Part III: The Nuclear Many-Body Problem

Transcription

1 Part III: The Nuclear Many-Body Problem To understand the properties of complex nuclei from first principles Microscopic Valence- Space Interactions Model spaces Many-body perturbation theory (MBPT) Calculating effective interaction In-medium Similarity RG Monopole part of interaction Deficiencies of this approach How will we approach this problem: QCD à NN (3N) forces à Renormalize à Solve many-body problem à Predictions

2 The Nuclear Many-Body Problem Nucleus strongly interacting many-body system how to solve A-body problem? Quasi-exact solutions only in light nuclei (GFMC, NCSM ) Large scale: controlled approximations to full Schrödinger Equation Valence space: diagonalize exactly with reduced number of degrees of freedom Medium-mass Medium-mass Large scale Valence space Coupled Cluster In-Medium SRG Green s Function Limited range: Closed shell ±1 Even-even Limited properties: Ground states only Some excited state H n = E n n Coupled Cluster In-Medium SRG Perturbation Theory All nuclei near closed-shell cores All properties: Ground states Excited states EW transitions

3 From Momentum Space to HO Basis To this point interaction matrix elements in momentum space, partial waves hkk, ll V k 0 K, l 0 Li To go to finite nuclei begin from Hamiltonian H n =(T + V ) n = E n n Assume many particles in the nucleus generate a mean field U: U a one-body potential simple to solve (typically Harmonic Oscillator) H = H 0 + H 1 ; H 0 = T + U; H 1 = V U So transform from momentum space to Harmonic Oscillator Basis nl, NL; i = Z p p k 2 dkk 2 dkr nl 2 k R NL 1/2 K kl, KL; i One more (ugly) transformation from center-of-mass to lab frame:!hab; JT V cd; JTi

4 Begin with degenerate HO levels Valence-Space Ideas 0h, 1f, 2p 0g,1d,2s 0f,1p d,1s 0p 0s hab; JT V cd; JTi Physics of V breaks HO degeneracy Problem: Can t solve Schrodinger equation in full Hilbert space

5 Valence-Space Ideas Nuclei understood as many-body system starting from closed shell, add nucleons Unperturbed HO spectrum Removes degeneracy in valence space only 0h, 1f, 2p 112 0h, 1f, 2p 112 0g,1d,2s 70 0g,1d,2s 70 0f,1p 0d,1s 0p Active nucleons occupy valence space sd -valence space 0f,1p 20 0p 40 0d 3/2 1s 1/2 0d 5/2 8 0s 2 Assume filled core 0s 2

6 Valence-Space Ideas Nuclei understood as many-body system starting from closed shell, add nucleons Valence-space Hamiltonian derived from nuclear forces: Single-particle energies Interaction matrix elements H v.s. = X i " i a i a i + V v.s. 0h, 1f, 2p 112 0g,1d,2s 70 0f,1p 40 c d 20 0p 0s 0d 3/2 1s 1/2 0d 5/2 8 2 sd valence space Inert a V b Active nucleons occupy valence space

7 Nuclei understood as many-body system starting from closed shell, add nucleons Valence-space Hamiltonian derived from nuclear forces: Single-particle energies Interaction matrix elements Valence-Space Philosophy H e = X i " ieff a i a i + V e 0h, 1f, 2p 0g,1d,2s H n = E n n! PH e P i = E i P i Effective valence space Hamiltonian: Sum all excitations outside valence space 0f,1p 40 c d c d 20 0p 0s 0d 3/2 1s 1/2 0d 5/2 8 2 sd valence space Inert a V b a V eff Decouple valence space from excitations b

8 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space 2) Self-consistent single-particle energies da ε eff a c = x V low-k d + a c +... N max 0h, 1f, 2p 112 ba a V b a 0g,1d,2s 70 c a d c d c d VˆQ = + eff V b a b a b c d c d f,1p 20 0p 40 0d 3/2 1s 1/2 0d 5/2 8 a b a b 0s 2

9 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies c d c d c d a VˆQ = eff b a V b + a b c d c d a b a b

10 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies 3) Harmonic-oscillator basis of major shells: converged! Single-Particle Energy (MeV) Neutron f f 5/2 5/2 2 p 1/2 f 7/2 p 3/ p 1/2 f 7/2 p 3/2 Proton Nh _ Nh _

11 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies 3) Harmonic-oscillator basis of major shells: converged! Energy (MeV) O V low k (1st) V low k (2nd) V low k (3rd) Ground-State Energy (MeV) Major Shells Nh _ Ca Ca st order 2nd order 3rd order Nh _

12 Aside: G-matrix Renormalization Standard method for softening interaction in nuclear structure for decades: Infinite summation of ladder diagrams Need two model spaces: 1) M space in which we will want to calculate (excitations allowed in M) 2) Large space Q in which particle excitations are allowed To avoid double counting, can t overlap matrix elements depend on M

13 Aside: G-matrix Renormalization Standard method for softening interaction in nuclear structure for decades: G ijkl (!) =V ijkl + X mn2q V ijmn Q! " m " n G mnkl (!) Iterative procedure Dependence on arbitrary starting energy!

14 G-matrix Renormalization Standard method for softening interaction in nuclear structure for decades: What happens as we keep increasing M? G ijkl (!) =V ijkl + X mn2q V ijmn Q! " m " n G mnkl (!)

15 G-matrix Renormalization Results of G-matrix renormalization vs. SRG AV18 SRG N 3 LO SRG G-mat SRG+ G-mat G-mat SRG+ G-mat Removes some diagonal high-momentum components Still large low-to-high coupling in both interactions No indication of universality Clear difference compared with SRG-evolved interactions!

16 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies 3) Harmonic-oscillator basis of major shells: converged! Energy (MeV) nd order 3 rd order V low k G-matrix 18 O Major Shells Major Shells Compare vs G-matrix (no sign of convergence) Clear benefit of low-momentum interactions!

17 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies 3) Harmonic-oscillator basis of major shells 4) Nuclear forces from chiral EFT 5) Requires extended valence spaces 50 0g 9/2 0f 5/2 1p 1/2 50 0g 9/2 0f 5/2 1p 1/2 Treat higher orbits nonperturbatively p 3/2 0f 7/2 0d 3/2 1s 1/2 0d 5/ p 3/2 0f 7/2 0d 3/2 1s 1/2 0d 5/2 8 0p 3/2 16 O 8 0p 3/2 0p 1/2 0p 1/2

18 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen)

19 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen) Regular dripline trend except oxygen Adding one proton binds 6 additional neutrons

20 Limits of Nuclear Existence: Oxygen Anomaly Where is the nuclear dripline? Limits defined as last isotope with positive neutron separation energy - Nucleons drip out of nucleus Neutron dripline experimentally established to Z=8 (Oxygen) Prediction with NN forces Microscopic picture: NN-forces too attractive Incorrect prediction of dripline

21 Monopole Part of Valence-Space Interactions Microscopic MBPT effective interaction in chosen model space Works near closed shells: deteriorates beyond this Deficiencies improved adjusting particular two-body matrix elements Monopoles: Angular average of interaction V T ab = PJ JT (2J + 1)Vabab (2J + 1) P J Determines interaction of orbit a with b: evolution of orbital energies V(ab;T) [MeV] V low k USDa USDb T=1 d5d5 d5d3 d5s1 d3d3 d3s1 s1s1 Δε a = V ab n b Microscopic low-momentum interactions Phenomenological USD interactions Clear shifts in low-lying orbitals: - T=1 repulsive shift

22 Physics in Oxygen Isotopes Calculate evolution of sd-orbital energies from interactions - 16 O d 3/2 1s 1/2 0d 5/2 0p 3/2 0p 1/ O O 28 - O - O 16 - O O 28 - O - O Fit to experiment Microscopic NN Theories d 3/2 orbit bound to 28 O Phenomenological Models d 3/2 orbit unbound

23 Physics in Oxygen Isotopes Calculate evolution of sd-orbital energies from interactions 20 0d 3/2 1s 1/2 0d 5/ O O 28 - O - O 16 - O O 28 - O - O - 16 O 8 0p 3/2 0p 1/2 Fit to experiment Energy (MeV) sd-shell sdf 7/2 p 3/2 shell USDb Mass Number A Microscopic NN Theories Phenomenological Models d 3/2 orbit bound to 28 O d 3/2 orbit unbound Dripline at 28 O Dripline at 24 O Oxygen anomaly unexplained with NN forces Origin of monopole shifts: Neglected 3N forces -- See lecture of A. Poves

24 Perturbative Approach 1) Effective Hamiltonian: sum excitations outside valence space to MBPT(3) 2) Self-consistent single-particle energies 3) Harmonic-oscillator basis of major shells 4) Nuclear forces from chiral EFT 5) Requires extended valence spaces c d Limitations Uncertain perturbative convergence VˆQ = Core eff physics inconsistent V or absent a b Degenerate valence space requires HO basis (HF requires nontrivial extension) Must treat additional orbitals nonperturbatively (extend valence space)

25 Particle/Hole Excitations Consider basis states as excitations from some reference state: Reference Slater Determinant 1p-1h excitation 2p-2h excitation Unoccupied (Particles) Occupied (Holes) " F i = NY i=1 a i 0i a i i = a aa i i ab ij = a a a i a b a j i Hamiltonian schematically given in terms of ph excitations hi H ji

26 Normal-Ordered Hamiltonian Now rewrite exactly the initial Hamiltonian in normal-ordered form N.O. = E 0 + X ij o f ij na i a j + 1 X 4 jkl ijkl o na i a j a la k X ijklmn W ijklmn n a i a j a k a la m a n o N.O. 0-body! 1-body 2-body 3-body EE0 0 = + + N.O. 1-body! N.O. 2-body! i i f = + + j j i j i j Γ = + k l k l Normal-ordered Hamiltonian w.r.t. reference state Loop = sum over occupied states Include dominant 1-,2-,3-body physics in NO i j two-body fo

27 Nonperturbative In-Medium SRG Tsukiyama, Bogner, Schwenk, PRL (2011) In-Medium SRG continuous unitary trans. drives off-diagonal physics to zero H(s) =U(s)HU (s) H d (s)+h od (s)! H d (1) From uncorrelated Hartree-Fock reference state (e.g., 16 O) define: H od = hp H hi + hpp H hhi + +h.c. hi H ji hnpnh H(1) corei =0 Drives all n-particle n-hole couplings to 0 decouples core from excitations

28 IM-SRG: Flow Equation Formulation Define U(s) implicitly from particular choice of generator: (s) (du(s)/ds) U (s) chosen for desired decoupling behavior e.g., I (s) = H d (s),h od (s) Wegner (1994) Solve flow equation for Hamiltonian (coupled DEs for 0,1,2-body parts) dh(s) ds =[ (s),h(s)] Hamiltonian and generator truncated at 2-body level: IM-SRG(2) 0-body flow drives uncorrelated ref. state to fully correlated ground state E 0 (1)! Core Energy H(s) =E 0 (s)+f(s)+ Ab initio method for energies of closed-shell systems (s)+

29 IM-SRG: Valence-Space Hamiltonians Open-shell systems Tsukiyama, Bogner, Schwenk, PRC (2012) Separate p states into valence states (v) and those above valence space (q) 50 p h g 9/2 0f 5/2 1p 1/2 1p 3/2 0f 7/2 0d 3/2 1s 1/2 0d 5/2 0p 3/2 0p 1/2 q v H(s = 0)! H(1) Redefine H od to decouple valence space from excitations outside v H od = hp H hi + hpp H hhi + hv H qi + hpq H vvi + hpp H hvi +h.c. E 0 (1)! Core Energy f(1)! SPEs (1)! V e

30 IM-SRG: Valence-Space Hamiltonians Open-shell systems Tsukiyama, Bogner, Schwenk, PRC (2012) Separate p states into valence states (v) and those above valence space (q) 50 p h g 9/2 0f 5/2 1p 1/2 1p 3/2 0f 7/2 0d 3/2 1s 1/2 0d 5/2 0p 3/2 0p 1/2 q v H(s = 0)! H(1) H eff Core physics included consistently (absolute energies, radii ) Inherently nonperturbative no need for extended valence space Non-degenerate valence-space orbitals

31 NN-only IM-SRG Monopoles Testing ab initio IM-SRG shell model monopoles Monopoles: Angular average of interaction V T ab = PJ JT (2J + 1)Vabab (2J + 1) P J Determines interaction of orbit a with b: evolution of orbital energies Improvements over MBPT? Δε a = V ab n b 0 T=1 NN-only significantly too attractive V(ab;T) (MeV) USDb NN-only NN+3N-ind NN+3N-ind improved but d 3/2 monopoles too attractive -2 d5d5 d5d3 d5s1 d3d3 d3s1 s1s1

32 Comparison with Large-Space Methods Results from SRG-evolved NN and NN+3N-ind forces Single-Particle Energy (MeV) d 3/2 s 1/2 d 5/2 NN+3N-ind (a) Mass Number A Energy (MeV) Exp. NN NN+3N-ind (b) Mass Number A Dripline still not reproduced

33 Comparison with Large-Space Methods Large-space methods with same SRG-evolved NN+3N-ind forces obtained in large many-body spaces NN+3N-ind Energy (MeV) Exp. NN+3N-ind Energy (MeV) MR-IM-SRG IT-NCSM SCGF CC AME Mass Number A Mass Number A Agreement between all methods with same input forces No reproduction of dripline in any case