The UNEDF Project. Dick Furnstahl. October, Collaborators: UNEDF ( Department of Physics Ohio State University

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1 Department of Physics Ohio State University October, 2007 Collaborators: UNEDF (

2 Topics Overview DFT RG 3NF DME Appendix Possible Topics Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

3 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Outline Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

4 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Universal Nuclear Energy Density Functional

5 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS SciDAC 2 Project: Building a Universal Nuclear Energy Density Functional Understand nuclear properties for element formation, for properties of stars, and for present and future energy and defense applications Scope is all nuclei, with particular interest in reliable calculations of unstable nuclei and in reactions Order of magnitude improvement over present capabilities = precision calculations Connected to the best microscopic physics Maximum predictive power with well-quantified uncertainties [See by Bertsch, Dean, and Nazarewicz]

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7 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Other SciDAC Science at the Petascale Projects Physics (Astro): Computational Astrophysics Consortium: Supernovae, Gamma Ray Bursts, and Nucleosynthesis, Stan Woosley (UC/Santa Cruz) [$1.9 Million per year for five years] Physics (QCD): National Computational Infrastructure for Lattice Gauge Theory, Robert Sugar (UC/Santa Barbara) [$2.2 Million per year for five years] Physics (Turbulence): Simulations of Turbulent Flows with Strong Shocks and Density Variations, Sanjiva Lele (Stanford) [$0.8 million per year for five years] Physics (Petabytes): Sustaining and Extending the Open Science Grid: Science Innovation on a PetaScale Nationwide Facility, Miron Livny (U. Wisconsin) [$6.1 Million per year for five years]

8 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Universal Nuclear Energy Density Functional

9 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS The Big Picture: From QCD to Nuclei Density Functional Theory A>100 Coupled Cluster, Shell Model A<100 Exact methods A 12 GFMC, NCSM Lattice QCD Low-mom. interactions Chiral EFT interactions (low-energy theory of QCD) QCD Lagrangian

10 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS UNEDF Mission 1 Find an optimal energy density functional using all knowledge of nucleonic, Hamiltonian, and basic nuclear properties 2 Apply DFT and its extensions to validate the functional using all the available relevant nuclear structure data 3 Apply the validated theory to properties of interest that cannot be measured, in particular the properties needed for reaction theory, such as cross sections relevant to NNSA programs

11 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Major Research Areas Ab initio structure Nuclear wf s from microscopic NN N NCSM/FCI, CC, GFMC/AFMC AV18/ILx, chiral EFT V low k Ab initio energy functionals DFT from microscopic NN N V low k MBPT DME Cold atoms superfluid LDA+ as prototype for nuclear DFT DFT applications Technology to calculate observables Skyrme HFB+ for all nuclei (solvers) Fitting the functional (e.g., correlation analysis) DFT extensions Long-range correlations, excited states LACM, GCM, TDDFT, QRPA, CI Reactions Low-energy reactions, fission,...

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13 Topics Overview DFT RG 3NF DME Appendix UNEDF SciDAC Areas CS Physics/Computer Science Partnerships Partnerships have been formed consisting of computer scientists and applied mathematicians linked with specific physicists. In each partnership, the mathematician/computer scientist is addressing a research topic in order to remove a specific barrier to progress on the computational/algorithmic physics side. FCI, NCSM linear algebra GFMC ADLB (load balancing) Gridded DFT solver wavelet basis; sinc basis Oscillator DFT solver efficiency and parallel execution Coupled Cluster NNN interactions Coupled Channel efficiency and parallel execution (proposed)

14 NDFT COMPUTATIONAL STRATEGY UNEDF Optimizer (linear regression?) Mean Field even even nuclei odd mass nuclei General purpose (symmetry free) Kohn Sham Fast axial Kohn Sham (HFB) code (HFB) code even even mass table Mean Field + Correlations all nuclei all nuclei Projected Kohn Sham Particle number Angular momentum Parity Isospin Generator Coordinate + Projection Quasiparticle RPA (deformed) mass table, even even + odd A mass table, excitations excited states Fission Pathway finder Action minimizer Real tiime propagation Imaginary time propagation fission data Time

15 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Outline Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

16 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Currently: Deformed Mass Table in One Day!

17 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Two-Neutron Separation Energies

18 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Quadrupole Deformations and B(E2)

19 Variety of phenomena: symmetry breaking and quantum corrections LACM: fission, fusion, coexistence phase transitional behavior new kinds of deformations Significant computational resources required: Generator Coordinate Method Projection techniques Imaginary time method (instanton techniques) QRPA and related methods TDHFB, ATDHF, and related methods Beyond Mean Field nuclear collective dynamics Shape coexistence Challenges: selection of appropriate degrees of freedom simultaneous treatment of symmetry coupling to continuum in weakly bound systems dynamical corrections; fundamental theoretical problems. rotational, vibrational, translational particle number isospin GCM M. Bender et al., PRC 69, (2004)

20 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Fission: Energy Surface from DFT

21 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons HFB Mass Formula: m 1 2 MeV

22 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons What are the theoretical error bars?

23 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Skyrme Energy Functionals (cf. Coulomb meta-gga) Minimize E = dx E[ρ(x), τ(x), J(x),...] (for N = Z ): E[ρ, τ, J] = 1 2M τ t 0ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W 0ρ J (t 1 t 2 )J 2 where ρ(x) = i φ i(x) 2 and τ(x) = i φ i(x) 2 (and J)

24 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Skyrme Energy Functionals (cf. Coulomb meta-gga) Minimize E = dx E[ρ(x), τ(x), J(x),...] (for N = Z ): E[ρ, τ, J] = 1 2M τ t 0ρ t 3ρ 2+α (3t 1 + 5t 2 )ρτ (9t 1 5t 2 )( ρ) W 0ρ J (t 1 t 2 )J 2 where ρ(x) = i φ i(x) 2 and τ(x) = i φ i(x) 2 (and J) Varying the (normalized) φ i s yields Kohn-Sham equation: ( 1 2M (x) + U(x) W 0 ρ 1 ) i σ φ i (x) = ɛ i φ i (x), U = 3 4 t 0ρ + ( 3 16 t t 2)τ + and 1 2M (x) = 1 2M + ( 3 16 t t 2)ρ Iterate until φ i s and ɛ i s are self-consistent In practice: other densities, pairing is very important (HFB), projection needed,...

25 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 70 for 8 Be

26 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 70 for 8 Be So for 8 Be there are 17,920 complex functions in 3A 3 = 21 dimensions!

27 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons The Many-Body Schrödinger Wave Function [adapted from Joe Carlson] How to represent the wave function for an A-body nucleus? Consider 8 Be (Z = 4 protons, N = 4 neutrons) Ψ = σ,τ χ σ χ τ φ(r) where R are the 3A spatial coordinates χ σ = 1 2 A (2 A terms) = 256 for A = 8 χ τ = n 1 n 2 p A ( A! N!Z! terms) = 70 for 8 Be So for 8 Be there are 17,920 complex functions in 3A 3 = 21 dimensions! Ab Initio possibilities: Monte Carlo sampling (GFMC, AFMC,... ) Reduced dof s (NCSM, coupled cluster,... ) Work with something other than wave functions

28 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Microscopic Nuclear Structure Methods Wave function methods (GFMC/AFMC, NCSM, CC,... ) many-body wave functions (in approximate form!) Ψ(x 1,, x A ) = everything (if operators known) limited to A < 100 (??) Green s functions (see W. Dickhoff, Many-Body Theory Exposed) response of ground state to removing/adding particles single-particle Green s function = expectation value of one-body operators, Hamiltonian energy, densities, single-particle excitations,... DFT (see C. Fiolhais et al., A Primer in Density Functional Theory) response of energy to perturbations of the density energy functional = plug in candidate density, get out trial energy, minimize (variational) energy and densities (TDFT = excitations)

29 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Full Configuration Interaction FCI: Diagonalize Hamiltonian in harmonic oscillator basis Basis of Slater determinants; configurations grow rapidly with # of orbitals K : K! (K A)!A! Low-momentum potentials need smaller basis Ground-State Energy [MeV] Li λ = 3.0 fm 1 λ = 2.0 fm h Ω [MeV] λ = 1.5 fm 1 N max = 2 N max = 4 N max = 6 N max = 8 N max = 10 λ = 1.0 fm h Ω [MeV]

30 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons No-Core Shell Model (NCSM) Recent results support chiral EFT ; ; hω= ; ; /2 - ; 3/2 5/2-5/2-7/2-1/2 - ; 3/2 5/2-5/2-2 + ; ; / /2-7/2 - ; ; 1 5/ / /2-1/2-3/ /2 - NN+NNN Exp NN NN+NNN Exp NN NN+NNN Exp NN ; 1 10 B 11 B 12 C 13 C 1 + 7/2-2 + ; /2 - ; 3/2 0 + ; ; /2-1/2-7/2-3/2-8 5/2-1/2-4 3/2-3/2-0 1/2-1/2 - NN+NNN Exp NN 3/2 - ; 3/2 5/2 -

31 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons No-Core Shell Model (NCSM) Li Quadrupole moment [e fm 2 ] Exp 8hΩ 6hΩ 4hΩ 2hΩ 0hΩ 10 NN+NNN Use sensitivities to nuclear structure to pin down LEC s gs to gs 1+ B(E2) ratio 2 12 C 10 B B(M1; ) [µ N 2 ] N max c D Exp NN+NNN NN

32 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Coupled Cluster Method (figures from G. Hagen) CC extended to 3-body forces by Hagen, Papenbrock, Dean,...

33 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Coupled Cluster Method (figures from G. Hagen) Improved convergence with low-resolution potentials

34 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons

35 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Low-Momentum 3N Forces in CC [T. Papenbrock]

36 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Configuration Interaction (Shell Model)

37 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Calculations of Medium to Heavy Nuclei How do we circumvent the computational barrier faced by wave function methods? Focus on response of ground state Remove and add particles = Green s functions energy, densities, single-particle properties Response of energy to perturbations of the density = DFT energy and density (or densities) Desired features: Microscopic from NN N interactions Compatible with existing technology (i.e., Skyrme HF) Systematically improvable Control of theoretical errors, model dependence

38 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT and Effective Actions (cf. Negele and Orland) External field Magnetization Helmholtz free energy F[H] Gibbs free energy Γ[M] Legendre = Γ[M] = F[H] + H M transform H = Γ[M] M ground state Γ[M] M = 0 Mgs

39 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT and Effective Actions (cf. Negele and Orland) External field Magnetization Helmholtz free energy F[H] Gibbs free energy Γ[M] Legendre = Γ[M] = F[H] + H M transform H = Γ[M] M ground state Γ[M] M = 0 Mgs Partition function with sources that adjust densities: Z[J] = e W [J] Tr e β(b H+J bρ) = path integral for W [J] Invert to find J[ρ] and Legendre transform from J to ρ: δw [J] ρ(x) = = Γ[ρ] = W [J] J ρ and J(x) = δγ[ρ] δj(x) δρ(x) = Γ[ρ] energy functional E[ρ], stationary at ρ gs (x)!

40 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Paths to a Nuclear Energy Functional Emulate Coulomb DFT: LDA based on precision calculation of uniform system E[ρ] = dr E(ρ(r)) plus constrained gradient corrections ( ρ factors) SLDA (Bulgac et al.) Fayans and collaborators (e.g., nucl-th/ ) E v = 2 3 ɛ Fρ 0 [a v + + a v 1 h v 1+ x 1/3 + x 1 h2+ v x 1/ h v 1 x 1/3 + x 1 h2 v x 1/3 2 + ] E/A (MeV) where x ± = (ρ n ± ρ p )/2ρ (fm -3 ) 0 FP81 WFF88 FaNDF 0 neutron matter RG approach (Polonyi and Schwenk, nucl-th/ ) nuclear matter Constructive Kohn-Sham DFT with low-momentum potentials

41 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Low Resolution = MBPT is Feasible! MBPT Many-Body Perturbation Theory Compare high resolution to low resolution: st order 2nd order pp ladder 3rd order pp ladder E/A [MeV] 50 0 Argonne v k f [fm -1 ]

42 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Low Resolution = MBPT is Feasible! MBPT Many-Body Perturbation Theory Compare high resolution to low resolution: st order 2nd order pp ladder 3rd order pp ladder E/A [MeV] 50 0 Argonne v V low k ( Λ=1.9 fm -1 ) k f [fm -1 ]

43 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Short-Term Roadmap for Microscopic Nuclear DFT Construct a chiral EFT to a given order (N 3 LO at present) Evolve Λ down with RG (to Λ 2 fm 1 for ordinary nuclei) NN interactions fully, NNN interactions approximately Generate density functional in effective action form Hartree-Fock plus second order, use DME in k-space 5 0 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 Λ = 2.1 fm -1 [no V 3N ] 5 0 Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 Λ = 2.1 fm -1 [no V 3N ] E/A [MeV] -5 E/A [MeV] -5 Hartree-Fock + " 2nd order" -10 Hartree-Fock k F [fm -1 ] k F [fm -1 ]

44 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Ab Initio DFT: Parallel Development 1 Momentum-space Renormalization Group (RG) methods to evolve chiral NN and NNN potentials to more perturbative forms as inputs to nuclear matter and ab initio methods (coupled cluster, NCSM). 2 Controlled nuclear matter calculations based on the RG-improved interactions, as ab initio input to Skyrme EDF benchmarking and microscopic functional. 3 Approximate DFT functional, initially by adapting density matrix expansion (DME) to RG-improved interactions. 4 Adaptation to Skyrme codes and allowance for fine tuning. Points of emphasis: Systematic upgrade path with existing and developing technology Theoretical error bars on interaction (vary EFT Λ and order of calculation) and on implementation (vary SRG λ or V low k Λ)

45 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Nuclear matter: Connecting to DFT HF + 2nd order NN-only V low k or SRG with Λ, λ 2 fm 1 doesn t saturate nuclear matter = as A, nuclei collapse Typical fit to NNN from chiral EFT at N 2 LO from A = 3, 4 = c D, c E (but not fit to SRG yet) Large uncertainty for c-terms from πn or NN Only symmetric nuclear matter so far E/A [MeV] K.E. + NN only NNN only K.E. + NN + NNN 15 SRG 20 λ = 2 fm ρ [fm 3 ]

46 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Nuclear matter: Connecting to DFT Coefficients of NNN force can be used to fine-tune nuclear matter within error bands Naturalness implies O(1) factors Should also be consistent with small A = need NNN fits (and eventually SRG running) In the short term: add short-distance counterterms for adjustments E/A [MeV] K.E. + NN only NNN only K.E. + NN + NNN 15 SRG 20 λ = 2 fm ρ [fm 3 ]

47 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Nuclear matter: Connecting to DFT Preliminary: may need more binding from NN to get naturalness and reasonable K sat What is missing in NN part at MeV/particle level? Need accurate nuclear matter calculation to E/A [MeV] K.E. + NN only NNN only K.E. + NN + NNN SRG λ = 2 fm 1 assess (coupled cluster!) ρ [fm 3 ]

48 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Construct W [J] and then Γ[ρ] order-by-order Diagrammatic expansion (i.e., use a power counting) LO : Inversion method = Split source J = J 0 + J cf. H = (H 0 + U) + (V U) with freedom to choose U J 0 chosen to get ρ(x) in noninteracting (Kohn-Sham) system: V trap = V trap J 0

49 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Construct W [J] and then Γ[ρ] order-by-order Diagrammatic expansion (i.e., use a power counting) LO : Inversion method = Split source J = J 0 + J cf. H = (H 0 + U) + (V U) with freedom to choose U J 0 chosen to get ρ(x) in noninteracting (Kohn-Sham) system: V trap = V trap J 0 Orbitals {ψ α (x)} in local potential J 0 ([ρ], x) = KS propagators A [ 2 /2m J 0 (x)]ψ α = ε α ψ α = ρ(x) = ψ α (x) 2 α=1 Self-consistency from J(x) = 0 = J 0 (x) = δγ int [ρ]/δρ(x)

50 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Orbital Dependent DFT (cf. LDA) Self-consistency from J(r) = 0 = J 0 (r) = δγ int [ρ]/δρ(r) i.e., Kohn-Sham potential is functional derivative of interacting energy functional (or E xc ) wrt densities How do we calculate this functional derivative?

51 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Orbital Dependent DFT (cf. LDA) Self-consistency from J(r) = 0 = J 0 (r) = δγ int [ρ]/δρ(r) i.e., Kohn-Sham potential is functional derivative of interacting energy functional (or E xc ) wrt densities How do we calculate this functional derivative? Orbital-dependent DFT = full derivative via chain rule: J 0 (r) = δγ int[φ α, ε α ] δρ(r) = dr δj 0(r ) δρ(r) { α dr [ δφ α (r ) δj 0 (r ) + δε α δj 0 (r ) ] δγ int δφ α(r ) + c.c. } Γ int ε α Solve the OPM equation for J 0 using χ s (r, r ) = δρ(r)/δj 0 (r ) d 3 r χ s (r, r ) J 0 (r ) = Λ xc (r) Λ xc (r) is functional of the orbitals φ α, eigenvalues ε α, and G 0 KS

52 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Orbital Dependent DFT (cf. LDA) Self-consistency from J(r) = 0 = J 0 (r) = δγ int [ρ]/δρ(r) i.e., Kohn-Sham potential is functional derivative of interacting energy functional (or E xc ) wrt densities How do we calculate this functional derivative? Orbital-dependent DFT = full derivative via chain rule: J 0 (r) = δγ int[φ α, ε α ] δρ(r) = dr δj 0(r ) δρ(r) { α dr [ δφ α (r ) δj 0 (r ) + δε α δj 0 (r ) ] δγ int δφ α(r ) + c.c. } Γ int ε α Solve the OPM equation for J 0 using χ s (r, r ) = δρ(r)/δj 0 (r ) d 3 r χ s (r, r ) J 0 (r ) = Λ xc (r) Λ xc (r) is functional of the orbitals φ α, eigenvalues ε α, and G 0 KS Approximation with explicit ρ(r), τ(r),... dependence?

53 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT vs. Solving Skyrme HF Kohn Sham Potentials Skyrme energy functional t 0, t1, t 2,... HFB solver Orbitals and Occupation # s J 0 (r) = δe int[ρ] δρ(r) [ 2 2m J 0(x)]ψ α = ε α ψ α = ρ(x) = α n α ψ α (x) 2

54 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT vs. Solving Skyrme HF Kohn Sham Potentials Orbital dependent functional Solve OPM HFB solver Orbitals and Occupation # s J 0 (r) = δe int[ρ] δρ(r) [ 2 2m J 0(x)]ψ α = ε α ψ α = ρ(x) = α n α ψ α (x) 2

55 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT vs. Solving Skyrme HF Kohn Sham Potentials DME energy functional A[ ρ], B[ ρ],... HFB solver Orbitals and Occupation # s J 0 (r) = δe int[ρ] δρ(r) [ 2 2m J 0(x)]ψ α = ε α ψ α = ρ(x) = α n α ψ α (x) 2

56 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons DFT Looks Like Low-Order HF Approximation WF: Best single Slater determinant in variational sense: Ψ HF = det{φ i (x), i = 1 A}, x = (r, σ, τ) where the φ i (x) satisfy non-local Schrödinger equations: ( ) 2 2M φ i(x) + V H (x) + v ext (x) φ i (x) + dy V E (x, y)φ i (y) = ɛ i φ i (x) with V H (x) = dy A φ j (y) 2 v(x, y), V E (x, y) = v(x, y) j=1 Self-consistent Green s function: same result from just + = A φ j (x)φ j (y) j=1 Kohn-Sham DFT equations always look like Hartree or zero-range Hartree-Fock ( multiplicative potential )

57 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Long-Range Chiral EFT = Enhanced Skyrme 2N forces 3N forces 4N forces Add the long-range (pion-exchange) contributions in the density matrix expansion (DME) Refit the Skyrme parameters Test for sensitities and improved observables Can we see the pion in medium to heavy nuclei?

58 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Observables Sensitive to 3N Interactions? Study systematics along isotopic chains Example: kink in radius shift r 2 (A) r 2 (208) isotopic shift r 2 (A)-r 2 (208) [fm 2 ] Pb isotopes total nucleon number A exp. SkI3 SLy6 NL3 Associated phenomenologically with behavior of spin-orbit isoscalar to isovector ratio fixed in original Skyrme Clues from chiral EFT contributions? (Kaiser et al.)

59 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Ratio of Isoscalar to Isovector Spin-Orbit Ratio fixed at 3:1 for short-range spin-orbit (usual Skyrme) Kaiser: DME spin-orbit from chiral two-body (left) and three-body (right) F so (k f ) F so (k f ) [MeVfm 5 ] [MeVfm 5 ] G so (k f ) G so (k f ) ρ [fm 3 ] ρ [fm -3 ] Systematic investigation needed. Stay tuned...

60 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Observables Sensitive to Missing Physics? Recent studies of tensor contributions [e.g., nucl-th/ ] 5 W p SO HFB+SLy4 ~ 2 ρ p + ρ n W p SO HFB+SkO ~ 2 ρ n + ρ p 5 exp 0-5 t e =0 E (MeV) g 7/2 1h 11/2 1h 11/2 1g 7/ Neutron Number N E (MeV) HFB+SLy4 W p SO ~ J n See also Brown et al., PRC 74 (2006) t e =200 1g 7/2 1h 11/2 HFB+SkO W p SO ~ J n 1h 11/2-1g 7/ Neutron Number

61 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Ab Initio WF s Ab Initio DFT Use external fields put the nuclei in (theoretical) traps Circumvent issues with self-bound nuclei More information about energy functional from each nucleus V trap = V trap J 0 Calculate with same Hamiltonian First calculations with simple 3NF (contact only) Fine-tuning may be needed Compare systematics of energy, radius vs. trap potentials

62 Topics Overview DFT RG 3NF DME Appendix Skyrme Methods CC/FCI DFT Comparisons Misconceptions vs. Correct Interpretations DFT is a Hartree-(Fock) approximation to an effective interaction DFT can accomodate all correlations in principle, but they are included perturbatively (which can fail for some V ) Nuclear matter is strongly nonperturbative in the potential perturbativeness is highly resolution dependent (fill in the blank) causes nuclear saturation another resolution-dependent inference Generating low-momentum interactions loses important information long-range physics is preserved relevant short-range physics encoded in potential Low-momentum NN potentials are just like G-matrices important distinction: conventional G-matrix still has high-momentum, off-diagonal matrix elements

63 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Outline Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

64 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling What Does the Nuclear Potential Look Like? Textbook answer (cf. force between atoms): Momentum units ( = c = 1): typical relative momentum in large nucleus 1 fm MeV (?)

65 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Consequences of a Repulsive Core S1 deuteron probability density uncorrelated correlated ψ(r) 2 [fm 3 ] 0.1 Argonne v 18 ψ(r) 2 [fm 3 ] V(r) [MeV] r [fm] r [fm] 100 Probability at short separations suppressed = correlations Greatly complicates expansion of many-body wave functions Short-distance structure high-momentum components

66 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling S-Wave (L = 0) NN Potential in Momentum Space Fourier transform in partial waves (Bessel transform) V L=0 (k, k ) = d 3 r j 0 (kr) V (r) j 0 (k r) = k V L=0 k Repulsive core = big high-k ( 2 fm 1 ) components

67 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling S-Wave (L = 0) NN Potential in Momentum Space Fourier transform in partial waves (Bessel transform) V L=0 (k, k ) = d 3 r j 0 (kr) V (r) j 0 (k r) = k V L=0 k Repulsive core = big high-k ( 2 fm 1 ) components

68 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Principle of Effective Low-Energy Theories

69 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Principle of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved

70 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Principle of Effective Low-Energy Theories If system is probed at low energies, fine details not resolved use low-energy variables for low-energy processes short-distance structure can be replaced by something simpler without distorting low-energy observables systematically achieved by effective field theory

71 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Try a Low-Pass Filter on V (r) = Set to zero high momentum (k 2 fm 1 ) matrix elements and see the effect on low-energy observables

72 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Use Phase Shifts to Test sin(kr+δ) 60 1 S0 0 R r phase shift (degrees) AV E lab (MeV) Here: 1 S 0 (spin-singlet, L = 0, J = 0) neutron-proton scattering Different phase shifts in each partial wave channel

73 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling A Low-Pass Filter on V (r) Fails At Low Energy 60 1 S0 0 R sin(kr+δ) r phase shift (degrees) AV18 AV18 [k max = 2.2 fm -1 ] E lab (MeV)

74 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling What Should We Conclude About the Potential? Does it mean that the physics of repulsive cores and correlations is important even for low-energy observables like phase shifts or bound-state energies? Are we stuck with keeping high momentum contributions for low-energy physics? Many say yes! Is the potential an observable? Intuition from Coulomb, classically and quantum mechanically and QED E.g., measure the energy at a fixed separation r Isn t V (r) = e2 r always? Answer to all questions: No!

75 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Why Did Our Low-Pass Filter Fail? Basic problem: low k and high k are coupled E.g., perturbation theory for (tangent of) phase shift: k V k + k k V k k V k (k 2 k 2 )/m S0 Solution: Unitary transformation of the H matrix = decouple! E n = Ψ n H Ψ n = ( Ψ n U )UHU (U Ψ n ) = Ψ n H Ψ n phase shift (degrees) AV18 AV18 [k max = 2.2 fm -1 ] Here: Use flow equations E lab (MeV)

76 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Why Did Our Low-Pass Filter Fail? Can t simply change short-distance structure without changing the low-energy results The reason is that there are virtual intermediate states But the effect of short-distance physics on low-energy physics can be absorbed by adjustments in the basic forces = Renormalization Group α(0) ; α(m W ) 1 128

77 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Computational Aside: Digital Potentials Although momentum is continuous in principle, in practice represented as discrete (gaussian quadrature) grid: = Calculations become just matrix multiplications! E.g., k V k + k k V k k V k (k 2 k 2 )/m + = V ii+ j V ij V ji 1 (k 2 i k 2 j )/m Resolution is sufficient for many significant figures

78 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

79 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

80 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

81 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

82 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

83 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

84 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

85 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

86 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

87 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

88 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

89 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

90 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

91 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow Equations in Action: NN Only In each partial wave with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k ) + q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k )

92 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Unitary Transformation: Bare vs. SRG phase shifts 0 δ [deg] S0 1 3 P1 P0 bare ps vsrg ps δ [deg] P1 F3 G E lab [MeV] E lab [MeV] E lab [MeV]

93 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Low-Pass Filters Work! phase shift [deg] [nucl-th/ ] Phase shifts with V s (k, k ) = 0 for k, k > k max S0 3 S P0 3 F E lab [MeV] D AV18 AV18 [k max = 2.2 fm -1 ] V s [k max = 2.2 fm -1 ]

94 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Consequences of a Repulsive Core Revisited S1 deuteron probability density 0.3 uncorrelated correlated 300 ψ(r) 2 [fm 3 ] Argonne v 18 ψ(r) 2 [fm 3 ] V(r) [MeV] r [fm] r [fm] Probability at short separations suppressed = correlations Greatly complicates expansion of many-body wave functions Short-distance structure high-momentum components 100

95 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Consequences of a Repulsive Core Revisited S1 deuteron probability density 0.3 uncorrelated correlated 300 ψ(r) 2 [fm -3 ] Argonne v 18 λ = 4.0 fm -1 λ = 3.0 fm -1 λ = 2.0 fm -1 ψ(r) 2 [fm 3 ] V(r) [MeV] r [fm] r [fm] Transformed potential = no short-range correlations in wf Potential is now non-local: V (r)ψ(r) d 3 r V (r, r )ψ(r ) Not a problem for many-body methods using ho matrix elements 100

96 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 5.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 5.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

97 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 4.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 4.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

98 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 3.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 3.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

99 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 2.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

100 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 2.0 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 2.0 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

101 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Run to Lower λ via SRG = Universality λ = 1.5 fm 1 1 S0 Diagonal V λ (k, k) Off-Diagonal V λ (k, 0) λ = 1.5 fm 1 1 S0 V λ (k,k) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ] V λ (k,0) [fm] /600 [E/G/M] 600/700 [E/G/M] 500 [E/M] 600 [E/M] k [fm 1 ]

102 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Band Diagonalizing with SRG Transform an initial hamiltonian, H = T + V: H s = U(s)HU (s) T + V s, [nucl-th/ ] where s is the flow parameter. Differentiating wrt s: dh s ds = [η(s), H s] with η(s) du(s) ds U (s) = η (s). η(s) is specified by the commutator with T diagonal = T D : which yields the flow equation, η(s) = [T D, H s ], dh s ds = dv s ds = [[T D, H s ], H s ]. T D determines flow: T D = T or T 2 or H D ( T + (V s ) D ) or...

103 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow in NN Momentum Basis with η(s) = [T, H s ] For NN only, project onto partial-wave rel. momentum k dv s ds = [[T rel, V s ], H s ] with ɛ k = 2 k 2 /M and λ 2 = 1/ s dv λ dλ (k, k ) (ɛ k ɛ k ) 2 V λ (k, k )+ q (ɛ k + ɛ k 2ɛ q )V λ (k, q)v λ (q, k ) V λ=2.5 (k, k ) 1st term 2nd term V λ=2.0 (k, k ) First term drives V λ toward diagonal: V λ (k, k ) = V λ= (k, k ) e [(ɛ k ɛ k )/λ 2 ] 2 +

104 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow of N 3 LO Chiral EFT Potentials 1 S 0 from N 3 LO (500 MeV) of Entem/Machleidt 1 S 0 from N 3 LO (550/600 MeV) of Epelbaum et al. See ntg/srg/ for more!

105 Topics Overview DFT RG 3NF DME Appendix Potential Low-Energy Filter Coupling Flow Decoupling Flow of N 3 LO Potentials 3 S 1 from N 3 LO (500 MeV) of Entem/Machleidt 3 S 1 from N 3 LO (550/600 MeV) of Epelbaum et al. See srg/ for more!

106 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Outline Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

107 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Variational Calculations in Three-Nucleon Systems Triton ground-state energy vs. size of harmonic oscillator basis (N max ω excitations) Rapid convergence as λ decreases See arxiv: for more examples Different binding energies = 3-body contribution E t [MeV] MeV N max initial λ = 4 fm 1 λ = 3 fm 1 λ = 2 fm 1 λ = 1 fm 1 550/600 MeV N max

108 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Flow Equations Lead to Many-Body Forces Schematically: dv s [[ ds = a a, ] a } a {{ aa }, ] a } a {{ aa } = + a } a {{ a aaa } + 2-body 2-body 3-body! so there will be A-body forces generated (!). Is this a problem? Normal ordering and truncation may be sufficient Ok if induced many-body forces are same size as natural ones Nuclear 3-body forces already needed in unevolved potential Natural hierarchy from chiral EFT = stop flow equations if unnatural Many-body methods must deal with them!

109 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Atomic 3-Body Forces: Axilrod-Teller Term (1943) Three-body potential for atoms/molecules from triple-dipole mutual polarization (3rd-order perturbation correction) V (i, j, k) = ν(1 + 3 cos θ i cos θ j cos θ k ) (r ij r ik r jk ) 3 Usually negligible in metals and semiconductors Can be important for ground-state energy of solids bound by van der Waals potentials Bell and Zuker (1976): 10% of energy in solid xenon

110 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E b ( 4 He) [MeV] how large is 4-body? E b ( 3 H) [MeV] A=3,4 binding energies SRG NN only, λ in fm 1 λ=1.0 N 3 LO λ=1.25 λ=1.75 λ=1.5 λ=2.0 λ=2.25 Expt. λ=2.5 λ=3.0 NN potentials SRG N 3 LO (500 MeV)

111 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E gs [MeV] Li N 3 SRG NCSM LO (500 MeV) how large is 4-body? λ [fm 1 ]

112 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body SRG 3-Body in Simple Test Cases See arxiv: E.g., two-level system with 2 or 3 particles SRG diagonalizes ( H (2) ɛa + V (2) aa V (2) ab = V (2) ba ɛ b + V (2) bb Diagonalize after evolving Can evolve V (2) + V (3) together here! 3-particle eigenvalues run if no 3-body force, but invariant with SRG running of V (3) ) Energy Energy λ = 1/s 1/ Lowest 3-particle eigenvalue (2) V bb (2) V aa (2) V ab 2-body matrix elements V (2) only V (2) + V (3) λ = 1/s 1/4

113 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body DFT Issues Organization about mean field: need convergent expansion like loop expansion: only mean field is nonperturbative can DFT deal with nuclear short-range correlations? claim: need low-momentum interactions DFT for self-bound systems does DFT even exist? (HK theorem for intrinic states?) symmetry breaking and zero modes game plans proposed: J. Engel, find intrinsic functional (one-d boson system) Giraud et al., use harmonic oscillator tricks methods to deal with soliton zero modes How to deal with long-range correlations? Effectiveness of approximations (e.g., DME)

114 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in relative k, k [sharp] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft?

115 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in relative k, k [e (k 2 /Λ 2 ) 8 ] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft?

116 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Low-Momentum Interactions from RG [AV18 3 S 1 ] V low k = Lower a cutoff Λ in k, k [e (k 2 /Λ 2 ) 4 ] SRG = Drive the Hamiltonian toward diagonal [λ 1/s 1/4 ] Other transformations also decouple (e.g., UCOM) Isn t chiral EFT already soft?

117 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Decoupling of N 3 LO Potentials ( 1 S 0 ) 1 S 0 from N 3 LO (500 MeV) of Entem/Machleidt 1 S 0 from N 3 LO (550/600 MeV) of Epelbaum et al. See srg/ for more!

118 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Decoupling of N 3 LO Potentials ( 1 S 0 ) 3 S 1 from N 3 LO (500 MeV) of Entem/Machleidt 3 S 1 from N 3 LO (550/600 MeV) of Epelbaum et al. See srg/ for more!

119 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Bethe-Brueckner-Goldstone Power Counting Strong short-range repulsion = Sum V ladders = G vs. V low k momentum dependence + phase space = perturbative Λ: P/2 ± k > k F and k < Λ 3 rd order / 2 nd order AV18 Λ = 2.1 fm k f [fm 1 ] Λ 0.5 k P/2 k F V(k,k) [fm] Λ = 2 fm 1 1 S0 V(k,k) 3 S1 V(k,k) 2.5 F: P/2 ± k < k F k [fm 1 ]

120 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Bethe-Brueckner-Goldstone Power Counting Strong short-range repulsion = Sum V ladders = G vs. V low k momentum dependence + phase space = perturbative Λ: P/2 ± k > k F and k < Λ 3 rd order / 2 nd order N 3 LO [600 MeV] λ = 2.0 fm -1 Λ = 2.0 fm k f [fm 1 ] Λ 0.5 k P/2 k F V(k,k) [fm] Λ = 2 fm 1 1 S0 V(k,k) 3 S1 V(k,k) 2.5 F: P/2 ± k < k F k [fm 1 ]

121 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: AV18

122 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: AV18

123 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: AV18

124 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: AV18

125 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: N 3 LO (500 MeV)

126 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: N 3 LO (500 MeV)

127 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: N 3 LO (500 MeV)

128 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare Potential and G Matrix: N 3 LO (500 MeV)

129 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Hole-Line Expansion Revisited (Bethe, Day,... ) Consider ratio of fourth-order diagrams to third-order: a l c k b m a l c k m b b n a l p c k q b m Conventional G matrix still couples low-k and high-k add a hole line = ratio n k F bn (1/e)G bn κ 0.15 no new hole line = ratio χ(r = 0) 1 = sum all orders Low-momentum potentials decouple low-k and high-k add a hole line = still suppressed no new hole line = also suppressed (limited phase space) freedom to choose single-particle U = use for Kohn-Sham = Density functional theory (DFT) should work!

130 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Two-Body Correlations at Nuclear Matter Density 0.2 Defect wf χ(r) for particular kinematics (k = 0, P cm = 0) AV18: Wound integral provides expansion parameter χ(r) S0 defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = 0) Argonne v r [fm]

131 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Two-Body Correlations at Nuclear Matter Density 0.2 Defect wf χ(r) for particular kinematics (k = 0, P cm = 0) AV18: Wound integral provides expansion parameter χ(r) S0 defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = 0) Argonne v 18 Λ = 4.5 fm r [fm]

132 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Two-Body Correlations at Nuclear Matter Density 0.2 Defect wf χ(r) for particular kinematics (k = 0, P cm = 0) AV18: Wound integral provides expansion parameter χ(r) S0 defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = 0) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm r [fm]

133 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Two-Body Correlations at Nuclear Matter Density 0.2 Defect wf χ(r) for particular kinematics (k = 0, P cm = 0) AV18: Wound integral provides expansion parameter Extreme case here, but same pattern in general Tensor ( 3 S 1 ) = larger defect χ(r) S0 defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = 0) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm -1 Λ = 2.5 fm r [fm]

134 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Two-Body Correlations at Nuclear Matter Density 0.2 Defect wf χ(r) for particular kinematics (k = 0, P cm = 0) AV18: Wound integral provides expansion parameter Extreme case here, but same pattern in general Tensor ( 3 S 1 ) = larger defect Still a sizable wound for N 3 LO χ(r) S0 defect χ(r) = Ψ(r) - Φ(r) (k F = 1.35 fm -1, k = 0) Argonne v 18 Λ = 4.5 fm -1 Λ = 3.5 fm -1 Λ = 2.5 fm -1 N 3 LO r [fm]

135 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter with NN Ladders Only [nucl-th/ ] st order 2nd order pp ladder 3rd order pp ladder Brueckner ladders order-by-order Repulsive core = series diverges E/A [MeV] 50 0 Argonne v k f [fm -1 ]

136 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter with NN Ladders Only [nucl-th/ ] st order 2nd order pp ladder 3rd order pp ladder Brueckner ladders order-by-order Repulsive core = series diverges E/A [MeV] 50 0 Argonne v 18 V low k converges No saturation in sight! V low k ( Λ=1.9 fm -1 ) k f [fm -1 ]

137 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Deja Vu All Over Again? There were active attempts to transform away hard cores and soften the tensor interaction in the late sixties and early seventies. But the requiem for soft potentials was given by Bethe (1971): Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required. Next 35+ years struggling to solve accurately with hard potential

138 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Deja Vu All Over Again? There were active attempts to transform away hard cores and soften the tensor interaction in the late sixties and early seventies. But the requiem for soft potentials was given by Bethe (1971): Very soft potentials must be excluded because they do not give saturation; they give too much binding and too high density. In particular, a substantial tensor force is required. Next 35+ years struggling to solve accurately with hard potential But the story is not complete: three-nucleon forces (3NF)!

139 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E b ( 4 He) [MeV] how large is 4-body? E b ( 3 H) [MeV] A=3,4 binding energies SRG NN only, λ in fm 1 λ=1.0 N 3 LO λ=1.25 λ=1.75 λ=1.5 λ=2.0 λ=2.25 Expt. λ=2.5 λ=3.0 NN potentials SRG N 3 LO (500 MeV)

140 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Observations on Three-Body Forces Three-body forces arise from eliminating dof s excited states of nucleon relativistic effects high-momentum intermediate states Omitting 3-body forces leads to model dependence observables depend on λ e.g., Tjon line 3-body contributions increase with density saturates nuclear matter E gs [MeV] Li N 3 SRG NCSM LO (500 MeV) how large is 4-body? λ [fm 1 ]

141 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body SRG from N 3 LO (500 MeV) 2.0 E calc / E expt He 12 C 16 O 14 N NM 2nd order 40 Ca λ [fm 1 ]

142 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body SRG from N 3 LO (500 MeV) E calc / E expt C vacuum 16 O vacuum NM HF 4 He 12 C 16 O NM 2nd order λ [fm 1 ]

143 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body How Does The 3N Contribution Scale? Saturation driven by 3NF 3N force perturbative for Λ < 2.5 fm 1 Coupled cluster results very promising Unnaturally large? Chiral: V 3N (Q/Λ) 3 V NN Four-body contributions? Power counting with NN + 3N HF at LO? V 3N / V low k Check ratios: Λ = 1.6 fm -1 Λ = 1.9 fm -1 Λ = 2.1 fm -1 Λ = 2.3 fm -1 3 H 4 He k F [fm -1 ]

144 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Compare 2nd-Order NN-Only to Empirical Point SRG Nuclear Matter E/A [MeV] λ = 2nd 0rder 3.0 fm 1 λ = 2nd 0rder 2.0 fm 1 λ = 2nd 0rder 1.5 fm 1 λ = 2nd 0rder 1.0 fm k F [fm 1 ]

145 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Does 3N/2N Ratio Scale Like 1/Λ 3 or 1/Λ? <V 3 >/<V 2 > Λ 3 / k F k F = fm 1 <V 3 >/<V 2 > f π 2 Λ / ρ k F = fm 1 SRG (T) V low k (sharp) V low k (n exp = 4) V low k (n exp = 8) V low k (sharp) [3N] λ or Λ [fm 1 ] λ or Λ [fm 1 ]

146 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Diagrams for SRG = Disconnected Cancels V (2) s = [T, V (2) s ] = [[T, V (2) s ], T ] = V (3) s = [T, V (3) s ] = [[T, V (3) s ], T ] = dv s (2) (a, b) = a b + a c b a c b ds (ɛ a ɛ b ) 2 V (2) s (a, b) c [(ɛ a ɛ c ) (ɛ c ɛ b )] V (2) s (a, c) V (2) s (c, b) dv (3) s ds =

147 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter Ladders [nucl-th/ ] Brueckner ladders order-by-order 5 0 2nd order 3rd order pp 4th order pp pp Ladder E/A [MeV] -5 Λ = 2.7 fm body approximated k F [fm -1 ]

148 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter Ladders [nucl-th/ ] Brueckner ladders order-by-order 5 0 2nd order 3rd order pp 4th order pp pp Ladder E/A [MeV] -5 Λ = 2.5(b) fm body approximated k F [fm -1 ]

149 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter Ladders [nucl-th/ ] Brueckner ladders order-by-order 5 0 2nd order 3rd order pp 4th order pp pp Ladder E/A [MeV] -5 Λ = 2.1 fm body approximated k F [fm -1 ]

150 Topics Overview DFT RG 3NF DME Appendix Model BBG 3-Body Nuclear Matter Ladders [nucl-th/ ] Brueckner ladders order-by-order 5 0 2nd order 3rd order pp 4th order pp pp Ladder E/A [MeV] -5 Λ = 1.9 fm body approximated k F [fm -1 ]

151 Topics Overview DFT RG 3NF DME Appendix Strategy Physics Nonlocal 3NF Vlowk Outline Overview of the UNEDF Project Density Functional Theory Changing the Resolution with Flow Equations 3-Body Forces Near-Term Gameplan for Microscopic Nuclear DFT

152 Topics Overview DFT RG 3NF DME Appendix Strategy Physics Nonlocal 3NF Vlowk Parallel Development Areas 1 Momentum-space Renormalization Group (RG) methods to evolve chiral NN and NNN potentials to more perturbative forms as inputs to nuclear matter and ab initio methods (coupled cluster, NCSM). 2 Controlled nuclear matter calculations based on the RG-improved interactions, as ab initio input to Skyrme EDF benchmarking and microscopic functional. 3 Approximate DFT functional, initially by adapting density matrix expansion (DME) to RG-improved interactions. 4 Adaptation to Skyrme codes and allowance for fine tuning. Points of emphasis: Systematic upgrade path with existing and developing technology Theoretical error bars on interaction (vary EFT Λ and order of calculation) and on implementation (vary SRG λ or V low k Λ)

153 Topics Overview DFT RG 3NF DME Appendix Strategy Physics Nonlocal 3NF Vlowk Sources of Theoretical Error Bars 1 EFT Hamiltonian estimate from order of EFT power counting lower bound from varying Λ EFT 2 RG (V low k, V SRG ) truncation: NN N contributions vary λ SRG or Λ Vlow k 3 Many-body approximations vary λ SRG or Λ Vlow k 4 Numerical approximations E/N [MeV] Hartree-Fock (NN+3N) virial FP T=3 MeV T=6 MeV T=10 MeV HF + 2nd-order NN vary basis size, etc ρ [fm -3 ] ρ [fm -3 ]