Quantum Monte Carlo calculations of medium mass nuclei
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1 Quantum Monte Carlo calculations of medium mass nuclei Diego Lonardoni FRIB Theory Fellow In collaboration with: J. Carlson, LANL S. Gandolfi, LANL X. Wang, Huzhou University, China A. Lovato, ANL & UniTN S. C. Pieper, ANL R. B. Wiringa, ANL INT, Program INT 17-2a, June 26, 2017
2 Outline 2 Introduction Quantum Monte Carlo methods Nuclear Hamiltonians Moving towards medium-mass nuclei Phenomenological potentials & QMC Chiral potentials & QMC Conclusions
3 Quantum Monte Carlo methods 3 Goal: solve the many-body problem for correlated systems in a non perturbative fashion 12 VMC CVMC GFMC AFDMC (C)VMC GFMC AFDMC light systems A apple CVMC AFDMC light to mediumheavy nuclei A 50 A E E 0 minimization E! E 0 propagation AFDMC infinite matter A!1 Pros: Work with bare interactions. Good for strongly correlated systems. Stochastic method: errors quantifiable and systematically improvable. 1/ p N Cons: Some limitations in A and/or in the interaction to be used.
4 Quantum Monte Carlo methods 4 CVMC V i = " 1+ X U ijk #" S Y i<j 1+U ij #" Y f c (r ij ) # A i 4 determinants: D i<j<k i<j minimization of E V = h V H V i h V V i E 0 closed-shell nuclei A apple 40 Example: one-body operator h V P i O i V i h V V i = X n i + X n ij + X n i,jk + X n ijk +... i i<j i6=j<k i<j<k 1+ X d ij + X d ijk + X d ij,kl +... i<j i<j<k cluster expansion for the spin-isospin dependent correlations i<j6=k<l i<k n i = ho i i D n ij = 1+U ij (O i + O j ) 1+U ij E n i n j every order in hxi f c (r ij ) included at
5 Quantum Monte Carlo methods 5 CVMC V i = " 1+ X U ijk #" S Y i<j 1+U ij #" Y f c (r ij ) # A i 4 determinants: D i<j<k i<j minimization of E V = h V H V i h V V i E 0 closed-shell nuclei A apple 40 Example: one-body operator h V P i O i V i h V V i = = X i cluster expansion for the spin-isospin dependent correlations X n i + X n ij + X n i,jk + X n ijk +... i i<j i6=j<k i<j<k 1+ X d ij + X d ijk + X d ij,kl +... i<j c i + X i<j i<j<k c ij + X i6=j<k i<j6=k<l i<k c i,jk + X i<j<k c ijk b 2b 3b up to 5b
6 Quantum Monte Carlo methods 6 AFDMC " V i = 1+ X i<j<k U ijk #" 1+ X i<j U ij #" Y i<j f c (r ij ) # A i N determinants: D J propagation in imaginary time: e H V i!1 0i nuclei A. 48 coordinate degrees of freedom: diffusion of positions in coordinate space spin-isospin degrees of freedom: Hubbard-Stratonovich transformation e 1 2 d O2 = p 1 Z dx e x2 2 +p d xo 2 auxiliary field rotation in spin-isospin space sign problem: constrained path approximation + release node ground-state energies within 1-2% with respect to GFMC
7 Nuclear Hamiltonians 7 Model: non-relativistic nucleons interacting with an effective nucleon-nucleon (NN) force and three-nucleon interaction (NNN) H = ~ 2 X 2m N i r 2 i + X i<j v ij + X i<j<k v ijk +... v ij v ijk fit to NN scattering data & deuteron fit to properties of (light?) nuclei + constraints Focus on two families of nuclear interactions: Phenomenological potentials: Argonne V18 (NN) + Urbana / Illinois (NNN) -EFT potentials: N 2 LO local Note: local vs non-local ( p =(p1 p 2 )/2 ( q = p 0 p local r p 0 =(p 0 1 p 0 2)/2 k =(p 0 + p)/2 non-local r r
8 Phenomenological potentials NN: Argonne V18 X X X 8 v ij = X p O p ij (r ij) X O p=1,8 ij =, i j,s ij, L ij S ij, i j NNN: Urbana / Illinois π π π π π π π π π π + short-range term Pros: Cons: Argonne interactions fit phase shifts up to high energies. Accurate up to (at least) 2-3 saturation density. Suitable for QMC calculations. Very good description of several observables in light nuclei (GFMC ground-state energies: uncertainties within 1-2%). Phenomenological interactions are phenomenological, not clear how to improve their quality. Theoretical uncertainties hard to quantify. 3-body forces?
9 Phenomenological potentials & QMC 9 3-body forces: UIX: fit to H3 binding energy & saturation density of SNM IL7: fit to ground- and excited-state energies of light nuclei (A<10) Energy (MeV) He He Li 7/2 5/2 5/2 7/2 1/2 3/2 7 Li Argonne v 18 with UIX or Illinois-7 GFMC Calculations 1 June 2011 AV He Li AV18 +UIX Be AV18 +IL7 7/2 + 5/2 + 7/2 7/2 3/2 3/2 + 5/2 + 1/2 5/2 1/2 + 3/2 9 Be 3 + Expt B C IL7 also needed to reproduce n- scattering 0 + E (MeV) 2.45(1)M obs : 2M P. Maris et al., Phys. Rev. C 87, (2013) 1.74(1)M Note: K. M. AV8 +UIX Nollett et al., Phys. and Rev. Lett. (almost) 99, (2007) AV8 are sti enough to support observed neutron stars, but AV8 +IL7 too soft.! How to reconcile with nuclei??? AV8' AV8' + UIX AV8' + IL7 PNM AFDMC ρ b (fm -3 )?
10 Phenomenological potentials & QMC O: variational energies, charge radii, charge form factors 10 0 obs potential CVMC exp F L (q) CVMC 16 O no MEC hei 16 O AV (2) AV18+UIX 5.15(2) p hr 2 AV (2) ch i AV18+UIX 2.745(2) (5) note: 12 C VMC : 2.484(2) 10-4 exp AV18 exp : 2.470(2) credit to R. B. Wiringa AV18+UIX q (fm -1 ) D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
11 Phenomenological potentials & QMC Ca: variational energies, charge radii, charge form factors 10 0 obs potential CVMC exp CVMC 40 Ca hei p hr 2 ch i AV (10) AV18+UIX 4.92(10) 40 Ca AV (2) AV18+UIX 3.617(2) (1) F L (q) 10-3 no MEC VMC 0 AV18 : exp AV18 AV18+UIX: A. Akmal, et al., Phys. Rev. C 58, 1804 (1998) 10-5 AV18+UIX q (fm -1 ) D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
12 Phenomenological potentials & QMC 13 Coulomb sum rules MEC S L (q) CVMC AV18 + UIX GFMC AV18 + IL7 4 He 16 O 40 Ca 4 He 12 C q (fm -1 ) D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
13 Phenomenological potentials & QMC 14 n p (k)/z (fm 3 ) single-nucleon momentum distributions 4 He 16 O 4 He 8 Be Li O 4 He Ca 16 O 12 C B Ca 2 H Ca k (fm -1 ) ρ p (k) (fm 3 ) AV18+UIX updated from R. B. Wiringa et al., Phys. Rev. C 89, (2014) k (fm -1 ) D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
14 Chiral potentials 15 NN NNN EFT: expansion in power of Q/ b LO O Q b Q m 100 MeV b m 800 MeV soft scale hard scale NLO 1 2 O Q b Long-range physics: given explicitly (no parameters to fit) by pion-exchanges 1 2 N 2 3 LO O Q b + Short-range physics: parametrized through contact interactions with low-energy constants (LECs) fit to low-energy data 1 2 N 3 4 LO O Q b + + Many-body forces enter systematically and are related via the same LECs
15 Chiral potentials 16 Pros: Cons: Chiral interactions have a theoretical derivation and they can be systematically improved (if proper power counting ). They are typically softer than the phenomenological forces, making most of the calculations easier to converge. Many-body forces are naturally accounted for. Standard formulation: momentum-space, non-local. Not suitable for QMC.
16 Chiral potentials 16 Pros: Cons: Chiral interactions have a theoretical derivation and they can be systematically improved (if proper power counting ). They are typically softer than the phenomenological forces, making most of the calculations easier to converge. Many-body forces are naturally accounted for. Standard formulation: momentum-space, non-local. Not suitable for QMC. local chiral N 2 LO potentials 2-body NN A. Gezerlis et al., Phys. Rev. Lett. 111, (2013) A. Gezerlis et al., Phys. Rev. C 90, (2014) J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) 3-body NNN I. Tews et al., Phys. Rev. C 93, (2016) J. E. Lynn et al., Phys. Rev. Lett. 116, (2016) -full local chiral N 3 LO potentials 2-body NN M. Piarulli et al., Phys. Rev. C 91, (2015) M. Piarulli et al., Phys. Rev. C 94, (2016) 3-body NNN in N 2 LO
17 Local chiral potentials 17 2-body N 2 LO pion exchanges up to N 2 LO depend only on p, p 0, q contact terms: 2 LO 7 NLO - N 2 LO no momentum dependence depend on q, q k local regulators in real space for both long and short range physics e (r/r 0) 4 coordinate cutoff: R 0 = fm momentum cutoff: MeV Fierz freedom: v ij = X p O p ij (r ij) O p=1,7 ij =, i j,s ij, i j + Lij S ij ( local (AV6) non-local included in DMC propagators, both GFMC and now AFDMC
18 Local chiral potentials 18 3-body N 2 LO π π π c 1,c 3,c 4 c D c E fit to: 4 He binding energy low energy n- scattering phase shifts same as NN need to be fit Note: regulator functions and finite cutoff in coordinate space different possible operator structures: V D V E D1, D2 E, E,EP suitable for GFMC and now AFDMC: light- to medium-heavy nuclei infinite matter TABLE I. Fit values for the couplings c D and c E for different choices of 3N forces and cutoffs. V 3N R 0 (fm) c E c D N 2 LO ðd1;eτþ N 2 LO ðd2;eτþ N 2 LO ðd2;e1þ N 2 LO ðd2;epþ J. E. Lynn et al., Phys. Rev. Lett. 116, (2016)
19 19 energies -2.0 GFMC -4.0 E/A (MeV) exp N 2 LO (1.0) N 2 LO (1.2) 2b 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
20 21 energies -2.0 GFMC AFDMC (preliminary) -4.0 E/A (MeV) exp N 2 LO (1.0) N 2 LO (1.2) 2b 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
21 GFMC energies (deg.) (b) n (a) NLO N 2 LO (D2,E ) N 2 LO (D2,EP) R matrix R 0 =1.0 fm R 0 =1.0 fm R 0 =1.2 fm E/A (MeV) -6.0 fit ce R 0 =1.2 fm exp R 3 4 E cm (MeV) 0 =1.0 fm N 2 LO (1.0) N 2 LO (1.2) 2b+3b c D 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
22 23 energies 18 N 2 LO (D2,E1) -2.0 GFMC N 2 LO (D2,EP) N 2 LO (D2,E ) -4.0 E/A (MeV) 10 8 E/A (MeV) -6.0 fit PNM AFDMC n (fm 3 ) -8.0 exp N 2 LO (1.0) N 2 LO (1.2) 2b+3b 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
23 energies phenom AV8' AV8' + UIX phenom E/A (MeV) δ JL (degrees) E c.m. (MeV) exp 3-2 AV18 AV18+UIX AV18+IL2 R-Matrix K. M. Nollett et al., Phys. Rev. Lett. 99, (2007) 1-2 E (MeV) 14 AV8' + IL PNM AFDMC ρ b (fm -3 ) P. Maris et al., Phys. Rev. C 87, (2013) N 2 LO (1.0) N 2 LO (1.2) 2b+3b 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
24 26 energies -2.0 GFMC AFDMC (preliminary) -4.0 E/A (MeV) -6.0 fit exp N 2 LO (1.0) N 2 LO (1.2) 2b+3b 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
25 27 energies GFMC E (MeV) q hr 2 pti (fm) E/A (MeV) -6.0 theory R 0 =1.0fm R 0 =1.2fm 3 H 3 He 4 He NLO N 2 LO (D2,E ) Exp. 3 H 3 He 4 He exp N 2 X N2LO max Q 4 X LO, LO (1.0) 2b+3b N 2 LO (1.2) Q 2 X LO X NLO, Q m b Q X NLO X N2 LO 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca E. Epelbaum et al., Eur. Phys. J. A 51 (2015) GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
26 28 energies -2.0 GFMC AFDMC (preliminary) MeV/A E/A (MeV) -6.0 theory -8.0 exp N 2 LO (1.0) N 2 LO (1.2) 2b+3b 4 MeV/A 3 H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
27 29 charge radii exp N 2 LO (1.0) N 2 LO (1.2) 2b r ch (fm) GFMC AFDMC (preliminary) H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
28 30 charge radii exp N 2 LO (1.0) N 2 LO (1.2) 2b+3b r ch (fm) GFMC AFDMC (preliminary) H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
29 31 charge radii exp N 2 LO (1.0) N 2 LO (1.2) 2b+3b r ch (fm) GFMC AFDMC (preliminary) H 3 He 4 He 6 He 6 Li 16 O 40 Ca 48 Ca GFMC: J. E. Lynn et al., Phys. Rev. Lett. 113, (2014) & Phys. Rev. Lett. 116, (2016)
30 32 charge form factors CVMC 16 O no MEC F L (q) AFDMC (preliminary) exp AV18+UIX N 2 LO (1.0) N 2 LO (1.2) q (fm -1 ) CVMC: D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
31 33 single-nucleon momentum distribution VMC AV18 AV6' n p (k)/z (fm 3 ) N 2 LO (1.0 fm) N 2 LO (1.2 fm) He 10-3 AFDMC (preliminary) k (fm -1 ) VMC: R. B. Wiringa et al., Phys. Rev. C 89, (2014)
32 34 single-nucleon momentum distribution AFDMC (preliminary) n p (k) integrated strenght % 93.1% 4 He 0.2 AV6' N 2 LO (1.2 fm) k (fm -1 )
33 35 single-nucleon momentum distribution CVMC AV18 AV6' n p (k)/z (fm 3 ) N 2 LO (1.0 fm) N 2 LO (1.2 fm) O 10-3 AFDMC (preliminary) k (fm -1 ) CVMC: D.L., A. Lovato, S. C. Pieper, R. B. Wiringa, arxiv:
34 36 single-nucleon momentum distribution He: AV6' 16 O: AV6' 4 He: N 2 LO (1.0 fm) 16 O: N 2 LO (1.0 fm) n p (k)/z (fm 3 ) AFDMC (preliminary) k (fm -1 )
35 37 single-nucleon momentum distribution He: AV6' 16 O: AV6' 4 He: N 2 LO (1.2 fm) 16 O: N 2 LO (1.2 fm) n p (k)/z (fm 3 ) AFDMC (preliminary) k (fm -1 )
36 38 two-nucleon momentum distribution (integrated Q) np, AV6' pp, AV6' np, N 2 LO (1.2 fm) pp, N 2 LO (1.2 fm) n 12 (q) (fm 3 ) He 10-2 AFDMC (preliminary) q (fm -1 )
37 39 two-nucleon momentum distribution % np, AV6' number of paris % 98.0% 99.5% pp, AV6' np, N 2 LO (1.2 fm) pp, N 2 LO (1.2 fm) 1 AFDMC (preliminary) q (fm -1 )
38 40 two-nucleon momentum distribution (integrated Q) np, AV6' pp, AV6' np, N 2 LO (1.2 fm) pp, N 2 LO (1.2 fm) n 12 (q) (fm 3 ) O AFDMC (preliminary) q (fm -1 )
39 Conclusions 41 progresses in QMC calculations moving towards an exact description of the medium region of the nuclear chart CVMC ground state properties of closed-shell nuclei up to A=40 investigation of phenomenological A Ca forces above A=16 AFDMC A O ground state properties of light- and medium-heavy nuclei with delta-less local chiral potentials at N 2 LO investigation of local chiral forces Atomic Number adapted from A. B. Balantekin et al., Mod. Phys. Lett. A 29, (2014) Next: complete the study at N 2 LO including truncation errors & other 3-body forms N 3 LO? employ delta-full local chiral potentials? symmetric nuclear matter? neutron matter delta-full? currents? single- and double-beta decay?
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