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CEA/SPhN, Saclay, France NSCL, Michigan State University, USA

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Presentation on theme: "CEA/SPhN, Saclay, France NSCL, Michigan State University, USA"— Presentation transcript:

1 CEA/SPhN, Saclay, France NSCL, Michigan State University, USA
Ab initio-driven nuclear energy density functional method Thomas DUGUET CEA/SPhN, Saclay, France IKS, KU Leuven, Belgium NSCL, Michigan State University, USA T. Duguet, A. Signoracci, J. Phys. G: Nucl. Part. Phys. 44 (2016) T. Duguet, M. Bender, J.-P. Ebran, T. Lesinski, V. Somà, Eur. Phys. J. A51 (2015) 162 P. Arthuis, A. Tichai, H. Hergert, R. Roth, J. P. Ebran, T. Duguet, in preparation B. Bally, T. Duguet, , arXiv: and arXiv: Bridging nuclear ab-initio and energy-density-functional theories IPN, Orsay, Oct

2 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions

3 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions The tower of effective (field) theories and the emergence of nuclear phenomena ESNT workshop, Saclay, January 2017

4 From a plurality of nuclear models…
→ Useful to identify relevant d.o.f and symmetries → Decent account of phenomena based on employed d.o.f BUT ① No systematic improvement towards accuracy ② No proper understanding of their intrinsic limitations ③ No clear path to connect them Conventional energy density functional method Liquid drop and mic-mac models Collective and algebraic models Conventional shell model Landau theory Cluster models ab initio methods based on conventional interactions Tension between reductionist and emerging viewpoints not appropriately articulated

5 …to an arborescence of nuclear effective (field) theories
Rationale of effective theories ① Identify appropriate energy scales / d.o.f / symmetries ② All interactions complying with symmetries are compulsory ③ Naturalness provides power counting (+ possible fine tuning) ④ Fix LECs from data or from underlying effective theory Reductionist description Emerging phenomena Effective theory-based energy density functional method Effective (field) theory for emerging symmetry breaking Effective theory-based shell model Halo effective field theory Effective theory-based Laudau theory Pion-less effective field theory ab initio methods based on c potentials c-Effective field theory in 2N, 3N… AN sectors Quantum Chromodynamics Appropriate epistemic scheme to articulate reductionist and emerging viewpoints

6 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions

7 Key concepts and shortcomings of current EDF method
Order parameters (SU(2), U(1)…) ➟ Use of product states essentially following an horizontal expansion ➟ Symmetry breaking (SR) = non-zero order parameters ➟ Symmetry restoration + GCM (MR) = fluctuation of phase + norm of order parameters ➟ Key ingredient = off-diagonal energy and norm kernels Empirically correlated -Density-dependent « operator » -Density (matrix) functional Uncorrelated Phenomenological mean-field form (See K. Bennaceur) ✪ Shortcomings ➟ MR calculations with density-dependent operator/density functional are ill defined ➟ Calculations with effective operator + mean-field kernels potentially lack flexibility ➟ Lack coupling to individual excitations/diabatic effects/vertical configurations in practice

8 Has made the fortune of the EDF method
Ab initio vs EDF approaches In which ways the EDF approach can be rooted into ab initio many-body methods? ✪ Pseudo data to fix param. of EDF kernels ➟ Complement data in unknown regions ➟ H+many-body must be mature enough ➟ NM EOS, static response… (cf. A. Gezerlis) Not yet mature enough in mid-mass nuclei that constitute the natural overlap with EDF method ✪ Analytical guidance to build EDF kernels ➟ Low-density neutron matter (cf. D. Lacroix) ➟ Must be based on same key principles ➟ Symmetry breaking and restoration Has made the fortune of the EDF method

9 EDF method in one slide - Focus on U(1) symmetry
Off-diagonal density matrix Gauge rotation Bogoliubov state Horizontal set of gauge-rotated Bogoliubov states ✪ Off-diagonal EDF kernels and their parametrisations Empirical choices break Pauli principle (self interaction/pairing) [Lacroix et al. 2009, Bender et al. 2009, Duguet et al. 2009] All h(j) are functionals of <= off-diagonal Wick theorem Pure (effective and phenomenological) mean-field kernels Norm kernel Energy kernel Classic choices 1970s’ 1990s’ ✪ SR implementation = HFB-like particle-number breaking scheme Diagonal kernels ✪ MR implementation = particle number restoration scheme Off-diagonal kernels Ill-defined for unappropriate h(j) [Dobaczewski et al. 2007] 2020s’?

10 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions Symmetry broken and restored coupled-cluster theory: II. Global gauge symmetry and particle number T. Duguet, A. Signoracci, J. Phys. G: Nucl. Part. Phys. 44 (2016) Ab initio-driven nuclear energy density functional method T. Duguet, M. Bender, J.-P. Ebran, T. Lesinski, V. Somà, Eur. Phys. J. A51 (2015) 162 Bogoliubov many-body perturbation theory calculations of open-shell nuclei P. Arthuis, A. Tichai, H. Hergert, R. Roth, J. P. Ebran, T. Duguet, in preparation

11 Exact diagonal kernels Exact off diagonal kernels
Correlated/improvable EDF kernels – Focus on U(1) symmetry Based on H = Ab initio Heff = EDF Recently -implemented -proposed Exact diagonal kernels Must be designed accordingly [Somà et al. 2011] [Signoracci et al. 2014] [Duguet, Signoracci, 2016] Recently proposed Exact off diagonal kernels [Duguet 2015] [Duguet, Signoracci 2016] Guidance of EDF so far Our proposal = BMBPT(2,3)-based off-diagonal EDF kernels

12 BMBPT of off-diagonal kernels – Focus on U(1) symmetry
✪ Normal-ordered grand potential (work on Fock space) Wij i creation/j annihilation QP operators ✪ Correlated off-diagonal kernels Evolution operator in imaginary time Linked-connected kernel of the operator = All connected diagrams linked to the operator Displays naturally terminating CC expansion Generalized BMBPT/CC 4 off-diagonal propagators in qp + Off-diagonal Wick theorem Diagrammatic, full expressions etc; see [T. Duguet, A. Signoracci, JPG 44 (2016) , P. Arthuis et al., in preparation]

13 From quartic down to quadratic polynomial in R(j) + no dep. in Ek
Off-diagonal kernels at BMBPT(2) – Focus on U(1) symmetry ✪ Off-diagonal density matrix ✪ Non-unitary transformation Angle-dependent part Polynomial in R--(j) Diagonal part ✪ Off-diagonal linked/connected grand-potential kernel at BMBPT(2) Mean-field Functional of both R(j) and Ek ✪ Mean-field kernel  Standard guidance of EDF much richer for given Heff qp/vertical mixing in energy kernel From quartic down to quadratic polynomial in R(j) + no dep. in Ek

14 Off-diagonal kernels at BMBPT(2) – Focus on U(1) symmetry
✪ Diagonal kernel at j=0  Diagonal BMBPT(2) Recover diagonal BMBPT(2) at j=0 (see P. Arthuis’ talk tomorrow for (much) more) ✪ Norm kernel First order ODE involving linked/connected kernel of A 2nd order correction Mean field Closed-form expression at any order Consistent correction Absent from empirical EDF

15 First step: MBPT calculations in closed-shell nuclei
Based on H = Ab initio Heff = EDF Ab initio calculations with low-momentum interactions [Tichai et al. 2016] MBPT(1,2,3) kernels

16 Second step: BMBPT calculations in open-shell nuclei
Based on H = Ab initio Heff = EDF Ab initio calculations with low-momentum interactions [Arthuis et al. unpublished] BMBPT(1,2,3) kernels

17 Second step: BMBPT calculations in open-shell nuclei
✪ Proof-of-principle BMBPT(1,2) calculations of 16-24O ➟ Hamiltonian: chiral N3LO 2N (500 MeV) + N2LO 3N (400 MeV), SRG-evolved to 2.0 fm-1 ➟Each order is typically a factor ~10 more CPU intensive (BMBPT(3) will remain well below BCCSD) Exp AO ~0.5 CPU hr ~200 CPU hrs + ~2000 CPU hrs 13 HO shells (hw = 20 MeV) ~500 CPU hrs ~20 CPU hrs [Arthuis, Tichai, Roth, Ebran, Duguet, 2017 unpublished] ➟ Systematic BMBPT(1,2,3) calculations to come « Regularized EDF generator » K. Bennaceur, J. Dobaczewski et al. See, e.g., arXiv: ✪ Next steps ►Implement with Heff and proceed to (re)fit at BMBPT(1,2,3) levels ►Implement off-diagonal kernels and perform PNR-BMBPT(1,2,3) calculations

18 Third step: PNR-BMBPT calculations in open-shell nuclei
Based on H = Ab initio Heff = EDF BMBPT(1,2,3) kernels Ab initio calculations with low-momentum interactions [Arthuis et al. to come]

19 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions

20 EDF scheme and workplan (exemplified for SU(2) here)
✪ Three-fold expansion Heff MR mixing = static correlations Kernels expansion = dynamical correlations Share the load Fixed Heff empirically considered here Towards an effective theory? Need consistent expansion eventually ✪ Short-term plan Deal with U(1) only [SU(2) later] Study safe character vs truncation Validate many-body method for Hpairing Heff = effective Gaussian vertex P. Arthuis Ph.D thesis Sup: J.-P. Ebran, T. Duguet Collab: A. Tichai

21 Outline Generic statements and motivations
Basics and shortcomings of current EDF method Many-body expansion of off-diagonal energy&norm kernels Resulting EDF scheme and workplan Conclusions

22 Conclusive remarks ✪ Evolution towards low-order BMBPT off-diagonal energy and norm kernels ➟ Dynamical correlations through vertical expansion = qp energy and density matrix functionals ➟ Energy and norm kernels must be treated consistently ✪ Implementions ➟ Diagonal, S.R., BMBPT(1,2,3) spherical code complete; calculations to come ➟ To be implemented and fitted with appropriate Heff at BMBPT(1,2,3) levels ►Optimize balance between complexity of many-body expansions and of Heff ➟ To be implemented in PNR calculations ➟ To be generalized to GCM-type horizontal mixing ✪ Norm kernels ➟ Flexible alternative to Pfaffian for arbitrary Bogoliubov states ➟ Method applicable to norm kernels beyond mean-field level

23 Vertical and horizontal expansions
✪ Vertical expansion ➟ Mix of orthogonal product states differing via non-collective (quasi-)ph excitations over one vacuum ➟ Efficienty capture « dynamical » correlations (in quantum chemistry language)  ➟ Dominant in ab initio philosophy (NCSM, MBPT, CC, IMSRG, D-SCGF…) ➟ Usually implemented on top of symmetry-conserving vacuum… but not always (e.g. G-SCGF, BCC)  ✪ Horizontal expansion ➟ Mix of non-orthogonal vacua differing via collective transformations ➟ Efficienty capture « non-dynamical » correlations associated with near degeneracies  ➟ Dominant in EDF philosophy (i.e. adiabatic GCM + symmetry restoration)   ➟ Inherently associated with symmetry breaking and restoration  

24 Background ✪ Correlated off-diagonal norm kernels within PNR-BCC and PNR-BMBPT theories Computable in closed form 1st order ODE Linked/connected kernel of A Involves ➟ First order ➟ Second order Depends on the dynamics Analytically scrutinized in On the norm overlap between many-body states. II. Correlated off-diagonal norm kernel, P. Arthuis, B. Bally, T. Duguet, in preparation

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