On the formulation of a functional theory for pairing with particle number restoration

1. On the formulation of a functional theory for pairing with particle number restoration Guillaume Hupin GANIL, Caen FRANCE • Collaborators : • M. Bender (CENBG) • D. Lacroix (GANIL) • D. Gambacurta (GANIL)

2. Brief summary of the SC-EDF functional SR-EDF and MR-EDF • SC-EDF • MR-EDFnon-regularized SC-EDF is No regularization possible The SC-EDF functional

3. Practice Expressing the 2-body densities a function of xi BCS occupation probabilities with with Or recurrence relation using

4. Variation After Projection in EDF (VAP) Pairing Hamiltonian Δε Single particle energy Correlation energy VAP Exact SR-EDF broken symmetry minimum E (MeV) Coupling strength PAV • Better reproduction of the energy. • Correct finite size effects (no threshold). • Optimization of the auxiliary state . • Flexibility of EDF. Threshold effects SC-EDF minimum VAP BCS Motivations

5. Variational principle With the SC-EDF functional Applied using the parameters of auxiliary state K. Dietrich et al. Phys. Rev. 135 (1964) MV Stoitsov et al. PRC (2007) …

6. Numerical methods New set of φi 1 - Imaginary time step method to diagonalized MF Hamiltonian. Solve minimization with gradient method 2 - Gradient method to solve the secular equations with respect to vi. New set of vi and MF potentials Preliminary : simplification of F(vk) to reduce the numeric to a minimum search. Evolve imaginary time 76Kr Convergence VAP VAP BCS BCS

7. Functional Theory : flexibility of SC-EDF Functional theory allows to modify the expression of the energy. → 1 Mean-Field. VAP VAP BCS BCS Original VAP VAP with ninj • Solves the BCS threshold problem. • Avoids complex numeric. • Easy to implement in existing Mean-Field codes.

8. Achievements of this work MR-EDF when regularized can be viewed as a DFT. A modification of the regularization makes possible to associate MR-EDF with a correlated auxiliary state. SC-EDF formalism allows to use density dependent interaction. DFT Here ? Question : Is it possible to express the energy as a function of ρ1 [N] It is already the case of the BCS theory :

9. Density Matrix Functional Theory (DMFT)- alternative path Focus on one body observables DFT DMFT Describes at the minimum of the functional energy T.L. Gilbert PRB 12 (1975) Full one body observables Correlation energy Exact Recently applied in electronic systems Example : Homogenous Electron Gas (HEG) N. N. Lathiotakis et al. PRB 75 (2007)

10. DMFT from a projected BCS state Δε Single particle energy with BCS PBCS ?

11. Applications and benchmarks of the new functional ? Objective : invert into A new systematic 1/N expansion beyond BCS: BCS Exact EHF - E Exact BCS Finite size effects OK when all terms are included Lacroix and Hupin, PRB 82 (2011) Coupling

12. Resummation into a compact functional All contributions can be approximately summed to give: with BCS 4 particles 16 particles 44 particles New func. Exact EHF - E EHF - E EHF - E BCS Coupling Coupling Coupling

13. Applications : more insights What is required for realistic situations in Condensed Matter and Nuclear physics ? Can be applied to odd system. Functional applicable to small and large systems while reproducing the desired physics (here the finiteness of systems). The single particle spectra upon which is applied the functional should not be constrained. Richardson model Any spectra Hupin et al. PRC83 (2011)

14. Applications : odd systems Richardson Odd systems have been described in terms of a blocked state – the last occupied state (i) of the Fermi sea. PBCS We define the mean gap (BCS gap in thermo. limit) Exact BCS Functional Great improvement over BCS + Energy of odd systems is better reproduced Particle number

15. Applications : thermodynamic limit Finiteness of physical systems is also of interest in condensed matter Superconducting Nanoscale grains Δε=d Single particle energy G. Sierra et al. PRB 61 (2000) Parameterization of the SP energy splitting and particle number (A) : Correlation energy Exact Lacks some correlations at small number of particles + The functional does as good as the PBCS ansatz Functional BCS ~1/A Dot = odd systems

16. Applications : random single particle spectra Generate SP energy levels ? New set of φi Functional Normalized to unity the average SP splitting Solve minimization with gradient method BCS For instance New set of vi and MF potentials Evolve imaginary time In a SC scheme This functional is efficient with any SP spectra can be used with self consistent methods

17. Extension : functional for finite temperature DMFT : information reduced to one body observables • D. Gambacurta (GANIL) Gibbs free energy Entropy reduced to a set of one body observables Balian, Amer. J. Phys. (1999) Functional build from Hamiltonian finite temperature Esebbag, NPA 552 (1993)

18. Conclusions and Perspectives Restoration of particle number in MR-EDF • Reanalyzed the MR-EDF method with its regularization. SC-EDF • Proposed and alternative method that is a functional of the projected state SC-EDF. • PAV : direct use of the SC-EDF functional • VAP : variation of the functional E (MeV) PAV VAP

19. Conclusions and Perspectives • SC-EDF (↔ MR-EDF regularized) is a framework for the restoration of particle number in functional theory. • However, the SC-EDF restores the functional flexibility (ρα ). • Refitting of the pairing functional. • Application to others symmetries ? • Neutron / Proton pairing with finite size correction. Exp Pairing energy BCS/HFB N/Z

20. Conclusions and Perspectives New DMFT functional for finite size systems with pairing • Proposition. • Benchmark with exact solution of Richardson model. • Check the applicability in realistic cases. • Large N • Odd even • Random spectra • Thermodynamics and dynamics of finite systems. • Quantum phase transition exploration.