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 The SC-EDF functional SC-EDF MR-EDF non-regularized No regularization possible SC-EDF is SR-EDF and MR-EDF

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

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

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

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

8 Achievements of this work 1.MR-EDF when regularized can be viewed as a DFT. 2.A modification of the regularization makes possible to associate MR-EDF with a correlated auxiliary state. 3.SC-EDF formalism allows to use density dependent interaction. DFT Here Question : Is it possible to express the energy as a function of ρ 1 [N] ? It is already the case of the BCS theory :

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

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

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

12 Resummation into a compact functional All contributions can be approximately summed to give: with BCS E HF - E Coupling E HF - E Coupling 4 particles16 particles44 particles BCS Exact New func.

13 Applications : more insights Hupin et al. PRC83 (2011) What is required for realistic situations in Condensed Matter and Nuclear physics ? 1.Can be applied to odd system. 2.Functional applicable to small and large systems while reproducing the desired physics (here the finiteness of systems). 3.The single particle spectra upon which is applied the functional should not be constrained. Richardson model Any spectra

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

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

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

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

18 Conclusions and Perspectives Restoration of particle number in MR-EDF  Reanalyzed the MR-EDF method with its regularization.  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 SC-EDF

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 (ρ α ).  Refitting of the pairing functional.  Application to others symmetries ?  Neutron / Proton pairing with finite size correction. Pairing energy N/Z Exp BCS/ HFB

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.  Thermodynamics and dynamics of finite systems.  Quantum phase transition exploration. Large N Odd even Random spectra