The ground state pairing correlations in finite fermionic systems are described with a high degree of accuracy within a variational approach based on a combined coupled-cluster and particle-number-projected BCS ansatz. The flexibility of this symmetry-preserving wavefunction enables a unified picture valid from weak to strong coupling, both in small and large systems. The present variational approach consistently yields an energy upper bound while operating at the same level of precision of the non-variational particle-number projected Bogoliubov-coupled-cluster theory [Phys. Rev. C 99, 044301 (2019)].
We study the structure of the number projected BCS (PBCS) wave function in the particle-hole basis, displaying its similarities with coupled clusters theory (CCT). The analysis of PBCS together with several modifications suggested by the CCT wave function is carried out for the exactly solvable Richardson model involving a pure pairing hamiltonian acting in a space of equally-spaced doubly-degenerate levels. We point out the limitations of PBCS to describe the non-superconducting regime and suggest possible avenues for improvement.
The particle number projected BCS (PBCS) approximation is tested against the exact solution of the SO(5) Richardson-Gaudin model for isovector pairing in a system of non-degenerate single particle orbits. Two isovector PBCS wave functions are considered. One is constructed as a single proton-neutron pair condensate, while the other corresponds to a product of a neutron pair condensate and a proton pair condensate. The PBCS equations are solved using a recurrence method and the analysis is performed for systems with an equal number of neutrons and protons distributed in a sequence of equally sp…
We analyze the accuracy of BCS-based approximations for calculating correlation energies and odd-even energy differences in 2-component fermionic systems with a small number of pairs. The analysis is focused on comparing BCS and projected BCS treatments with the exact solution of the pairing Hamiltonian, considering parameter ranges appropriate for nuclear pairing energies. We find that the projected BCS is quite accurate over the entire range of coupling strengths in spaces of up to about 20 doubly degenerate orbitals. It is also quite accurate for two cases we considered with a more realisti…
The mean values of a many-body Hamiltonian including a proton-neutron pairing term and matrix elements of one-, two- and four-body operators within a basis of particle number projected BCS states, are analytically expressed in terms of a single function Q(N) depending on the number of particles, $N$. The function Q(N) is calculated using a recursion in $N$ in which the shells and the BCS angles are kept the same for any step of iteration. An illustrative example is numerically considered in a restricted single particle space. Some specific features for the standard BCS, the projection after va…
We present a similarity transformation theory based on a polynomial form of a particle-hole pair excitation operator. In the weakly correlated limit, this polynomial becomes an exponential, leading to coupled cluster doubles. In the opposite strongly correlated limit, the polynomial becomes an extended Bessel expansion and yields the projected BCS wavefunction. In between, we interpolate using a single parameter. The effective Hamiltonian is non-hermitian and this Polynomial Similarity Transformation Theory follows the philosophy of traditional coupled cluster, left projecting the transformed …
The projected shell model analysis is carried out using the triaxial Nilsson+BCS basis. It is demonstrated that, for an accurate description of the moments of inertia in the transitional region, it is necessary to take the triaxiality into account and perform the three-dimensional angular-momentum projection from the triaxial Nilsson+BCS intrinsic wavefunction.
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.
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 propose a scheme to perform the variational principle directly on the coherent pair condensate (VDPC). The result is equivalent to that of the so-called variation after particle-number projection, but now the particle number is always conserved and the time-consuming projection is avoided. This work considers VDPC+BCS. We derive analytical expressions for the average energy and its gradient in terms of the coherent pair structure. In addition, we give analytically the pair structure at the energy minimum, and discuss its asymptotic behavior away from the Fermi surface, which looks quite sim…
We present in this paper the general framework of a method which permits to restore the rotational and particle number symmetries of wave functions obtained in Skyrme HF+BCS calculations. This restoration is nothing but a projection of mean-field intrinsic wave functions onto good particle number and good angular momentum. The method allows also to mix projected wave functions. Such a configuration mixing is discussed for sets of HF+BCS intrinsic states generated in constrained calculations with suitable collective variables. This procedure gives collective states which are eigenstates of the …
The BCS and/or HFB theories are extended by treating the effect of four quasi-particle states perturbatively. The approach is tested on the pairing hamiltonian, showing that it combines the advantage of standard perturbation theory valid at low pairing strength and of non-perturbative approaches breaking particle number valid at higher pairing strength. Including the restoration of particle number, further improves the description of pairing correlation. In the presented test, the agreement between the exact solution and the combined perturbative + projection is almost perfect. The proposed me…
While coupled cluster theory accurately models weakly correlated quantum systems, it often fails in the presence of strong correlations where the standard mean-field picture is qualitatively incorrect. In many cases, these failures can be largely ameliorated by permitting the mean-field reference to break physical symmetries. Symmetry-broken coupled cluster, e.g. Bogoliubov coupled cluster, theory can indeed provide reasonably accurate energetic predictions, but the broken symmetry can compromise the quality of the resulting wave function and predictions of observables other than the energy. M…
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.
Generalized BCS equations which consistently include the projection on the particle-number parity are derived from a systematic variational method. Numerical solutions are given that are illustrative of ultra-small metallic grains. Compared to the BCS approximation, the pairing is slightly enhanced in even and noticeably inhibited in odd systems. In the latter case, the gap increases with T near T=0; for a suitably small pairing this can result in a reentrance phenomenon. Paramagnetic effects are evaluated as functions of the temperature and the magnetic field.
Applying a variational multiparticle-multihole configuration mixing method whose purpose is to include correlations beyond the mean field in a unified way without particle number and Pauli principle violations, we investigate pairing-like correlations in the ground states of $ ^{116}$Sn,$ ^{106}$Sn and $ ^{100}$Sn. The same effective nucleon-nucleon interaction namely, the D1S parameterization of the Gogny force is used to derive both the mean field and correlation components of nuclear wave functions. Calculations are performed using an axially symetric representation. The structure of correl…
We present the numerical code TAURUS_vap that solves the variation after particle-number projection equations for symmetry-unrestricted real Bogoliubov quasiparticle states represented in a spherical harmonic oscillator basis. The model space considered is invariant under spatial and isospin rotations but no specific set of orbits is assumed such that the code can carry out both valence-space and no-core calculations. In addition, no number parity is assumed for the Bogoliubov quasiparticle states such that the code can be used to describe even-even, odd-even and odd-odd nuclei. The variationa…
Recently we proposed a particle-number-conserving theory for nuclear pairing [Jia, Phys. Rev. C 88, 044303 (2013)] through the generalized density matrix formalism. The relevant equations were solved for the case when each single-particle level has a distinct set of quantum numbers and could only pair with its time-reversed partner (BCS-type Hamiltonian). In this work we consider the more general situation when several single-particle levels could have the same set of quantum numbers and pairing among these levels is allowed (HFB-type Hamiltonian). The pair condensate wavefunction (the HFB wav…
We construct a coherent state of q-deformed zero coupled nucleon pairs distributed in several single-particle orbits. Using a variational approach, the set of equations of qBCS theory, to be solved self consistently for occupation probabilities, gap parameter Delta, and the chemical potential lambda, is obtained. Results for valence nucleons in nuclear degenerate sdg major shell show that the strongly coupled zero angular momentum nucleon pairs can be substituted by weakly coupled q-deformed zero angular momentum nucleon pairs. A study of Sn isotopes reveals a well defined universe of (G, q) v…
We apply a variational method devised for the nuclear many--body problem to the 1-dimensional Hubbard--model with nearest neighbor hopping and periodic boundary conditions. The test wave function consist for each state out of a single Hartree--Fock determinant mixing all the sites (or momenta) as well as the spin--projections of the electrons. Total spin and linear momentum are restored by projection methods before the variation. It is demonstrated that this approach reproduces the results of exact diagonalisations for half--filled N=12 and N=14 lattices not only for the energies and occupatio…
We report a theoretical analysis of variational wave functions for the BCS pairing problem. Starting with a Jastrow-Feenberg (or, in a more recent language "fixed-node") wave function for the superfluid state, we develop the full optimized Fermi-Hypernetted Chain (FHNC-EL) equations which sum a local approximation of the parquet-diagrams. Close examination of the procedure reveals that it is essential to go beyond the usual Jastrow-Feenberg approximation to guarantee the correct stability range.
As a model for a deformed nucleus the many level pairing model (picket fence model with ~100 levels) is considered in four approximations and compared to the exact solution given by Richardson long time ago. It is found that, as usual, the number projected BCS method improves over standard BCS but that it is much less accurate than the more sophisticated many-body-approaches which are Coupled Cluster Theory (CCT) in its SUB2 version or Self-Consistent Random Phase Approximation (SCRPA).
Comparative study of quartet superfluid state: Quartet Bardeen-Cooper-Schrieffer theory and generalized Nambu-Gor'kov formalism
RIKEN Nishina Center for Accelerator-Based Science（理化学研究所西村加速器科学中心）；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）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论数学科学中心）
A mean-field model with a generalized pairing interaction that accounts for neutron-proton pairing is presented. Both the BCS as well as number-projected solutions of the model are presented. For the latter case the Lipkin-Nogami projection technique was extended to encompass the case of non-separable proton-neutron systems. The influence of the projection on various pairing phases is discussed. In particular, it is shown that number-projection allows for mixing of different pairing phases but, simultanously, acts destructively on the proton-neutron correlations. The basic implications of prot…
We investigate the coexistence of spherical, deformed and superdeformed states at low spin in S32, Ar36, Ar38 and Ca40. The microscopic states are constructed by configuration mixing of BCS states projected on good particle number and angular momentum. The BCS states are themselves obtained from Hartree-Fock BCS calculations using the Skyrme interaction SLy6 for the particle-hole channel, and a density-dependent contact force in the pairing channel. The same interaction is used within the Generator Coordinate Method to determine the configuration mixing and calculate the properties of even-spi…
The framework of relativistic self-consistent mean-field models is extended to include correlations related to the restoration of broken symmetries and to fluctuations of collective variables. The generator coordinate method is used to perform configuration mixing of angular-momentum and particle-number projected relativistic wave functions. The geometry is restricted to axially symmetric shapes, and the intrinsic wave functions are generated from the solutions of the relativistic mean-field + Lipkin-Nogami BCS equations, with a constraint on the mass quadrupole moment. The model employs a rel…
The projected random phase approximation (PRPA) for charge-exchange excitations is derived from the time-dependent variational principle. Explicit results for the unperturbed energies (including the self-energy corrections), the PRPA matrices, and the transition matrix elements are presented. The effect of the projection procedure on the two-neutrino $ββ$ decay in $^{76}Ge$ is briefly discussed.
The variational reduced density matrix theory has been recently applied with great success to models within the truncated doubly-occupied configuration interaction space, which corresponds to the seniority zero subspace. Conservation of the seniority quantum number restricts the Hamiltonians to be based on the SU(2) algebra. Among them there is a whole family of exactly solvable Richardson-Gaudin pairing Hamiltonians. We benchmark the variational theory against two different exactly solvable models, the Richardson-Gaudin-Kitaev and the reduced BCS Hamiltonians. We obtain exact numerical result…
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 …
