Introduction Separable TgDFT Summary Two ideas for nuclear DFT (and friends) Thomas Lesinski Dept. of Physics & Institute for Nuclear Theory University of Washington, Seattle GSI, May 9, 2012, EMMI program “The Extreme Matter Physics of Nuclei: From Universal Properties to Neutron-Rich Extremes” Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Nuclear structure: methods Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Outline 1 Separable approximations of few-nucleon forces Introduction V low k , NN, particle-hole channel Chiral V 3N at N3LO, particle-hole channel 2 DFT with configuration mixing Hohenberg-Kohn scheme Form of the functional 3 Summary Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Outline 1 Separable approximations of few-nucleon forces Introduction V low k , NN, particle-hole channel Chiral V 3N at N3LO, particle-hole channel 2 DFT with configuration mixing Hohenberg-Kohn scheme Form of the functional 3 Summary Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Separable V NN ■ Principle � � λ a g a ∗ ij g a a g ′ a ik g ′ a λ ′ v ijkl = kl = jl a a ■ If λ a << λ 1 for a > n: truncate a g ′ a ij λ ′ ij g ′ a ■ HF: Γ ij = � a ˘ ρ a , with ˘ ρ a = � ij ρ ij a g a ij g a ■ HFB: ∆ ij = � κ a = � ij λ a ˘ κ a , with ˘ ij κ ij ■ Cost O ( nN ) down from O ( N 2 ) = O ( n 4 o ) ■ The story so far ■ K. Hebeler, T. Duguet, TL, A. Schwenk, Phys. Rev. C 80 044321 (2009) ■ Y. Tian, Z. Ma, P. Ring Phys. Rev. C 79 064301 (2009) ■ T. Nik˘ sić, P. Ring, D. Vretenar, Yuan Tian and Zhong-yu Ma, Phys. Rev. C 81, 054318 (2010) ■ L. M. Robledo, Phys. Rev. C 81, 044312 (2010) ■ TL, K Hebeler, T Duguet and A Schwenk, J. Phys. G 39 015108 (2012) Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Separable V NN ■ Principle � � λ a g a ∗ ij g a a g ′ a ik g ′ a λ ′ v ijkl = kl = jl a a ■ If λ a << λ 1 for a > n: truncate a g ′ a ij λ ′ ij g ′ a ■ HF: Γ ij = � a ˘ ρ a , with ˘ ρ a = � ij ρ ij a g a ij g a ■ HFB: ∆ ij = � κ a = � ij λ a ˘ κ a , with ˘ ij κ ij ■ Cost O ( nN ) down from O ( N 2 ) = O ( n 4 o ) ■ The story so far ■ K. Hebeler, T. Duguet, TL, A. Schwenk, Phys. Rev. C 80 044321 (2009) ■ Y. Tian, Z. Ma, P. Ring Phys. Rev. C 79 064301 (2009) ■ T. Nik˘ sić, P. Ring, D. Vretenar, Yuan Tian and Zhong-yu Ma, Phys. Rev. C 81, 054318 (2010) ■ L. M. Robledo, Phys. Rev. C 81, 044312 (2010) ■ TL, K Hebeler, T Duguet and A Schwenk, J. Phys. G 39 015108 (2012) Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Separable V NN ■ Principle � � λ a g a ∗ ij g a a g ′ a ik g ′ a λ ′ v ijkl = kl = jl a a ■ If λ a << λ 1 for a > n: truncate a g ′ a ij λ ′ ij g ′ a ■ HF: Γ ij = � a ˘ ρ a , with ˘ ρ a = � ij ρ ij a g a ij g a ■ HFB: ∆ ij = � κ a = � ij λ a ˘ κ a , with ˘ ij κ ij ■ Cost O ( nN ) down from O ( N 2 ) = O ( n 4 o ) ■ The story so far ■ K. Hebeler, T. Duguet, TL, A. Schwenk, Phys. Rev. C 80 044321 (2009) ■ Y. Tian, Z. Ma, P. Ring Phys. Rev. C 79 064301 (2009) ■ T. Nik˘ sić, P. Ring, D. Vretenar, Yuan Tian and Zhong-yu Ma, Phys. Rev. C 81, 054318 (2010) ■ L. M. Robledo, Phys. Rev. C 81, 044312 (2010) ■ TL, K Hebeler, T Duguet and A Schwenk, J. Phys. G 39 015108 (2012) Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary Separable V NN : pairing systematics TL, K Hebeler, T Duguet and A Schwenk, J. Phys. G 39 015108 (2012) Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V low k , NN, (Λ = 2 . 0 fm − 1 ) particle-hole channel ■ Basis: spherical Bessel, R box = 18 fm, k cut = 2 . 5 fm − 1 , l cut = 20 ■ N = 3744 for J π = 0 + ■ Obtain ph-separable form 1 ) − 1 | V NN , T z | ( n 2 l 2 j 2 ) − 1 n ′ 2 � ( J ) � n 1 l 1 j 1 ( n ′ 1 l ′ 1 j ′ 2 l ′ 2 j ′ � g JT z , a 1 ) g JT z , a λ JT z = a ( n 1 l 1 j 1 , n ′ 1 l ′ 1 j ′ ( n 2 l 2 j 2 , n ′ 2 l ′ 2 j ′ 2 ) a ■ Eigenvalue decomposition: ScaLAPACK PDSYEV Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V low k , NN, particle-hole channel 10000 nn np 100 1 |eigenvalue| 0.01 0.0001 1e-06 1e-08 1e-10 0 500 1000 1500 2000 2500 3000 3500 4000 index Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N ? ■ Can we use a similar technique for 3N forces ? Higher-Order Singular Value Decomposition (HOSVD) ■ L. De Lathauwer, B. De Moor and J. Vandewalle, SIAM J. Matrix Anal. Appl. 21, 1253 (2000) ■ For a symmetric rank-3 tensor: T pqr = � stu S stu U sp U tq U ur with ■ U an orthogonal (unitary) matrix prs S qrs = σ 2 rs S ∗ ■ All-orthogonality: � p δ pq ■ Ordering: σ 1 ≥ σ 2 ≥ ··· ≥ σ N ■ ph and pp factorizations: HF scaling O ( N 3 ) = O ( n 6 o ) → O ( n 3 )+ O ( nN ) v (3) � λ abc g a ∗ ij g b lm g ′ c � abc g ′ a il g ′ b jm g ′ c λ ′ ijklmn = kn = kn abc abc 1 Define A pI = T pqr , with I = ( q , r ) and SVD A pI = � s U ps σ s V ∗ Is 2 Core tensor: ru T stu , cost O ( N 5 ) ■ Use S pqr = � stu U ∗ ps U ∗ qt U ∗ stu U sp U tq U ur S stu = T pqr , cost O ( n 9 ) ■ Invert � Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N ? ■ Can we use a similar technique for 3N forces ? Higher-Order Singular Value Decomposition (HOSVD) ■ L. De Lathauwer, B. De Moor and J. Vandewalle, SIAM J. Matrix Anal. Appl. 21, 1253 (2000) ■ For a symmetric rank-3 tensor: T pqr = � stu S stu U sp U tq U ur with ■ U an orthogonal (unitary) matrix prs S qrs = σ 2 rs S ∗ ■ All-orthogonality: � p δ pq ■ Ordering: σ 1 ≥ σ 2 ≥ ··· ≥ σ N ■ ph and pp factorizations: HF scaling O ( N 3 ) = O ( n 6 o ) → O ( n 3 )+ O ( nN ) v (3) � λ abc g a ∗ ij g b lm g ′ c � abc g ′ a il g ′ b jm g ′ c λ ′ ijklmn = kn = kn abc abc 1 Define A pI = T pqr , with I = ( q , r ) and SVD A pI = � s U ps σ s V ∗ Is 2 Core tensor: ru T stu , cost O ( N 5 ) ■ Use S pqr = � stu U ∗ ps U ∗ qt U ∗ stu U sp U tq U ur S stu = T pqr , cost O ( n 9 ) ■ Invert � Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N ? ■ Can we use a similar technique for 3N forces ? Higher-Order Singular Value Decomposition (HOSVD) ■ L. De Lathauwer, B. De Moor and J. Vandewalle, SIAM J. Matrix Anal. Appl. 21, 1253 (2000) ■ For a symmetric rank-3 tensor: T pqr = � stu S stu U sp U tq U ur with ■ U an orthogonal (unitary) matrix prs S qrs = σ 2 rs S ∗ ■ All-orthogonality: � p δ pq ■ Ordering: σ 1 ≥ σ 2 ≥ ··· ≥ σ N ■ ph and pp factorizations: HF scaling O ( N 3 ) = O ( n 6 o ) → O ( n 3 )+ O ( nN ) v (3) � λ abc g a ∗ ij g b lm g ′ c � abc g ′ a il g ′ b jm g ′ c λ ′ ijklmn = kn = kn abc abc 1 Define A pI = T pqr , with I = ( q , r ) and SVD A pI = � s U ps σ s V ∗ Is 2 Core tensor: ru T stu , cost O ( N 5 ) ■ Use S pqr = � stu U ∗ ps U ∗ qt U ∗ stu U sp U tq U ur S stu = T pqr , cost O ( n 9 ) ■ Invert � Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N ? ■ Can we use a similar technique for 3N forces ? Higher-Order Singular Value Decomposition (HOSVD) ■ L. De Lathauwer, B. De Moor and J. Vandewalle, SIAM J. Matrix Anal. Appl. 21, 1253 (2000) ■ For a symmetric rank-3 tensor: T pqr = � stu S stu U sp U tq U ur with ■ U an orthogonal (unitary) matrix prs S qrs = σ 2 rs S ∗ ■ All-orthogonality: � p δ pq ■ Ordering: σ 1 ≥ σ 2 ≥ ··· ≥ σ N ■ ph and pp factorizations: HF scaling O ( N 3 ) = O ( n 6 o ) → O ( n 3 )+ O ( nN ) v (3) � λ abc g a ∗ ij g b lm g ′ c � abc g ′ a il g ′ b jm g ′ c λ ′ ijklmn = kn = kn abc abc 1 Define A pI = T pqr , with I = ( q , r ) and SVD A pI = � s U ps σ s V ∗ Is 2 Core tensor: ru T stu , cost O ( N 5 ) ■ Use S pqr = � stu U ∗ ps U ∗ qt U ∗ stu U sp U tq U ur S stu = T pqr , cost O ( n 9 ) ■ Invert � Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N ? ■ Define A pI = T pqr , with I = ( q , r ) and SVD A pI = � s U ps σ s V Is s U ps σ 2 ■ Use EVD of � I A pI A ∗ qI = � s U ∗ qs ■ Choose convenient representation � I A pI A ∗ qI = � IJK A pI W ∗ JI W JK A ∗ qK 1 ) − 1 ( � 2 ) − 1 | V 3 N | ( l 3 j 3 n 3 ) − 1 l ′ 3 � ( J ) � � k 1 σ 1 � k 2 σ 2 ( � k ′ 1 σ ′ k ′ 2 σ ′ 3 j ′ 3 n ′ ■ Basis: spherical Bessel, R box = 15 fm, k cut = 2 . 5 fm − 1 , l cut = 12 ■ N = 1909 for J π = 0 + ■ V 3 N chiral N2LO, Λ χ = 700 MeV, Λ 3 N = 2 . 0 fm − 1 ( c 1 = − 0 . 76, c 3 = − 4 . 78, c 4 = 3 . 96, c D = − 2 . 785, c E = − 0 . 822) Two ideas for nuclear DFT (and friends) T. Lesinski
Introduction Separable TgDFT Summary V 3N (PRELIMINARY) 1 nn[n] pn[n] 0.1 0.01 | σ / σ max | 0.001 0.0001 1e-05 1e-06 0 100 200 300 400 500 600 700 800 900 1000 index Two ideas for nuclear DFT (and friends) T. Lesinski
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