Symmetry and Supersymmetry in Nuclear Pairing: Exact Solutions
A. B. Balantekin, Y. Pehlivan
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6 pages of LATEX, to be published in the Proceedings of the "10th Int. Spring Seminar on Nuclear Physics: New Quests in Nuclear Structure", Vietri Sul Mare, May 21-25, 2010
Pairing plays a crucial role in nuclear spectra and attempts to describe it has a long history in nuclear physics. The limiting case in which all single particle states are degenerate, but with different s-wave pairing strengths was only recently solved. In this strong coupling limit the nuclear pairing Hamiltonian also exhibits a supersymmetry. Another solution away from those limits, namely two non-degenerate single particle states with different pairing strengths, was also given. In this contribution these developments are summarized and difficulties with possible generalizations to more single particle states and d-wave pairing are discussed.
We show that nuclear pairing Hamiltonian exhibits supersymmetry in the strong-coupling limit. The underlying supersymmetric quantum mechanical structure explains the degeneracies between the energies of the N and Nmax-N+1 pair eigenstates. The supersymmetry transformations connecting these states are given.
We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersy…
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The screening potential in the $^1S_0$ and $^3S_1$ pairing channels in neutron and nuclear matter in different approximations is discussed. It is found that the vertex corrections to the potential are much stronger in nuclear matter than in neutron matter.
We discuss the possible manifestation of pairing dynamics in nuclear collisions beyond the standard quasi-static treatment of pairing correlations. These involve solitonic excitations induced by pairing phase difference of colliding nuclei and pairing dynamic enhancement in the di-nuclear system formed by merging nuclei.
Nuclear supersymmetry is reviewed and some of its applications and extensions are discussed, together with a proposal for new, more stringent and precise tests to probe the supersymmetry classification, in particular, correlations between nuclei that belong to the same supermultiplet. The combination of these theoretical and experimental studies may play a unifying role in nuclear phenomena
In this contribution, I discuss the role of symmetries and algebraic methods in nuclear structure physics. In particular, I review some recent developments in nuclear supersymmetry and indicate possible applications for light nuclei in the sd- and pf-shell.
Extensions of nuclear supersymmetry are discussed, together with a proposal for new, more stringent and precise tests that probe the susy classification and specific two-particle correlations among supersymmetric partners. The combination of these theoretical and experimental studies may play a unifying role in nuclear phenomena.
We determine the influence of the three-body force and the medium modification of meson masses on pairing in nuclear and neutron matter. A reduction of the pairing gap is found and increasing with density.
Nuclear pairing correlations are known to play an important role in various single-particle and collective aspects of nuclear structure. After the first idea by A. Bohr, B. Mottelson and D. Pines on similarity of nuclear pairing to electron superconductivity, S.T. Belyaev gave a thorough analysis of the manifestations of pairing in complex nuclei. The current revival of interest in nuclear pairing is connected to the shift of modern nuclear physics towards nuclei far from stability; many loosely bound nuclei are particle-stable only due to the pairing. The theoretical methods borrowed from mac…
The 1S0 pairing gap in isospin-symmetric nuclear matter and finite nuclei is investigated using the chiral nucleon-nucleon potential at the N3LO order in the two-body sector, and the N2LO order in the three-body sector. To include realistic nuclear forces in RHB (Relativistic Hartree Bolgoliubov) calculations we rely on a separable form of the pairing interaction adjusted to the bare nuclear force. The separable pairing force is applied to the analysis of pairing properties for several isotopic and isotonic chains of spherical nuclei.
We give the exact solution of orbit dependent nuclear pairing problem between two non-degenerate energy levels using the Bethe ansatz technique. Our solution reduces to previously solved cases in the appropriate limits including Richardson's treatment of reduced pairing in terms of rational Gaudin algebra operators.
The coexistence of neutron-neutron (n-n), proton-proton (p-p), and neutron-proton (n-p) pairings is investigated by adopting an effective density-dependent contact pairing potential. These three types of pairings can coexist only if the n-p pairing is stronger than the n-n and p-p pairings for isospin asymmetric nuclear matter. In addition, the existence of n-n and p-p pairs might enhance n-p pairings in asymmetric nuclear matter when the n-p pairing strength is significantly stronger than the n-n and p-p ones. Conversely, the n-p pairing is reduced by the n-n and p-p pairs when the n-p pairin…
We show that the nuclear supersymmetry model (n-susy) in its extended version, predicts correlations in the nuclear structure matrix elements which characterize transfer reactions between nuclei that belong to the same supermultiplet. These correlations are related to the fermionic generators of the superalgebra and if verified experimentally can provide a direct test of the model.
Surface properties of neutron-neutron (T=1) pairing in semi-infinite nuclear matter in a hard wall potential are investigated in BCS approximation using the Gogny force. Surface enhancement of the gap function, pairing tensor and correlation energy density is put into evidence.
We simulate a heavy-ion collision using the canonical-basis time-dependent Hartree-Fock-Bogoliubov theory (Cb-TDHFB) treating pairing correlation in the three-dimensional coordinate space. We apply the Cb-TDHFB to 22O+22O collision with a contact-type pairing energy functional, and compare results of Cb-TDHFB and TDHF to investigate the effects of pairing correlations in nuclear fusion. Our results seem to indicate that pairing effects do not increase the fusion cross section in this system.
The overview of the Exact Pairing technique based on the quasispin symmetry is presented. Extensions of this method are discussed in relation to mean field, quadrupole collectivity, electromagnetic transitions, and many-body level density. Realistic calculations compared with experimental data are used to support the methods as well as to emphasize the manifestations of pairing correlations in nuclear many-body systems.
I review the phenomena associated with pairing in nuclear physics, most prominently the ubiquitous presence of odd-even mass differences and the properties of the excitation spectra, very different for even-even and odd-A nuclei. There are also significant dynamical effects of pairing, visible in the inertias associated with nuclear rotation and large-amplitude shape deformation.
The pairing gap for $^{53}$Ca obtained from new experimental data on the masses of $^{52-54}$Ca has the smallest value yet observed. This is explained in the framework of the nuclear shell model with schematic and realistic Hamiltonians as being due to shell gaps around the low-$ j $ orbital $ 1p_{1/2} $. Minima in the pairing gaps for all nuclei are shown and discussed
A survey of algebraic approaches to various problems in nuclear physics is given. Examples are chosen from pairing of many-nucleon systems, nuclear structure, fusion reactions below the Coulomb barrier, and supernova neutrino physics to illustrate the utility of group-theoretical and related algebraic methods in nuclear physics.
Using the Dirac-Hartree-Fock-Bogoliubov approximation to study nuclear pairing, we have found the short-range correlations of the Dirac $^1$S$_0$ pairing fields to be essentially identical to those of the two-nucleon virtual state at all values of the baryon density. We make use of this fact to develop a relativistic separable potential that correctly describes the pairing fields.
We discuss several pairing-related phenomena in nuclear systems, ranging from superfluidity in neutron stars to the gradual breaking of pairs in finite nuclei. We describe recent experimental evidence that points to a relation between pairing and phase transitions (or transformations) in finite nuclear systems. A simple pairing interaction model is used in order to study and classify an eventual pairing phase transition in finite fermionic systems such as nuclei. We show that systems with as few as 10-16 fermions can exhibit clear features reminiscent of a phase transition.
A microscopic theory for nuclear pairing is proposed through the generalized density matrix formalism. The analytical equations are as simple as that of the BCS theory, and could be solved within a similar computer time. The current theory conserves the exact particle number, and is valid at arbitrary pairing strength (including those below the BCS critical strength). These are the two main advantages over the conventional BCS theory. The theory is also of interests to other mesoscopic systems.
We use the Gorkov formulation of the Dirac-Hartree-Fock-Bogoliubov approximation to nuclear pairing to study the ^1S_0 nucleon-nucleon correlations in nuclear matter. We find the short-range correlations of the ^1S_0 pairing fields to be almost identical to those of the two-nucleon virtual state. We obtain mutually consistent results for the pairing fields, using several different sets of effective interaction parameters, when we demand that each of these sets places the virtual-state pole at its physical location.
We investigate the impact of pairing correlations on the behavior of unstable asymmetric nuclear matter at low temperature. We focus on the relative role of the pairing interaction, coupling nucleons of the same type (neutrons or protons), with respect to the symmetry potential, which enhances the neutron-proton attraction, along the clusterization process driven by spinodal instabilities. It is found that, especially at the transition temperature from the normal to the superfluid phase, pairing effects may induce significant variations in the isotopic content of the clusterized matter. This a…
Using equations, governing np pairing correlations in S=1, T=0 pairing channel (PRC 63 (2001) 021304(R)), it is shown that at low densities equations for the energy gap in the spectrum of quasiparticles and chemical potentials of protons and neutrons allow solutions with negative chemical potential. This corresponds to appearance of Bose--Einstein condensate (BEC) of deuterons in low density region of nuclear matter.
The main ideas behind nuclear supersymmetry are presented, starting from the basic concepts of symmetry and the methods of group theory in physics. We propose new, more stringent experimental tests that probe the supersymmetry classification in nuclei and point out that specific correlations should exist for particle transfer intensities among supersymmetric partners. We also discuss possible ways to generalize these ideas to cases where no dynamical symmetries are present. The combination of these theoretical and experimental studies may play a unifying role in nuclear phenomena.
The role of superfluidity in the incompressibility and in the symmetry energy is studied in nuclear matter and finite nuclei. Several pairing interactions are used: surface, mixed and isovector dependent. Pairing has a small effect on the nuclear matter incompressibility at saturation density, but the effects are significant at lower densities. The pairing effect on the centroid energy of the isoscalar Giant Monopole Resonance (GMR) is also evaluated for Pb and Sn isotopes by using a microscopic constrained-HFB approach, and found to change at most by 10% the nucleus incompressibility $K_A$. I…
We consider 1S0 pairing in infinite neutron matter and nuclear matter and show that in the lowest order approximation, where the pairing interaction is taken to be the bare nucleon-nucleon (NN) interaction in the 1S0 channel, the pairing interaction and the energy gap can be determined directly from the 1S0 phase shifts. This is due to the almost separable character of the NN interaction in this partial wave. Since the most recent NN interactions are charge-dependent, we have to solve coupled gap equations for proton-proton, neutron-neutron, and neutron-proton pairing in nuclear matter. The re…
The properties of pairing correlation in nuclear matter are investigated by using various versions of Skyrme forces. Truncation of states involving pairing correlation, necessary due to zero range nature of the Skyrme force, is discussed in detail. A plateau appears in pairing gap versus cutoff for each force. We propose to choose the cutoff parameter in the middle of the plateau so that the parameterization is independent of nuclides.
The pairing properties of nuclear systems are a sensitive probe of the effective nucleon-nucleon interactions. We compare the 1S0 pairing gaps in nuclear and neutron matter derived from the phenomenological Gogny interaction and a renormalization group motivated low-momentum V_low-k interaction extracted from realistic interactions. We find that the pairing gaps predicted by these interactions are in an excellent agreement in a wide range of sub-nuclear densities. The close agreement between the predictions of the effective forces remains intact in the case where the single particle spectra in…
Transfer reactions constitute a stringent test for nuclear supersymmetry, a theory that simultaneously describes neighboring nuclei with bosonic and fermionic character. We construct and analytically evaluate one-nucleon transfer matrix elements between supersymmetric partners with the U(6/4) case as an example, and stress the need for a careful treatment of bosonic and fermionic operators in the construction of mixed tensor operators.
A shell-model study of proton-neutron pairing in f - p shell nuclei using a parametrized hamiltonian that includes deformation and spin-orbit effects as well as isoscalar and isovector pairing is reported. By working in a shell-model framework we are able to assess the role of the various modes of proton-neutron pairing in the presence of nuclear deformation without violating symmetries. Results are presented for $^{44}$Ti, $^{46}$Ti and $^{48}$Cr.
A realistic nuclear mean-field hamiltonian with pairing has been diagonalized using Fock space representation that allows for nearly exact treatment of the problem. Calculations were performed for all the even-even nuclei with Z in (20, 100), whose pairing gaps were possible to extract out of the experimental masses. The optimal values of the pairing strength constants for the protons and neutrons have been found.
The odd-even staggering of nuclear masses was recognized in the early days of nuclear physics. Recently, a similar effect was discovered in other finite fermion systems, such as ultrasmall metallic grains and metal clusters. It is believed that the staggering in nuclei and grains is primarily due to pairing correlations (superconductivity), while in clusters it is caused by the Jahn-Teller effect. We find that, for light and medium-mass nuclei, the staggering has two components. The first one originates from pairing while the second, comparable in magnitude, has its roots in the deformed mean …
Pairing effects in non-uniform nuclear matter, surrounded by electrons, are studied in the protoneutron star early stage and in other conditions. The so-called nuclear pasta phases at subsaturation densities are solved in a Wigner-Seitz cell, within the Thomas-Fermi approximation. The solution of this problem is important for the understanding of the physics of a newly born neutron star after a supernova explosion. It is shown that the pasta phase is more stable than uniform nuclear matter on some conditions and the pairing force relevance is studied in the determination of these stable phases…
The structure and density dependence of the pairing gap in infinite matter is relevant for astrophysical phenomena and provides a starting point for the discussion of pairing properties in nuclear structure. Short-range correlations can significantly deplete the available single-particle strength around the Fermi surface and thus provide a reduction mechanism of the pairing gap. Here, we study this effect in the singlet and triplet channels of both neutron matter and symmetric nuclear matter. Our calculations use phase-shift equivalent interactions and chiral two-body and three-body interactio…
Low-lying nuclear states of Sm isotopes are studied in the framework of a collective Hamiltonian based on covariant energy density functional theory. Pairing correlation are treated by both BCS and Bogoliubov methods. It is found that the pairing correlations deduced from relativistic Hartree-Bogoliubov (RHB) calculations are generally stronger than those by relativistic mean-field plus BCS (RMF+BCS) with same pairing force. By simply renormalizing the pairing strength, the diagonal part of the pairing field is changed in such a way that the essential effects of the off-diagonal parts of the p…
We analyse how the spatial localisation properties of pairing correlations are changing in a major neutron shell of heavy nuclei. It is shown that the radial distribution of the pairing density depends strongly on whether the chemical potential is close to a low or a high angular momentum level and has very little sensitivity to whether the pairing force acts in the surface or in the bulk. The averaged pairing density over one major shell is however rather flat, practically independent of the pairing force. Hartree-Fock-Bogoliubov calculations for the isotopic chain $^{100-132}$Sn are presente…
The properties of pairing correlations in symmetric nuclear matter are studied in the relativistic mean field (RMF) theory with the effective interaction PK1. Considering well-known problem that the pairing gap at Fermi surface calculated with RMF effective interactions are three times larger than that with Gogny force, an effective factor in the particle-particle channel is introduced. For the RMF calculation with PK1, an effective factor 0.76 give a maximum pairing gap 3.2 MeV at Fermi momentum 0.9 fm$^{-1}$, which are consistent with the result with Gogny force.
We apply the algebraic Bethe ansatz technique to the nuclear pairing problem with orbit dependent coupling constants and degenerate single particle energy levels. We find the exact energies and eigenstates. We show that for a given shell, there are degeneracies between the states corresponding to less and more than half full shell. We also provide a technique to solve the equations of Bethe ansatz.
A shell-model study of proton-neutron pairing in $2p1f$ shell nuclei using a parametrized hamiltonian that includes deformation and spin-orbit effects as well as isoscalar and isovector pairing is reported. By working in a shell-model framework we are able to assess the role of the various modes of proton-neutron pairing in the presence of nuclear deformation without violating symmetries. Results are presented for $^{44}$Ti, $^{45}$Ti, $^{46}$Ti, $^{46}$V and $^{48}$Cr to assess how proton-neutron pair correlations emerge under different scenarios. We also study how the presence of a one-body …
Effects of pairing with isospin $T=0$ and $T=1$ are systematically studied in a model, which is based on a realistic nucleon-nucleon interaction and allows to describe the transition from infinite nuclear matter to finite nuclei. Special attention is paid to the development of the spin-orbit term in the mean field of nucleons in finite nuclei. The spin-orbit term yields a drastic suppression of $T=0$ proton-neutron pairing but does not lead to a complete disappearance in finite nuclei. Arguments are presented, why no clear evidence of $T=0$ pairing can be observed in the binding energies of fi…
We discuss various properties of singlet S pairing in neutron and nuclear matter in terms of the bare nuclon-nucleon interaction. Relations to the NN phase shifts are discussed, as well as various properties of the BCS wave function and the coherence length for fermion superfluids in infinite matter.
The effect of pairing on small and large amplitude dynamics is discussed. Pairing correlation is treated in a fully microscopic transport theory using a simplified BCS version of the Time-Dependent Hartree-Fock Bogolyubov (TDHFB) theory. The approach is applied to study the Giant Dipole and Giant Quadrupole Resonance in spherical and/or deformed magnesium isotopes showing very good agreement with QRPA theory. The present framework also provides a consistent approach to describe nuclear reactions around the Coulomb barrier. As an illustration, the influence of pairing correlations on multi-nucl…
The spectroscopic strengths of two-nucleon transfer reactions constitute a stringent test for two-nucleon correlations in the nuclear wave functions. A comparison between the recently measured 198Hg(d,alpha)196Au reaction and the predictions from the nuclear quartet supersymmetry lends further support to the validity of supersymmetry in nuclear physics.
We present solutions of the coupled, non-relativistic $^3S_1$-$^3D_1$ gap equations for neutron-proton pairing in symmetric nuclear matter, and estimate relativistic effects by solving the same gap equations modified according to minimal relativity and using single-particle energies from a Dirac-Brueckner-Hartree-Fock calculation. As a main result we find that relativistic effects decrease the value of the gap at the saturation density $k_F=1.36 fm^{-1}$ considerably, in conformity with the lack of evidence for strong neutron-proton pairing in finite nuclei.
Pairing correlations in nuclei play a decisive role in determining nuclear drip-lines, binding energies, and many collective properties. In this work a new Configuration-Space Monte-Carlo (CSMC) method for treating nuclear pairing correlations is developed, implemented, and demonstrated. In CSMC the Hamiltonian matrix is stochastically generated in Krylov subspace, resulting in the Monte-Carlo version of Lanczos-like diagonalization. The advantages of this approach over other techniques are discussed; the absence of the fermionic sign problem, probabilistic interpretation of quantum-mechanical…
The $^1S_0$-pairing gap in semi-infinite nuclear matter is evaluated microscopically using the effective pairing interaction recently found explicitly in the coordinate representation starting from the separable form of the Paris NN-potential. Instead of direct iterative solution of the gap equation, a new method proposed by V.A.Khodel, V.V.Khodel and J.W.Clark was used which simplifies the procedure significantly. The gap $Δ$ obtained in our calculations exibits a strong variation in the surface region with a pronounced maximum near the surface.
We propose a modified pairing functional for nuclear structure calculations which avoids the abrupt phase transition between pairing and non-pairing states. The intended application is the description of nuclear collective motion where the smoothing of the transition is compulsory to remove singularities. The stabilized pairing functional allows a thoroughly variational formulation, unlike the Lipkin-Nogami (LN) scheme which is often used for the purpose of smoothing. First applications to nuclear ground states and collective excitations prove the reliability and efficiency of the proposed sta…
The influence of pairing correlations on the nuclear size and in particular on the formation of nuclear halos is studied in the framework of relativistic continuum Hartree-Bogoliubov (RCHB) theory. It turns out that the contributions from the weakly-bound orbits with low orbital angular momenta $l$ play an important role. As an example, we investigate the neutron-rich Mg isotopes as a function of the pairing strength in situations, where the neuron Fermi surface are below, between and above two weakly-bound $2p$ levels. We find that the size of the pairing correlations has a two-fold influence…
To describe quantal collective phenomena, it is useful to requantize the time-dependent mean-field dynamics. We study the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory for the two-level pairing Hamiltonian, and compare results of different quantization methods. The one constructing microscopic wave functions, using the TDHFB trajectories fulfilling the Einstein-Brillouin-Keller quantization condition, turns out to be the most accurate. The method is based on the stationary-phase approximation to the path integral. We also examine the performance of the collective model which assumes th…
Ab initio gap equation for ^1S_0 pairing in a nuclear slab is solved for the Argonne v18 NN-potential. The gap function is compared in detail with the one found previously for the separable form of the Paris potential. The difference between the two gaps turned out to be about 10%. Dependence of the gap on the chemical potential mu is analyzed.
The uncertainty in the nuclear matrix elements (NMEs) of $0νββ$ decay for $^{76}$Ge, $^{82}$Se, $^{128}$Te, $^{130}$Te, and $^{136}$Xe in the self-consistent quasiparticle random phase approximation (QRPA) method is investigated by using eighteen Skyrme interactions supplemented with either a volume- or surface-type of pairing interactions. The NMEs for the isotopes concerned (except $^{136}$Xe) are less sensitive to the particle-hole ($ph$) interactions, while strongly dependent on the employed isovector particle-particle ($pp$) pairing interactions even though the pairing strengths are optim…
We use a recently improved version of the chiral nucleon-nucleon potential at next-to-next-to-leading order to calculate the $^1S_0$ pairing gap in isospin-symmetric nuclear matter. The pairing potential consists of the long-range one- and two-pion exchange terms and two short-distance NN-contact couplings. We find that the inclusion of the two-pion exchange at next-to-next-to-leading order reduces substantially the cut-off dependence of the $^1S_0$ pairing gap determined by solving a regularised BCS equation. Our results are close to those obtained with the universal low-momentum nucleon-nucl…
I describe the foundation of a Density Functional Theory approach to include pairing correlations, which was applied to a variety of systems ranging from dilute fermions, to neutron stars and finite nuclei. Ground state properties as well as properties of excited states and time-dependent phenomena can be achieved in this manner within a formalism based on microscopic input.
We discuss the pairing gap, a measure for nuclear pairing correlations, in chains of spherical, semi-magic nuclei in the framework of self-consistent nuclear mean-field models. The equations for the conventional BCS model and the approximate projection-before-variation Lipkin-Nogami method are formulated in terms of local density functionals for the effective interaction. We calculate the Lipkin-Nogami corrections of both the mean-field energy and the pairing energy. Various definitions of the pairing gap are discussed as three-point, four-point and five-point mass-difference formulae, average…
The surface behaviour of the pairing gap previously studied for semi-infinite nuclear matter is analyzed in the slab geometry. The gap-shape function is calculated in two cases: (a) pairing with the Gogny force in a hard-wall potential and (b) pairing with the separable Paris interaction in a Saxon-Woods mean-field potential. It is shown that the surface features are preserved in the case of slab geometry, being almost independent of the width of the slab. It is also demonstrated that the surface enhancement is strengthened as the absolute value of chemical potential $|μ|$ decreases which simu…
We present a new Skyrme-Hartree-Fock-Bogoliubov nuclear-mass model in which the contact-pairing force is constructed from microscopic pairing gaps of symmetric nuclear matter and neutron matter calculated from realistic two- and three-body forces, with medium-polarization effects included. With the pairing being treated more realistically than in any of our earlier models, the rms deviation with respect to essentially all the available mass data falls to 0.581 MeV, the best value ever found within the mean-field framework. Since our Skyrme force is also constrained by the properties of pure ne…
