical way that polynomially-scaling expansion methods based on unperturbed states that possibly break (and restore) symmetries constitute an optimal route to extend ab initio calculations to heavy closed- and open-shell nuclei.
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Submitted 17 October, 2024; v1 submitted 5 June, 2024;
originally announced June 2024.
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Anisotropic flow in fixed-target $^{208}$Pb+$^{20}$Ne collisions as a probe of quark-gluon plasma
Authors:
Giuliano Giacalone,
Wenbin Zhao,
Benjamin Bally,
Shihang Shen,
Thomas Duguet,
Jean-Paul Ebran,
Serdar Elhatisari,
Mikael Frosini,
Timo A. Lähde,
Dean Lee,
Bing-Nan Lu,
Yuan-Zhuo Ma,
Ulf-G. Meißner,
Govert Nijs,
Jacquelyn Noronha-Hostler,
Christopher Plumberg,
Tomás R. Rodríguez,
Robert Roth,
Wilke van der Schee,
Björn Schenke,
Chun Shen,
Vittorio Somà
Abstract:
The System for Measuring Overlap with Gas (SMOG2) at the LHCb detector enables the study of fixed-target ion-ion collisions at relativistic energies ($\sqrt{s_{\rm NN}}\sim100$ GeV in the centre-of-mass). With input from \textit{ab initio} calculations of the structure of $^{16}$O and $^{20}$Ne, we compute 3+1D hydrodynamic predictions for the anisotropic flow of Pb+Ne and Pb+O collisions, to be t…
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The System for Measuring Overlap with Gas (SMOG2) at the LHCb detector enables the study of fixed-target ion-ion collisions at relativistic energies ($\sqrt{s_{\rm NN}}\sim100$ GeV in the centre-of-mass). With input from \textit{ab initio} calculations of the structure of $^{16}$O and $^{20}$Ne, we compute 3+1D hydrodynamic predictions for the anisotropic flow of Pb+Ne and Pb+O collisions, to be tested with upcoming LHCb data. This will allow the detailed study of quark-gluon plasma (QGP) formation as well as experimental tests of the predicted nuclear shapes. Elliptic flow ($v_2$) in Pb+Ne collisions is greatly enhanced compared to the Pb+O baseline due to the shape of $^{20}$Ne, which is deformed in a bowling-pin geometry. Owing to the large $^{208}$Pb radius, this effect is seen in a broad centrality range, a unique feature of this collision configuration. Larger elliptic flow further enhances the quadrangular flow ($v_4$) of Pb+Ne collisions via non-linear coupling, and impacts the sign of the kurtosis of the elliptic flow vector distribution ($c_2\{4\}$). Exploiting the shape of $^{20}$Ne proves thus an ideal method to investigate the formation of QGP in fixed-target experiments at LHCb, and demonstrates the power of SMOG2 as a tool to image nuclear ground states.
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Submitted 26 February, 2025; v1 submitted 30 May, 2024;
originally announced May 2024.
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Ab initio description of monopole resonances in light- and medium-mass nuclei: III. Moments evaluation in ab initio PGCM calculations
Authors:
Andrea Porro,
Thomas Duguet,
Jean-Paul Ebran,
Mikael Frosini,
Robert Roth,
Vittorio Somà
Abstract:
The paper is the third of a series dedicated to the ab initio description of monopole giant resonances in mid-mass closed- and open-shell nuclei via the so-called projected generator coordinate method. The present focus is on the computation of the moments $m_k$ of the monopole strength distribution, which are used to quantify its centroid energy and dispersion. First, the capacity to compute low-…
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The paper is the third of a series dedicated to the ab initio description of monopole giant resonances in mid-mass closed- and open-shell nuclei via the so-called projected generator coordinate method. The present focus is on the computation of the moments $m_k$ of the monopole strength distribution, which are used to quantify its centroid energy and dispersion. First, the capacity to compute low-order moments via two different methods is developed and benchmarked for the $m_1$ moment. Second, the impact of the angular momentum projection on the centroid energy and dispersion of the monopole strength is analysed before comparing the results to those obtained from consistent quasi-particle random phase approximation calculations. Next, the so-called energy weighted sum rule (EWSR) is investigated. First, the appropriate ESWR in the center-of-mass frame is derived analytically. Second, the exhaustion of the intrinsic EWSR is tested in order to quantify the (unwanted) local-gauge symmetry breaking of the presently employed chiral effective field theory ($χ$EFT) interactions. Finally, the infinite nuclear matter incompressibility associated with the employed $χ$EFT interactions is extracted by extrapolating the finite-nucleus incompressibility computed from the monopole centroid energy.
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Submitted 22 April, 2024;
originally announced April 2024.
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Tensor factorization in ab initio many-body calculations: Triaxially-deformed (B) MBPT calculations in large bases
Authors:
M. Frosini,
T. Duguet,
P. Tamagno
Abstract:
Whether for fundamental studies or nuclear data evaluations, first-principle calculations of atomic nuclei constitute the path forward. Today, performing \textit{ab initio} calculations (a) of heavy nuclei, (b) of doubly open-shell nuclei or (c) with a sub-percent accuracy is at the forefront of nuclear structure theory. While combining any two of these features constitutes a major challenge, addr…
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Whether for fundamental studies or nuclear data evaluations, first-principle calculations of atomic nuclei constitute the path forward. Today, performing \textit{ab initio} calculations (a) of heavy nuclei, (b) of doubly open-shell nuclei or (c) with a sub-percent accuracy is at the forefront of nuclear structure theory. While combining any two of these features constitutes a major challenge, addressing the three at the same time is currently impossible. From a numerical standpoint, these challenges relate to the necessity to handle (i) very large single bases and (ii) mode-6, \textit{i.e.} three-body, tensors (iii) that must be stored repeatedly. Performing second-order many-body perturbation theory(ies) calculations based on triaxially deformed and superfluid reference states of doubly open-shell nuclei up to mass $A=72$, the present work achieves a significant step forward by addressing challenge (i). To do so, the memory and computational cost associated with the handling of large tensors is scaled down via the use of tensor factorization techniques. The presently used factorization format is based on a randomized singular value decomposition that does not require the computation and storage of the very large initial tensor. The procedure delivers an inexpensive and controllable approximation to the original problem, as presently illustrated for calculations that could not be performed without tensor factorization. With the presently developed technology at hand, one can envision to perform calculations of yet heavier doubly open-shell nuclei at sub-percent accuracy in a foreseeable future.
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Submitted 9 July, 2024; v1 submitted 12 April, 2024;
originally announced April 2024.
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Ab initio description of monopole resonances in light- and medium-mass nuclei: II. Ab initio PGCM calculations in $^{46}$Ti, $^{28}$Si and $^{24}$Mg
Authors:
Andrea Porro,
Thomas Duguet,
Jean-Paul Ebran,
Mikael Frosini,
Robert Roth,
Vittorio Somà
Abstract:
Giant resonances (GRs) are a striking manifestation of collective motions in atomic nuclei. The present paper is the second in a series of four dedicated to the use of the projected generator coordinate method (PGCM) for the ab initio determination of the isoscalar giant monopole resonance (GMR) in closed- and open-shell mid-mass nuclei.
While the first paper was dedicated to quantifying various…
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Giant resonances (GRs) are a striking manifestation of collective motions in atomic nuclei. The present paper is the second in a series of four dedicated to the use of the projected generator coordinate method (PGCM) for the ab initio determination of the isoscalar giant monopole resonance (GMR) in closed- and open-shell mid-mass nuclei.
While the first paper was dedicated to quantifying various uncertainty sources, the present paper focuses on the first applications to three doubly-open shell nuclei, namely $^{46}$Ti, $^{28}$Si and $^{24}$Mg. In particular, the goal is to investigate from an ab initio standpoint (i) the coupling of the GMR with the giant quadrupole resonance (GQR) in intrinsically-deformed nuclei, (ii) the possible impact of shape coexistence and shape mixing on the GMR, (iii) the GMR based on shape isomers and (iv) the impact of anharmonic effects on the monopole response. The latter is studied by comparing PGCM results to those obtained via the quasi-particle random phase approximation (QRPA), the traditional many-body approach to giant resonances, performed in a consistent setting.
Eventually, PGCM results for sd-shell nuclei are in excellent agreement with experimental data, which is attributed to the capacity of the PGCM to capture the important fragmentation of the monopole response in light, intrinsically-deformed systems. Still, the comparison to data in $^{28}$Si and $^{24}$Mg illustrates the challenge (and the potential benefit) of extracting unambiguous experimental information.
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Submitted 24 February, 2024;
originally announced February 2024.
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The unexpected uses of a bowling pin: exploiting $^{20}$Ne isotopes for precision characterizations of collectivity in small systems
Authors:
Giuliano Giacalone,
Benjamin Bally,
Govert Nijs,
Shihang Shen,
Thomas Duguet,
Jean-Paul Ebran,
Serdar Elhatisari,
Mikael Frosini,
Timo A. Lähde,
Dean Lee,
Bing-Nan Lu,
Yuan-Zhuo Ma,
Ulf-G. Meißner,
Jacquelyn Noronha-Hostler,
Christopher Plumberg,
Tomás R. Rodríguez,
Robert Roth,
Wilke van der Schee,
Vittorio Somà
Abstract:
Whether or not femto-scale droplets of quark-gluon plasma (QGP) are formed in so-called small systems at high-energy colliders is a pressing question in the phenomenology of the strong interaction. For proton-proton or proton-nucleus collisions the answer is inconclusive due to the large theoretical uncertainties plaguing the description of these processes. While upcoming data on collisions of…
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Whether or not femto-scale droplets of quark-gluon plasma (QGP) are formed in so-called small systems at high-energy colliders is a pressing question in the phenomenology of the strong interaction. For proton-proton or proton-nucleus collisions the answer is inconclusive due to the large theoretical uncertainties plaguing the description of these processes. While upcoming data on collisions of $^{16}$O nuclei may mitigate these uncertainties in the near future, here we demonstrate the unique possibilities offered by complementing $^{16}$O$^{16}$O data with collisions of $^{20}$Ne ions. We couple both NLEFT and PGCM ab initio descriptions of the structure of $^{20}$Ne and $^{16}$O to hydrodynamic simulations of $^{16}$O$^{16}$O and $^{20}$Ne$^{20}$Ne collisions at high energy. We isolate the imprints of the bowling-pin shape of $^{20}$Ne on the collective flow of hadrons, which can be used to perform quantitative tests of the hydrodynamic QGP paradigm. In particular, we predict that the elliptic flow of $^{20}$Ne$^{20}$Ne collisions is enhanced by as much as 1.170(8)$_{\rm stat.}$(30)$_{\rm syst.}$ for NLEFT and 1.139(6)$_{\rm stat.}$(39)$_{\rm syst.}$ for PGCM relative to $^{16}$O$^{16}$O collisions for the 1% most central events. At the same time, theoretical uncertainties largely cancel when studying relative variations of observables between two systems. This demonstrates a method based on experiments with two light-ion species for precision characterizations of the collective dynamics and its emergence in a small system.
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Submitted 8 February, 2024;
originally announced February 2024.
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Ab initio description of monopole resonances in light- and medium-mass nuclei: I. Technical aspects and uncertainties of ab initio PGCM calculations
Authors:
Andrea Porro,
Thomas Duguet,
Jean-Paul Ebran,
Mikael Frosini,
Robert Roth,
Vittorio Somá
Abstract:
Giant resonances (GRs) are a striking manifestation of collective motions in mesoscopic systems such as atomic nuclei. Until recently, theoretical investigations have essentially relied on the (quasiparticle) random phase approximation ((Q)RPA), and extensions of it, based on phenomenological energy density functionals (EDFs). As part of a current effort to describe GRs within an ab initio theoret…
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Giant resonances (GRs) are a striking manifestation of collective motions in mesoscopic systems such as atomic nuclei. Until recently, theoretical investigations have essentially relied on the (quasiparticle) random phase approximation ((Q)RPA), and extensions of it, based on phenomenological energy density functionals (EDFs). As part of a current effort to describe GRs within an ab initio theoretical scheme, the present work promotes the use of the projected generator coordinate method (PGCM). This method, which can handle anharmonic effects while satisfying symmetries of the nuclear Hamiltonian, displays a favorable (i.e. mean-field-like) scaling with system's size. Presently focusing on the isoscalar giant monopole resonance (GMR) of light- and medium-mass nuclei, PGCM's potential to deliver wide-range ab initio studies of GRs in closed- and open-shell nuclei encompassing pairing, deformation, and shape coexistence effects is demonstrated. The comparison with consistent QRPA calculations highlights PGCM's unique attributes and sheds light on the intricate interplay of nuclear collective excitations. The present paper is the first in a series of four and focuses on technical aspects and uncertainty quantification of ab initio PGCM calculations of GMR using the doubly open-shell $^{46}$Ti as an illustrative example. The second paper displays results for a set of nuclei of physical interest and proceeds to the comparison with consistent (deformed) ab initio QRPA calculations. While the third paper analyzes useful moments of the monopolar strength function and different ways to access them within PGCM calculations, the fourth paper focuses on the effect of the symmetry restoration on the monopole strength function.
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Submitted 3 February, 2024;
originally announced February 2024.
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Benchmark of many-body approaches for magnetic dipole transition strength
Authors:
M. Frosini,
W. Ryssens,
K. Sieja
Abstract:
The low-energy enhancement observed recently in the deexcitation gamma-ray strength functions, suggested to arise due to the magnetic dipole radiation, motivates theoretical efforts to improve the description of M1 strength in available nuclear structure models. Reliable theoretical predictions of nuclear dipole excitations are of interest for different nuclear applications and in particular for n…
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The low-energy enhancement observed recently in the deexcitation gamma-ray strength functions, suggested to arise due to the magnetic dipole radiation, motivates theoretical efforts to improve the description of M1 strength in available nuclear structure models. Reliable theoretical predictions of nuclear dipole excitations are of interest for different nuclear applications and in particular for nuclear astrophysics, where the calculations of radiative capture cross sections often resort to theoretical strength functions. We aim to benchmark many-body methods in their description of the M1 strength functions, with a special emphasis on the low-energy effects observed in the deexcitation strength. We investigate the zero-temperature and finite-temperature magnetic dipole strength functions computed within the quasiparticle random-phase approximation and compare them to those from exact diagonalizations of the same Hamiltonian in restricted orbital spaces. The study is carried out for a sample of 25 spherical and deformed nuclei which can be described by diagonalization of the respective effective Hamiltonian in three different valence spaces. A reasonable agreement is found for the total photoabsorption strengths while the QRPA distributions are shown to be systematically shifted down in energy with respect to exact results. Photoemission strengths obtained within the FT-QRPA appear insufficient to explain the low-energy enhancement of the M1 strength functions. The problems encountered in QRPA calculations are ascribed to the lack of correlations in the nuclear ground state and to the truncation of the many-body space. In particular, the latter prevents obtaining the sufficiently high level density to produce the low-energy enhancement of the strength function, making the (FT-)QRPA approach unsuitable for predictions of such effects across the nuclear chart.
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Submitted 18 December, 2023;
originally announced December 2023.
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Generative deep-learning reveals collective variables of Fermionic systems
Authors:
Raphaël-David Lasseri,
David Regnier,
Mikaël Frosini,
Marc Verriere,
Nicolas Schunck
Abstract:
Complex processes ranging from protein folding to nuclear fission often follow a low-dimension reaction path parameterized in terms of a few collective variables. In nuclear theory, variables related to the shape of the nuclear density in a mean-field picture are key to describing the large amplitude collective motion of the neutrons and protons. Exploring the adiabatic energy landscape spanned by…
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Complex processes ranging from protein folding to nuclear fission often follow a low-dimension reaction path parameterized in terms of a few collective variables. In nuclear theory, variables related to the shape of the nuclear density in a mean-field picture are key to describing the large amplitude collective motion of the neutrons and protons. Exploring the adiabatic energy landscape spanned by these degrees of freedom reveals the possible reaction channels while simulating the dynamics in this reduced space yields their respective probabilities. Unfortunately, this theoretical framework breaks down whenever the systems encounters a quantum phase transition with respect to the collective variables. Here we propose a generative-deep-learning algorithm capable of building new collective variables highly representative of a nuclear process while ensuring a differentiable mapping to its Fermionic wave function. Within this collective space, the nucleus can evolve continuously from one of its adiabatic quantum phase to the other at the price of crossing a potential energy barrier. This approach applies to any Fermionic system described by a single Slater determinant, which encompasses electronic systems described within the density functional theory.
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Submitted 14 June, 2023;
originally announced June 2023.
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Rooting the EDF method into the ab initio framework. PGCM-PT formalism based on MR-IMSRG pre-processed Hamiltonians
Authors:
T. Duguet,
J. -P. Ebran,
M. Frosini,
H. Hergert,
V. Somà
Abstract:
Recently, ab initio techniques have been successfully connected to the traditional valence-space shell model. In doing so, they can either explicitly provide ab initio shell-model effective Hamiltonians or constrain the construction of empirical ones. In the present work, the possibility to follow a similar path for the nuclear energy density functional (EDF) method is analyzed. For this connectio…
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Recently, ab initio techniques have been successfully connected to the traditional valence-space shell model. In doing so, they can either explicitly provide ab initio shell-model effective Hamiltonians or constrain the construction of empirical ones. In the present work, the possibility to follow a similar path for the nuclear energy density functional (EDF) method is analyzed. For this connection to be actualized, two theoretical techniques are instrumental: the recently proposed ab initio PGCM-PT many-body formalism and the MR-IMSRG pre-processing of the nuclear Hamiltonian. Based on both formal arguments and numerical results, possible new lines of research are briefly discussed, namely to compute ab initio EDF effective Hamiltonians at low computational cost, to constrain empirical ones or to produce them directly via an effective field theory that remains to be invented.
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Submitted 7 September, 2022;
originally announced September 2022.
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Zero- and finite-temperature electromagnetic strength distributions in closed- and open-shell nuclei from first principles
Authors:
Y. Beaujeault-Taudière,
M. Frosini,
J. -P. Ebran,
T. Duguet,
R. Roth,
V. Somà
Abstract:
Ab initio approaches to the nuclear many-body problem have seen their reach considerably extended over the past decade. However, collective excitations have been scarcely addressed so far due to the prohibitive cost of solving the corresponding equations of motion. Here, a numerically efficient method to compute electromagnetic response functions at zero- and finite-temperature in superfluid and d…
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Ab initio approaches to the nuclear many-body problem have seen their reach considerably extended over the past decade. However, collective excitations have been scarcely addressed so far due to the prohibitive cost of solving the corresponding equations of motion. Here, a numerically efficient method to compute electromagnetic response functions at zero- and finite-temperature in superfluid and deformed nuclei from an ab initio standpoint is presented and applied to $^{16}$O, $^{28}$Si, $^{46}$Ti and $^{56}$Fe. This work opens the path to systematic ab initio calculations of nuclear responses to electroweak probes across a significant portion of the nuclear chart.
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Submitted 24 August, 2022; v1 submitted 25 March, 2022;
originally announced March 2022.
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Multi-reference many-body perturbation theory for nuclei III -- Ab initio calculations at second order in PGCM-PT
Authors:
Mikael Frosini,
Thomas Duguet,
Jean-Paul Ebran,
Benjamin Bally,
Heiko Hergert,
Tomás R. Rodríguez,
Robert Roth,
Jiangming Yao,
Vittorio Somà
Abstract:
In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first rea…
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In spite of missing dynamical correlations, the projected generator coordinate method (PGCM) was recently shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing absolute binding energies and reaching high accuracy eventually requires the inclusion of dynamical correlations on top of the PGCM. In this context, the present work discusses the first realistic results of a novel multi-reference perturbation theory (PGCM-PT) that can do so within a symmetry-conserving scheme for both ground and low-lying excited states. First, proof-of-principle calculations in a small ($e_{\mathrm{max}}=4$) model space demonstrate that exact binding energies of closed- (\nucl{O}{16}) and open-shell (\nucl{O}{18}, \nucl{Ne}{20}) nuclei are reproduced within $0.5-1.5\%$ at second order, i.e. through PGCM-PT(2). Moreover, profiting from the pre-processing of the Hamiltonian via multi-reference in-medium similarity renormalization group transformations, PGCM-PT(2) can reach converged values within smaller model spaces than with an unevolved Hamiltonian. Doing so, dynamical correlations captured by PGCM-PT(2) are shown to bring essential corrections to low-lying excitation energies that become too dilated at leading order, i.e., at the strict PGCM level. The present work is laying the foundations for a better understanding of the optimal way to grasp static and dynamical correlations in a consistent fashion, with the aim of accurately describing ground and excited states of complex nuclei via ab initio many-body methods.
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Submitted 27 January, 2022; v1 submitted 2 November, 2021;
originally announced November 2021.
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Multi-reference many-body perturbation theory for nuclei II -- Ab initio study of neon isotopes via PGCM and IM-NCSM calculations
Authors:
Mikael Frosini,
Thomas Duguet,
Jean-Paul Ebran,
Benjamin Bally,
Tobias Mongelli,
Tomás R. Rodríguez,
Robert Roth,
Vittorio Somà
Abstract:
The neon isotopic chain displays a rich phenomenology, ranging from clustering in the ground-state of the self-conjugate doubly open-shell stable $^{20}$Ne isotope to the physics of the island of inversion around the neutron-rich $^{30}$Ne isotope. This second (i.e. Paper II) of the present series proposes an extensive ab initio study of neon isotopes based on two complementary many-body methods,…
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The neon isotopic chain displays a rich phenomenology, ranging from clustering in the ground-state of the self-conjugate doubly open-shell stable $^{20}$Ne isotope to the physics of the island of inversion around the neutron-rich $^{30}$Ne isotope. This second (i.e. Paper II) of the present series proposes an extensive ab initio study of neon isotopes based on two complementary many-body methods, i.e. the quasi-exact in-medium no-core shell model (IM-NCSM) and the projected generator coordinate method (PGCM) that is ideally suited to capturing strong static correlations associated with shape deformation and fluctuations. Calculations employ a state-of-the-art generation of chiral effective field theory Hamiltonians and evaluate the associated systematic uncertainties. In spite of missing so-called dynamical correlations, which can be added via the multi-reference perturbation theory proposed in the first paper (i.e. Paper I) of the present series, the PGCM is shown to be a suitable method to tackle the low-lying spectroscopy of complex nuclei. Still, describing the physics of the island of inversion constitutes a challenge that seems to require the inclusion of dynamical correlations. This is addressed in the third paper (i.e. Paper III) of the present series.
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Submitted 27 January, 2022; v1 submitted 1 November, 2021;
originally announced November 2021.
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Multi-reference many-body perturbation theory for nuclei I -- Novel PGCM-PT formalism
Authors:
Mikael Frosini,
Thomas Duguet,
Jean-Paul Ebran,
Vittorio Somà
Abstract:
Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of…
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Perturbative and non-perturbative expansion methods already constitute a tool of choice to perform ab initio calculations over a significant part of the nuclear chart. In this context, the categories of accessible nuclei directly reflect the class of unperturbed state employed in the formulation of the expansion. The present work generalizes to the nuclear many-body context the versatile method of Ref. \cite{burton20a} by formulating a perturbative expansion on top of a multi-reference unperturbed state mixing deformed non-orthogonal Bogoliubov vacua, i.e. a state obtained from the projected generator coordinate method (PGCM). Particular attention is paid to the part of the mixing taking care of the symmetry restoration, showing that it can be exactly contracted throughout the expansion, thus reducing significantly the dimensionality of the linear problem to be solved to extract perturbative corrections.
While the novel expansion method, coined as PGCM-PT, reduces to the PGCM at lowest order, it reduces to single-reference perturbation theories in appropriate limits. Based on a PGCM unperturbed state capturing (strong) static correlations in a versatile and efficient fashion, PGCM-PT is indistinctly applicable to doubly closed-shell, singly open-shell and doubly open-shell nuclei. The remaining (weak) dynamical correlations are brought consistently through perturbative corrections. This symmetry-conserving multi-reference perturbation theory is state-specific and applies to both ground and excited PGCM unperturbed states, thus correcting each state belonging to the low-lying spectrum of the system under study.
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Submitted 27 January, 2022; v1 submitted 29 October, 2021;
originally announced October 2021.
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In-medium $k$-body reduction of $n$-body operators
Authors:
Mikael Frosini,
Thomas Duguet,
Benjamin Bally,
Yann Beaujeault-Taudière,
Jean-Paul Ebran,
Vittorio Somà
Abstract:
The computational cost of ab initio nuclear structure calculations is rendered particularly acute by the presence of (at least) three-nucleon interactions. This feature becomes especially critical now that many-body methods aim at extending their reach beyond mid-mass nuclei. Consequently, state-of-the-art ab initio calculations are typically performed while approximating three-nucleon interaction…
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The computational cost of ab initio nuclear structure calculations is rendered particularly acute by the presence of (at least) three-nucleon interactions. This feature becomes especially critical now that many-body methods aim at extending their reach beyond mid-mass nuclei. Consequently, state-of-the-art ab initio calculations are typically performed while approximating three-nucleon interactions in terms of effective, i.e. system-dependent, zero-, one- and two-nucleon operators. While straightforward in doubly closed-shell nuclei, existing approximation methods based on normal-ordering techniques involve either two- and three-body density matrices or a symmetry-breaking one-body density matrix in open-shell systems. In order to avoid such complications, a simple, flexible, universal and accurate approximation technique involving the convolution of the initial operator with a sole symmetry-invariant one-body matrix is presently formulated and tested numerically. Employed with a low-resolution Hamiltonian, the novel approximation method is shown to induce errors below $2-3\%$ across a large range of nuclei, observables and many-body methods.
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Submitted 19 February, 2021;
originally announced February 2021.
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Bogoliubov many-body perturbation theory under constraint
Authors:
Pepijn Demol,
Mikael Frosini,
Alexander Tichai,
Vittorio Somà,
Thomas Duguet
Abstract:
In order to solve the A-body Schrödinger equation both accurately and efficiently for open-shell nuclei, a novel many-body method coined as Bogoliubov many-body perturbation theory (BMBPT) was recently formalized and applied at low orders. Based on the breaking of U(1) symmetry associated with particle-number conservation, this perturbation theory must operate under the constraint that the average…
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In order to solve the A-body Schrödinger equation both accurately and efficiently for open-shell nuclei, a novel many-body method coined as Bogoliubov many-body perturbation theory (BMBPT) was recently formalized and applied at low orders. Based on the breaking of U(1) symmetry associated with particle-number conservation, this perturbation theory must operate under the constraint that the average number of particles is self-consistently adjusted at each perturbative order. The corresponding formalism is presently detailed with the goal to characterize the behavior of the associated Taylor series. BMBPT is, thus, investigated numerically up to high orders at the price of restricting oneself to a small, i.e. schematic, portion of Fock space. While low-order results only differ by 2 - 3 % from those obtained via a configuration interaction (CI) diagonalization, the series is shown to eventually diverge. The application of a novel resummation method coined as eigenvector continuation further increase the accuracy when built from low-order BMBPT corrections and quickly converges towards the CI result when applied at higher orders. Furthermore, the numerically-costly self-consistent particle number adjustment procedure is shown to be safely bypassed via the use of a computationally cheap a posteriori correction method. Eventually, the present work validates the fact that low order BMBPT calculations based on an a posteriori (average) particle number correction deliver controlled results and demonstrates that they can be optimally complemented by the eigenvector continuation method to provide results with sub-percent accuracy. This approach is, thus, planned to become a workhorse for realistic ab initio calculations of open-shell nuclei in the near future.
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Submitted 9 December, 2020; v1 submitted 7 February, 2020;
originally announced February 2020.
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Improved many-body expansions from eigenvector continuation
Authors:
Pepijn Demol,
Thomas Duguet,
Andreas Ekström,
Mikael Frosini,
Kai Hebeler,
Sebastian König,
Dean Lee,
Achim Schwenk,
Vittorio Somà,
Alexander Tichai
Abstract:
Quantum many-body theory has witnessed tremendous progress in various fields, ranging from atomic and solid-state physics to quantum chemistry and nuclear structure. Due to the inherent computational burden linked to the ab initio treatment of microscopic fermionic systems, it is desirable to obtain accurate results through low-order perturbation theory. In atomic nuclei however, effects such as s…
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Quantum many-body theory has witnessed tremendous progress in various fields, ranging from atomic and solid-state physics to quantum chemistry and nuclear structure. Due to the inherent computational burden linked to the ab initio treatment of microscopic fermionic systems, it is desirable to obtain accurate results through low-order perturbation theory. In atomic nuclei however, effects such as strong short-range repulsion between nucleons can spoil the convergence of the expansion and make the reliability of perturbation theory unclear. Mathematicians have devised an extensive machinery to overcome the problem of divergent expansions by making use of so-called resummation methods. In large-scale many-body applications such schemes are often of limited use since no a priori analytical knowledge of the expansion is available. We present here eigenvector continuation as an alternative resummation tool that is both efficient and reliable because it is based on robust and simple mathematical principles.
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Submitted 28 November, 2019;
originally announced November 2019.