In-medium Similarity Renormalization Group for nuclear many-body systems

Transcription

1 In-medium Similarity Renormalization Group for nuclear many-body systems Koshiroh Tsukiyama (CNS/ U. Tokyo) Dec HPCI project Large-scale quantum many-body calculations for nuclear property and it s application In collaboration with Achim Schwenk (GSI/EMMI, TUD) and Scott K. Bogner (NSCL/MSU)

2 Description of nuclei from NN/NNN Methods should be controlled and improvable Medium-mass nuclei Light nuclei Density functional theory Systematic computationally efficient Universal EDF unknown NN/NNN ab initio binding energy, radius accuracy, feasibility configuration interaction (v) =(E E c ) (v) H (v) e shell evolution, deformation, double-bata decay,..

3 Λ QCD V(k,k ) AV18 H( )=T rel + V 2N ( )+V 3N ( )+V 4N ( )+ χeft pion-exchange and short-range contact power counting (still open) - systematic expansion - many-body forces automatically RG evolution to lower resolution/λ hard still hard NN observables kept unchanged Low-momentum interactions Vlow-k, SRG Bogner-Schwenk-Furnstahl, PPNP, 65, 94 (2010)

4 Similarity Renormalization Group Glazek and Wilson, Phys. Rev. D48, 5863(1993), or Wegner, Ann. Phys. (Leipzig) 3, 77 (1994) The unitary evolution of the Hamiltonian via flow equation d H(s) =[,H(s)] ds (s) = du(s) ds U (s) d (s) =[H(s),H od (s)] by F. Wegner ds Tr Hod 2 0 Flexibility of choosing the H d (s) for a particular problem. can arbitrarily be defined cutoff λ s -1/4 as evolution variable

5 RG and many-body interactions Free-space SRG, evolving consistent 3N interactions => exact method Ground-State Energy [MeV] "24 "25 "26 "27 "28 4 He N 3 LO (500 MeV) NN-only NN+NNN-induced +NNN-initial Expt. NN only => λ-dependent + induced NNN => almost λ-independent Jurgenson, Furnstahl and Navratil PRL103, (2009) same trend for heavier systems Roth et al., PRL107, 07201(2011) " ! [fm "1 ] In-medium SRG Defined in many-body system (finite density) Approximate evolution of 3-,.. A-body operators within 2b machinery. Different SRG evolutions for different mass regions. K.T., S. Bogner and A. Schwenk, PRL106, (2011)

6 Normal-ordered Hamiltonian Ĥ = ij T ij a i a j + 1 2! 2 ijkl V (2) ijkl a i a j a la k + 1 3! 2 V (3) ijklnm a i a j a k a na m a l + ijklmn Normal order w.r.t. a finite-density Fermi vacuum Φ, e.g. HF. H = E 0 + ij f ij {a i a j} + 1 2! 2 ijkl ijkl{a i a j a la k } + 1 3! 2 W ijklmn {a i a j a k a na m a l } ijklmn {A i A j } =0 where coefficients of normal-ordered operators are given by E 0 = H = f ij = T ij + k k T kk n k ij V (2) ijij n in j ijk V (3) ijkijk n in j n k V (2) ikjk n k kl V (3) ikljkl n kn l ijkl = V (2) ijkl m V (3) ijmklm n m W ijklmn = V (3) ijklmn 3-body and higher-body interactions through density- dependent coefficients. => may be efficient truncation scheme

7 d H(s) =[,H(s)] (s) ds Decoupling (schematic picture) is determined s as to eliminate H od (s) Φ 0 Φ p Φ 0 Φ p decouples ground state H od = fph + Γpphh vertices connecting reference state and np-nh excited states Φ 0 Φ v Φ q decouples valence space H od = f ph + Γ pphh + f vq + f vh + Γ vv qq + Γ vhpp + Γ vv v h vertices connecting valence space and outside of it.

8 In-medium SRG flow equation dh(s) ds =[,H(s)] = [ (1) + (2) + (3) +,f + + W + ] Flow eqns. d ds E 0(s) =2 d ds ijkl (s) = commutator form => no unlinked diagram => size extensive: energy scales linearly w/ # of particles a ab n a n b (1) ab f ba abcd (2) abcd cdab(s)n a n b n c n d abcdef (3) abcdef W defabcn a n b n c n d n e n f (1 P ij )( (1) ia ajkl f ia (2) ajkl ) (1 P kl)( (1) ai ijal f ai (2) ijal ) ab (1 n a n b )( (2) ijab abkl ijab (2) abkl ) IM-SRG(2): n 6, IM-SRG(3): n 8 pp or hh ab (n a n b ) (1 P ij )(1 P kl ) (2) bjal + (n n ) (3) f W (1) aibk ph

9 Non-Perturbative feature: Schematic The flow equation can essentially be seen as Γ[n+1]= + + Γ[n] With the initial condition Γ[0]= =: V(bare two-body coupling)

10 Non-Perturbativeness of IM-SRG: Schematic Γ[1]= + + Solving the flow equation step by step V Γ[2]= + + Correlations to all order

11 IM-SRG(2), 95% Flow equation by perturbative analysis Ė 0 (s) =[ (2), ] [2] +[ (1),f] [4] +[ (3),W] [4] IM-SRG(3), 99% f(s) =[ (1),f] [2] +[ (2), ] [2] +[ (1), ] [3] +[ (2),f] [3] +[ (2),W] [3] +[ (3), ] [3] +[ (3),W] [4] (s) =[ (2),f] [1] +[ (2), ] [2] +[ (1), ] [3] +[ (2),W] [3] +[ (3), ] [3] +[ (1),W] [4] +[ (3),f] [4] +[ (3),W] [4] Ẇ (s) =[ (3),f] [2] +[ (2), ] [2] +[ (2),W] [3] +[ (3), ] [3] +[ (1),W] [4] +[ (3),f] [4] IM-SRG(2): 3rd-order exact for GS energy and 2nd-order exact for V eff. IM-SRG(3): 4th-order exact for GS energy and 3rd-order exact for V eff. IM-SRG is controlled and improvable method

12 Numerical calculations N 3 LO (Λ=500MeV) from χeft Entem-Machleidt, PRC 68, (R) (2003) Free-space SRG evolved version Vsrg (λ) Bogner-Perry-Furnstahl, PRC75, (2008) N=emax N=2 N=1 N=0 N=emax N=2 N=1 N=0 emax # SP dim (f)

13 4 He with two different generators Input Vsrg λ=2.0fm -1 95% 99% Truncation up to normal-ordered 2-body level is a good approximation.

14 CCSD Λ-CCSD(T) Agrees well with CCSD (95% of correlation) MBPT(2,3) break down IM-SRG(2) work for 16 O and 40 Ca

15 H od gets suppressed.

16 Evolution of operators Arbitrary operator evolved on equal footing d ds H(s) =[,H(s)] d ds O r(s) =[, O r (s)] Or(s) =O r (0) (s)+o r (1) (s)+o r (2) (s) E.g., RMS radius O r (0) 1 A i (r i R cm ) 2 r = O r (0) = lim s O r (0) (s) Joint benchmark is ongoing (NCFC, IT-NCSM,CCM,MBGF,UMOA,IM-SRG) Results agree within uncertainty Next step from H. Kamada et al., PRC64:044001(2001)

17 Ground-state convergence in 6 Li ( 4 He+ 2 vs 6) Effective 2-body problem 6-body problem

18 6 Li Spectra: IM-SRG vs NCSM Effective 2-body problem 6-body problem Works for 18-body as well

19 Summary Summary " We introduced SRG evolution of Hamiltonian in many-body medium (IM-SRG). " We numerically demonstrated the features of in-medium SRG. " Decoupling of a Hamiltonian, Size-extensivity, Non-perturbative feature. " Radius (arbitral operators can be evolved). " Contamination of center of mass excitation is very small. " Shell-model effective interactions for valence nucleons (p and sd). Work in Progress " Derivation of effective operator. " Systematic improvement; 3-body flow equations.