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) 015103 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:1704.05324 and arXiv:1706.04553 Bridging nuclear ab-initio and energy-density-functional theories IPN, Orsay, Oct. 2-6 2017

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

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 http://esnt.cea.fr/Phocea/Page/index.php?id=71

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

…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

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

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

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

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’?

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) 015103 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

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

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) 015103, P. Arthuis et al., in preparation]

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

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

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

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

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:1611.09311 ✪ 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

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]

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

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

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

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

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  

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