Using ideas from BCS descriptions of systems of fermions, a covariant extension of the relativistic Fermi gas model is presented as a way to incorporate correlation effects in nuclei. The model is developed for the BCS nuclear ground state and for final states consisting of a single plane-wave nucleon plus a BCS recoiling daughter nucleus. The nuclear spectral function is obtained and from it the superscaling function for use in treating high-energy quasielastic electroweak processes. Interestingly, this model has the capability to yield the asymmetric tail seen in the experimental scaling function.
The scaling function $f(ψ')$ for medium and heavy nuclei with $Z\neq N$ for which the proton and neutron densities are not similar is constructed within the coherent density fluctuation model (CDFM) as a sum of the proton and neutron scaling functions. The latter are calculated in the cases of $^{62}$Ni, $^{82}$Kr, $^{118}$Sn, and $^{197}$Au nuclei on the basis of the corresponding proton and neutron density distributions which are obtained in deformed self-consistent mean-field Skyrme HF+BCS method. The results are in a reasonable agreement with the empirical data from the inclusive electron …
We present a covariant extension of the relativistic Fermi gas model which incorporates correlation effects in nuclei. Within this model, inspired by the BCS descriptions of systems of fermions, we obtain the nuclear spectral function and from it the superscaling function for use in treating high-energy quasielastic electroweak processes. Interestingly, this model has the capability to yield the asymmetric tail seen in the experimental scaling function.
Superscaling analyses of inclusive electron scattering from nuclei are extended from the quasielastic processes to the delta excitation region. The calculations of $(e,e^\prime)$ cross sections for the target nucleus $^{12}$C at various incident electron energies are performed using scaling functions $f(ψ^{\prime})$ obtained in approaches going beyond the mean-field approximation, such as the coherent density fluctuation model (CDFM) and the one based on the light-front dynamics (LFD) method. The results are compared with those obtained using the relativistic Fermi gas (RFG) model and the exte…
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…
The nature of pairing correlations in neutron matter is re-examined. Working within the conventional approximation in which the $nn$ pairing interaction is provided by a realistic bare $nn$ potential fitted to scattering data, it is demonstrated that the standard BCS theory fails in regions of neutron number density where the pairing constant $λ$, depending crucially on density, has a non-BCS negative sign. We are led to propose a non-BCS scenario for pairing phenomena in neutron matter that involves the formation of a hidden dineutron state. In low-density neutron matter where the pairing con…
We construct a BCS-like model that combines nucleonic pairing correlations and possible quartic correlations of alpha-type in a single variational wave function and derive corresponding gap equations. In the approximation of large logarithms typical for the BCS approach, we show that the system reveals two possible types of a condensate which cannot coexist. If the alpha-type condensate prevails at N=Z, the growth of the neutron excess will naturally lead to the first order phase transition to the nucleon condensates.
The density, spin and isospin correlation functions in nuclear matter with a neutron-proton ($np$) condensate are calculated to study the possible signatures of the BEC-BCS crossover in the low-density region. It is shown that the criterion of the crossover (Phys. Rev. Lett. {\bf 95}, 090402 (2005)), consisting in the change of the sign of the density correlation function at low momentum transfer, fails to describe correctly the density-driven BEC-BCS transition at finite isospin asymmetry or finite temperature. As an unambiguous signature of the BEC-BCS transition, there can be used the prese…
The effect of nucleon-nucleon correlations in symmetric nuclear matter at finite temperature is studied beyond BCS theory. Starting from a Hartree-Fock description of nuclear matter with the Gogny effective interaction, we add correlations corresponding to the formation of preformed pairs and scattering states above the superfluid critical temperature within the in-medium T-matrix approach, which is analogous to the Nozieres-Schmitt-Rink theory. We calculate the critical temperature for a BEC superfluid of deuterons, of a BCS superfluid of nucleons, and in the crossover between these limits. T…
We introduce a BCS-like theory for the quartet correlations induced by the isovector pairing interaction. It is based on a coherent state of BCS type and, unlike usual mean field approaches, it displays a vanishing pair anomalous density $\langle c^\dagger c^\dagger\rangle =0$. We find good agreement between our theory and the exact results. We discuss how the pairing and quarteting correlations share some similar qualitative features within the BCS approach. However, there is no sharp quarteting phase transition. We also present various ways in which our theory may be further developed.
We evaluate correlation functions of the BCS model for finite number of particles. The integrability of the Hamiltonian relates it with the Gaudin algebra ${\cal G}[sl(2)]$. Therefore, a theorem that Sklyanin proved for the Gaudin model, can be applied. Several diagonal and off-diagonal correlators are calculated. The finite size scaling behavior of the pairing correlation function is studied.
The exploration of the non-perturbative regime of QCD, that is the low-energy portion of the hadron spectrum, requires the adoption of theoretical methods more frequently applied to other, more conventional, quantum many body systems, like the atomic nucleus, solid state systems, etc. In this work we have adopted, as a first step, the well-known BCS method to describe correlations between pairs of quarks and the associated ground state. Going beyond the BCS method would imply the inclusion of correlations by means of the TDA or RPA approximations. Since, we are interested in analyzing the role…
A new consistent analysis of the renormalized proton--neutron quasiparticle random phase approximation based on the simultaneous recalculation of the one--body density matrix and the pairing tensor has been used to study the double beta decay. We demonstrated that inclusion of the quasiparticle correlations at the BCS level reduces the ground state correlations in the particle--particle channel of the proton--neutron interaction. We also simplified the RQRPA equations significantly obtaining a low--dimensioned set of linear equations for the quasiparticle densities. The formalism was applied t…
S-wave pairing in neutron matter is studied within an extension of correlated basis function (CBF) theory to include the strong, short range spatial correlations due to realistic nuclear forces and the pairing correlations of the Bardeen, Cooper and Schrieffer (BCS) approach. The correlation operator contains central as well as tensor components. The correlated BCS scheme of Ref. [Nucl. Phys. A363 (1981) 383], developed for simple scalar correlations, is generalized to this more realistic case. The energy of the correlated pair condensed phase of neutron matter is evaluated at the two--body or…
We consider the development of Cooper pairs in a self-consistent Hartree Fock mean field for the even Sm isotopes. Results are presented at the level of a BCS treatment, a number-projected BCS treatment and an exact treatment using the Richardson ansatz. While projected BCS captures much of the pairing correlation energy that is absent from BCS, it still misses a sizable correlation energy, typically of order $1 MeV$. Furthermore, because it does not average over the properties of the fermion pairs, the exact Richardson solution permits a more meaningful definition of the Cooper wave function …
We develop a relativistic mean field (RMF) description of deformed nuclei with the pairing correlations in the BCS approximation. The treatment of the pairing correlations for nuclei with the Fermi surface being close to the threshold of unbound states needs a special attention. To this end, we take the delta function interaction for the pairing interaction with the hope to pick up those states with the wave function being concentrated in the nuclear region and perform the standard BCS approximation for the single particle states generated by the RMF theory with deformation. We apply the RMF +…
We discuss the HFB equations in coordinate representation,a suitable method for handling the full effects of the continuous quasiparticle spectrum. We show how the continuum HFB equations can be solved with the correct asymptotic conditions instead of the discretization conditions which are commonly used in the literature. The continuum HFB method is illustrated with a model where the mean field and pairing field have simple forms. The relationship with the continuum Hartree-Fock-BCS (HF-BCS) approximation is also discussed. Realistic HFB and HF-BCS calculations based on Skyrme interactions ar…
Understanding the properties of drip line nuclei requires to take into account the correlations with the continuum spectrum of energy of the system. This paper has the purpose to show that the continuum single particle level density is a convenient way to consider the pairing correlation in the continuum. Isospin mean-field and isospin pairing strength are used to find the Bardeen-Cooper-Schrieffer (BCS) and Lipkin-Nogami (LN) approximate solutions of the pairing Hamiltonian. Several physical properties of the whole chain of the Tin isotope, as gap parameter, Fermi level, binding energy, and o…
The dineutron correlations and the crossover from superfluidity of neutron Cooper pairs in the $^1S_0$ pairing channel to Bose-Einstein condensation (BEC) of dineutron pairs in both symmetric and neutron matter are studied within the relativistic Hartree-Bogoliubov theory, with the effective interaction PK1 of the relativistic mean-field approach in the particle-hole channel and the finite-range Gogny force in the particle-particle channel. The influence of the pairing strength on the behaviors of dineutron correlations is investigated. It is found that the neutron pairing gaps at the Fermi su…
Recent calculations with an effective isospin dependent contact interaction show the possibility of the crossover from superfluidity of neutron Cooper pairs in $^1S_0$ pairing channel to Bose-Einstein condensation (BEC) of di-neutron bound states in dilute nuclear matter. The density and spin correlation functions are calculated for a di-neutron condensate in asymmetric nuclear matter with the aim to find the possible features of the BCS-BEC crossover. It is shown that the zero-momentum transfer spin correlation function satisfies the sum rule at zero temperature. In symmetric nuclear matter, …
We develop a perturbative model to treat the off-diagonal components in the Hartree-Fock-Bogoliubov (HFB) transformation matrix, which are neglected in the BCS approximation. Applying the perturbative model to a weakly bound nucleus $^{84}$Ni, it is shown that the perturbative approach reproduces well the solutions of the HFB method both for the quasi-particle energies and the radial dependence of quasi-particle wave functions. We find that the non-resonant part of the continuum single-particle state can acquire an appreciable occupation probability when there exists a weakly bound state close…
The influence of short-range correlations on the spectral distribution of neutrons is incorporated in the solution of the gap equation for the ${}^3P_2-{}^3F_2$ coupled channel in pure neutron matter. This effect is studied for different realistic interactions including one based on chiral perturbation theory. The gap in this channel vanishes at all relevant densities due to the treatment of these correlations. We also consider the effect of long-range correlations by including polarization terms in addition to the bare interaction which allow the neutrons to exchange density and spin fluctuat…
We consider inter- and intra-species pairing interactions in an asymmetrical Fermi system. Using equation of motion method, we obtain coupled mean-field equations for superfluid gap functions and population densities. We construct a phase diagram across BCS-BEC regimes. Due to intra-species correlations, the BCS singlet superfluid state can sustain finite polarizations, $P$. For larger $P$, we find phase separations in BCS and BEC regimes. A superfluid phase exists for all $P$ deep in BEC regime. Our results may apply to pairing in ultracold fermions, nuclear and quark matter physics.
The Bardeen-Cooper-Schrieffer (BCS) mean-field theory of the pairing interaction breaks down for nuclei and ultra-small metallic grains (nanoparticles). Finite-temperature pairing correlations in such finite-size systems can be calculated beyond the BCS theory in an auxiliary-field Monte Carlo approach. We identify thermal signatures of pairing correlations in both nuclei and nanoparticles that depend on the particle-number parity.
Strongly correlated Fermi systems with pairing interactions become superfluid below a critical temperature $T_c$. The extent to which such pairing correlations alter the behavior of the liquid at temperatures $T > T_c$ is a subtle issue that remains an area of debate, in particular regarding the appearance of the so-called pseudogap in the BCS-BEC crossover of unpolarized spin-$1/2$ nonrelativistic matter. To shed light on this, we extract several quantities of crucial importance at and around the unitary limit, namely: the odd-even staggering of the total energy, the spin susceptibility, the …
We study the correlation energy associated with the pair fluctuations in BCS theory. We use a schematic two-level pairing model and discuss the behavior of the correlation energy across shell closures, including the even-odd differences. It is shown that the random phase approximation (RPA) to the ground state energy reproduces the exact solutions quite well both in the normal fluid and in the superfluid phases. In marked contrast, other methods to improve the BCS energies such as Lipkin-Nogami method, only work when the strength of a pairing interaction is large. We thus conclude that the RPA…
Pairing correlations and their influence on nuclear properties has been studied within BCS model. Using this theoretical model with inclusion of pairing interaction between nucleons, nuclear level density and entropy of Pd(105,106,107) have been extracted. The results well coincide with the empirical values of the nuclear level densities obtained by Oslo group. Then the entropy excess of Pd107 and Pd105 compared to Pd106 as a function of the temperature has been studied. Also the role of neutron and proton system in entropy excess have been investigated by the using of the entropy excess ratio…
Based on the relativistic Hartree-Bogoliubov theory, the influence of the pairing interaction strength on the di-neutron correlations and the crossover from superfluidity of neutron Cooper pairs in the $^{1}S_{0}$ channel to Bose-Einstein condensation of di-neutron pairs is systematically investigated in the nuclear matter. The bare nucleon-nucleon interaction Bonn-B is taken in the particle-particle channel with an effective factor to simulate the medium effects and take into account the possible ambiguity of pairing force, and the effective interaction PK1 is used in the particle-hole channe…
We study the isospin-singlet neutron-proton pairing in bulk nuclear matter as a function of density and isospin asymmetry within the BCS formalism. In the high-density, weak-coupling regime the neutron-proton paired state is strongly suppressed by a minor neutron excess. As the system is diluted, the BCS state with large, overlapping Cooper pairs evolves smoothly into a Bose-Einstein condensate of tightly bound neutron-proton pairs (deuterons). In the resulting low-density system a neutron excess is ineffective in quenching the pair correlations because of the large spatial separation of the d…
Neutral current quasielastic (anti)neutrino scattering cross sections on a $^{12}$C target are analyzed using a realistic spectral function $S(p,E)$ that gives a scaling function in accordance with the ($e,e'$) scattering data. The spectral function accounts for the nucleon-nucleon (NN) correlations by using natural orbitals (NOs) from the Jastrow correlation method and has a realistic energy dependence. The standard value of the axial mass $M_A= 1.032$ GeV is used in all calculations. The role of the final-state interaction (FSI) on the spectral and scaling functions, as well as on the cross …
We have performed a systematic study of the ground state for even-even 1,002 nuclei in which 28 octupole-deformed nuclei are found. The interplay between the spatial deformation and the pairing correlation, plays important roles in nuclear structure. Our model is based on the Skyrme Hartree-Fock plus BCS model represented in the three-dimensional Cartesian coordinate space which can describe any kind of nuclear shape. The quadrupole and octupole deformed nuclei appear in the mass region with characteristic neutron and proton numbers which are consistent with previous studies. In our results, t…
Two-neutron correlation in the $^1S$ channel in quasi two-dimensional (2D) neutron systems at zero temperature is studied by means of the BCS theory with finite-range effective nuclear forces. The dineutron correlation in low density neutron systems confined in an infinite slab is investigated in a simplified model that neutron motion of one direction is frozen. When the slab is thin enough, two neutrons form a tightly bound dineutron with a small size in the quasi-2D system, and a Bose dineutron gas is found in low density limit. With increase of Fermi momentum, the neutron system changes fro…
Recently it has been demonstrated, considering Ni and Ca isotopes as prototypes, that the relativistic mean-field plus BCS (RMF+BCS) approach wherein the single particle continuum corresponding to the RMF is replaced by a set of discrete positive energy states for the calculation of pairing energy provides a good approximation to the full relativistic Hartree-Bogoliubov (RHB) description of the ground state properties of the drip-line neutron rich nuclei. The applicability of RMF+BCS is essentially due to the fact that the main contribution to the pairing correlations is provided by the low-ly…
We investigate the quartet correlations in four-component fermionic systems at the thermodynamic limit within a variational many-body theory. The Bardeen-Cooper-Schrieffer (BCS)-type variational wave function is extended to the systems with the coexistence of pair and quartet correlations at zero temperature. Special attention is paid to the application of the present framework to an alpha-particle condensation in symmetric nuclear matter, where the coexistence of deuteron and alpha condensations is anticipated. We also discuss how physical properties, such as quasiparticle dispersion, can be …
In this note, a novel BCS-type formalism is constructed in the framework of a schematic QCD inspired quark model, having in mind the description of color symmetrical superconducting states. The physical properties of the BCS vacuum (average numbers of quarks of different colors) remain unchanged under an arbitrary color rotation. In the usual approach to color superconductivity, the pairing correlations affect only the quasi-particle states of two colors, the single particle states of the third color remaining unaffected by the pairing correlations. In the theory of color symmetrical supercond…
Growth of quartet correlations in neutron-rich tellurium isotopes within quartet Bardeen-Cooper-Schrieffer theory
RIKEN Nishina Center for Accelerator-Based Science（RIKEN理化学研究所西村加速器科学中心）；Center for Mathematics and Physics, University of Aizu（会津大学数理物理中心）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）；Quark Nuclear Science Institute, The University of Tokyo（东京大学夸克核科学研究所）
We study theoretically the di-neutron spatial correlations and the crossover from superfluidity of neutron Cooper pairs in the $^1S_0$ pairing channel to Bose-Einstein condensation (BEC) of di-neutron pairs for both symmetric and neutron matter in the microscopic relativistic pairing theory. We take the bare nucleon-nucleon interaction Bonn-B in the particle-particle channel and the effective interaction PK1 of the relativistic mean-field approach in the particle-hole channel. It is found that the spatial structure of neutron Cooper pair wave function evolves continuously from BCS-type to BEC-…
Based on previous works, analytical calculational procedures for dealing with the strongly interacting fermions ground state are further developed through a medium dependent potential in terms of the Bethe-Peierls contact interaction model. The methods are exact in the unitary limit regime and they lead to the self-consistent equations analogous to the Hartree ones. The single particle energy spectrum rearrangement effects on the thermodynamics due to the Hugenholtz-van Hove theorem constraint are addressed. These effects lead to an additional instantaneous correlation potential contribution t…
We present an exploratory study of structure of 13C, 15C, 17C and 19C, showing that the simple one-quasiparticle projected BCS (PBCS) model is capable to account for several important properties of these nuclei. Next we discuss the importance of the Pauli Principle in the particle-core models of normal-parity states in 13C and 11Be. This is done by considering the pairing interaction between nucleons moving in an over-all deformed potential. To assess the importance of pairing correlations in these light nuclei we use both the simple BCS and the PBCS approximations. We show that the Pauli Prin…
The antisymmetrized quasi-cluster model (AQCM) was proposed to describe α-cluster and $jj$-coupling shell models on the same footing. In this model, the cluster-shell transition is characterized by two parameters; $R$ representing the distance between α clusters and α describing the breaking of α clusters, and the contribution of the spin-orbit interaction, very important in the $jj$-coupling shell model, can be taken into account starting with the α cluster model wave function. Not only the closure configurations of the major shells, but also the subclosure configurations of the $jj$-coupling…
Charge-current quasielastic (CCQE) (anti)neutrino scattering cross sections on a $^{12}$C target are analyzed using a spectral function $S(p,{\cal E})$ that gives a scaling function in accordance with the ($e,e'$) scattering data. The spectral function accounts for the nucleon-nucleon (NN) correlations, it has a realistic energy dependence and natural orbitals (NO's) from the Jastrow correlation method are used in its construction. In all calculations the standard value of the axial mass $M_A= 1.032$ GeV/c$^2$ is used. The results are compared with those when NN correlations are not included, …
The equation for the gap parameter represents the main equation of the pairing theory of superconductivity. Although it is formally defined through a single-particle property, physically it reflects the pairing correlations between opposite-spin fermions. Here, we exploit this physical connection and cast the gap equation in an alternative form which explicitly highlights these two-particle correlations, by showing that it is equivalent to a Hugenholtz-Pines condition for fermion pairs. At a formal level, a direct connection is established in this way between the treatment of the condensate fr…
We investigate the spatial structure of the two-neutron wave function in the Borromean nucleus $^{11}$Li, using a three-body model of $^9$Li $+n+n$, which includes many-body correlations stemming from the Pauli principle. The behavior of the neutron pair at different densities is simulated by calculating the two-neutron wave function at several distances between the core nucleus $^9$Li and the center of mass of the two neutrons. With this representation, a strong concentration of the neutron pair on the nuclear surface is for the first time quantitatively established for neutron-rich nuclei. T…
We present a truly canonical theory of superconductivity in ultrasmall metallic grains by variationally optimizing fixed-N projected BCS wave-functions, which yields the first full description of the entire crossover from the bulk BCS regime (mean level spacing $d \ll$ bulk gap $\tildeΔ$) to the ``fluctuation-dominated'' few-electron regime ($d\gg\tildeΔ$). A wave-function analysis shows in detail how the BCS limit is recovered for $d\ll \tilde Δ$, and how for $d \gg \tilde Δ$ pairing correlations become delocalized in energy space. An earlier grand-canonical prediction for an observable parit…
Fermion systems with more than two components can exhibit pairing condensates of much more complex structure than the well-known single BCS condensate of spin-up and spin-down fermions. In the framework of the exactly solvable SO(8) Richardson-Gaudin model with SU(4)-symmetric Hamiltonians, we show that the BCS approximation remains valid in the thermodynamic limit of large systems for describing the ground state energy and the canonical and quasiparticle excitation gaps. Correlations beyond BCS pairing give rise to a spectrum of collective excitations, but these do not affect the bulk energy …
We study the static correlation functions of the Richardson pairing model (also known as the reduced or discrete-state BCS model) in the canonical ensemble. Making use of the Algebraic Bethe Ansatz formalism, we obtain exact expressions which are easily evaluated numerically for any value of the pairing strength up to large numbers of particles. We provide explicit results at half-filling and extensively discuss their finite-size scaling behavior.
In this work, a new functional is introduced to treat pairing correlations in finite many-body systems. Guided by the projected BCS framework, the energy is written as a functional of occupation numbers. It is shown to generalize the BCS approach and to provide an alternative to Variation After Projection framework. Illustrations of the new approach are given for the pairing Hamiltonian for various particle numbers and coupling strengths. In all case, a very good agreement with the exact solution is found.
