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Momentum distributions and pair distribution functions Stefano Gandolfi Los Alamos National Laboratory (LANL) Quantitative challenges in EMC and SRC Research and Data-Mining Massachusetts Institute of Technology, December 2-5, 2016

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SLIDE 1

Momentum distributions and pair distribution functions

Stefano Gandolfi Los Alamos National Laboratory (LANL)

Quantitative challenges in EMC and SRC Research and Data-Mining Massachusetts Institute of Technology, December 2-5, 2016 www.computingnuclei.org

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 1 / 17

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SLIDE 2

Nucleon-nucleon correlations

Illustration of back-to-back proton-neutron pairs in Jefferson Lab Experiment, Subedi et al., Science (2008)

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 2 / 17

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SLIDE 3

Nucleon-nucleon correlations

Ratio of np to pp pairs in light (upper panel) and heavy (lower panel) nuclei, Subedi et al., Science (2008), Hen et al., Science (2014)

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 3 / 17

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SLIDE 4

Nucleon-nucleon correlations

Overarching questions: What is the origin of nucleon-nucleon correlations? Are nucleons more correlated at low- or high-momenta? Can we extract information from momentum distributions? Can we extract information from pair correlation functions? Which are observables and data related to those correlations? In this talk I am not giving answers, but showing what we can calculate! One-body momentum distributions Two-body momentum distributions Contact parameter in cold atoms but not in nuclei

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 4 / 17

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SLIDE 5

Nuclear Hamiltonian

Most common models: non-relativistic nucleons interacting with an effective nucleon-nucleon force (NN) and three-nucleon interaction (TNI). H = − 2 2m

A

  • i=1

∇2

i +

  • i<j

vij +

  • i<j<k

Vijk vij NN, Argonne, chiral EFT, CD-Bonn, etc. Vijk TNI Urbana, Illinois, chiral EFT, etc. Many-body methods: GFMC, AFDMC, CC, SCGF, ...

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 5 / 17

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SLIDE 6

Momentum distribution - 4He

Preliminary! AFDMC calculations:

0.5 1 1.5 2 2.5 3 3.5 4 4.5 k (fm

  • 1)

0.01 1 100 n(k) (fm

3)

AV6’ AV7’ N

2LO (1.0 fm)

N

2LO (1.2 fm) 4He

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 KM (fm

  • 1)

1 2 3 4 5 number of particles AV6’ N

2LO (1.2 fm) 4He

Useful to “check” the contact(s)?

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 6 / 17

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SLIDE 7

Universality

Preliminary! By rescaling the momentum distributions of 4He and 16O, at large momenta they seem universal:

0.5 1 1.5 2 2.5 3 3.5 4 4.5 k (fm

  • 1)

0.01 1 100 n(k) (fm

3) 4He, AV6’ 4He, N 2LO (1.2 fm) 16O, AV6’ 16O, N 2LO (1.2 fm)

Again, useful to “check” the contact(s)?

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 7 / 17

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SLIDE 8

Momentum distribution - 40Ca

Preliminary! AFDMC calculations using AV6’ compared to Coupled-Cluster using N2LOSAT:

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 k (fm

  • 1)

10

  • 4

10

  • 2

10 10

2

n(k) (fm

3)

N

2LOSAT

AV6’

40Ca

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 KM (fm

  • 1)

5 10 15 20 25 30 35 40 number of particles AV6’ N

2LOSAT 40Ca

CC calculations provided by G. Hagen (ORNL). Implications???

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 8 / 17

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SLIDE 9

Symmetric nuclear matter

Self Consistent Green’s Function calculations of nuclear matter: Rios, Polls, Dickhoff, PRC (2014)

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 9 / 17

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SLIDE 10

Two-body momentum distributions in light nuclei

VMC calculations using AV18+UIX:

1 2 3 4 5 10-1 101 103 105

12C

1 2 3 4 5 10-1 101 103 105

10B

1 2 3 4 5 10-1 101 103 105

8Be

1 2 3 4 5 10-1 101 103 105

6Li

1 2 3 4 5 10-1 101 103 105 q (fm-1) ρpN(q,Q=0) (fm3)

4He

Carlson, et al., RMP (2015). Blue symbols are T=0 pairs, and red symbols are T=1 pairs. Strong dominance of T=0 pairs!

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 10 / 17

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SLIDE 11

Two-body momentum distributions of 16O

Preliminary! AFDMC calculations using different NN interactions (Q integrated):

0.5 1 1.5 2 2.5 3 3.5 4 4.5 k (fm

  • 1)

0.0001 0.001 0.01 0.1 1 10 100 1000 n2 (k) (fm

  • 1)

T=0 (np), AV6’ T=1 (pp), AV6’ T=0 (np), N

2LO (1.2 fm)

T=1 (pp), N

2LO (1.2 fm) 4He

0.5 1 1.5 2 2.5 3 3.5 4 4.5 k (fm

  • 1)

0.001 0.01 0.1 1 10 100 1000 10000 n2 (k) (fm

  • 1)

T=0 (np), AV6’ T=1 (pp), AV6’ T=0 (np), N

2LO (1.2 fm)

T=1 (pp), N

2LO (1.2 fm) 16O

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 KM (fm

  • 1)

1 2 3 number of pairs T=0 (np), AV6’ T=1 (pp), AV6’ T=0 (np), N

2LO (1.2 fm)

T=1 (pp), N

2LO (1.2 fm) 4He

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 KM (fm

  • 1)

4 8 12 16 20 24 28 32 36 number of pairs T=0 (np), AV6’ T=1 (pp), AV6’ T=0 (np), N

2LO (1.2 fm)

T=1 (pp), N

2LO (1.2 fm) 16O

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 11 / 17

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SLIDE 12

Pair distribution functions

Preliminary! AFDMC calculations using different NN interactions:

1 2 3 4 5 6 7 8 r (fm) 0.05 0.1 0.15 0.2 0.25 g(r) (fm

  • 3)

4He, AV6’ 4He, N2LO (1.0 fm) 40Ca, AV6’ 40Ca, N2LO (1.0 fm)

Very similar at the origin, but more statistics needed. For the last time ... useful to “check” the contact(s)?

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 12 / 17

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SLIDE 13

Contact parameter

Tan universal relations: Tan’s contact E EFG = ξ − ζ kFa − 5ν 3(kFa)2 + . . . , C NkF = 6 5πζ N(k) → 8 10π ζ k4

F

k4 = 2 3π2 C NkF k4

F

k4 g↑↓(r) → 9π 20 ζ(kFr)−2 = 3 8 C NkF (kFr)−2 S↑↓(k) → 3π 10 ζ kF k = 1 4 C NkF kF k Shina Tan, Ann. Phys. (2008).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 13 / 17

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SLIDE 14

Contact parameter

  • 0.2
  • 0.1

0.1 0.2

1 / kFa

0.1 0.2 0.3 0.4 0.5 0.6

E / EFG

0.015 0.03 0.045 re / r0 0.38 0.39 0.4 0.41 E / EFG 1 / kFa = 0

Equation of state

0.2 0.4 0.6 0.8 1

kF r

50 100 150 200

g↑↓(kF r)

VMC DMC ext. f(kFr)=a+b/(kFr)2

0.4 0.8 1.2 kF r 0.5 1 1.5 (kF r)

2 g↑↓(kF r)

Pair distribution function

0.5 1 1.5 2 2.5 3 3.5 4

k / kF

0.1 0.2 0.3 0.4 0.5

(k/kF)

4 n(k/kF)

VMC DMC ext.

Momentum distribution

1 2 3 4 5 6 7

kF r

  • 1
  • 0.5

0.5 1

ρ

(1) (kF r)

VMC DMC ext. f(kFr)=1 - c kFr

One-body density matrix

C/NkF = 3.39(1), Gandolfi, Schmidt, Carlson, PRA 83, 041601 (2011).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 14 / 17

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SLIDE 15

Contact parameter

  • 2
  • 1

1 2 1 / a kF

  • 6
  • 4
  • 2

ξ

fit QMC

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 1 / a kF 5 10 15 20 C / NkF

QMC exp exp, new

  • Enss. et al.

Hoinka, Lingham, Fenech, Hu, Vale, Drut, Gandolfi, PRL 110, 055305 (2013).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 15 / 17

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SLIDE 16

Contact parameter in nuclei?

Preliminary!

2 4 6 8 k (fm

  • 1)

5 10 15 20 25 k

4n(k) (fm

  • 1)

4He, AV6’ 4He, N 2LO (1.2 fm) 16O, AV6’ 16O, N 2LO (1.2 fm) 40Ca, AV6’ 40Ca, N 2LOSAT

Some flat region for harder interactions that is “maybe” universal for different A (but not using different interactions).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 16 / 17

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SLIDE 17

Summary Thank you! Discussion ...

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 17 / 17

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SLIDE 18

Extra slides

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 1 / 27

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SLIDE 19

Nuclear Hamiltonian

Model: non-relativistic nucleons interacting with an effective nucleon-nucleon force (NN) and three-nucleon interaction (TNI). H = − 2 2m

A

  • i=1

∇2

i +

  • i<j

vij +

  • i<j<k

Vijk vij NN fitted on scattering data. Sum of operators: vij =

  • Op=1,8

ij

v p(rij) , Op

ij = (1,

σi · σj, Sij, Lij · Sij) × (1, τi · τj) Argonne AV8’. Local chiral forces up to N2LO has the similar spin/isospin operatorial structure of AV8’ - Gezerlis, Tews, et al. PRL (2013), PRC (2014)

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 2 / 27

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SLIDE 20

Phase shifts, AV8’

100 200 300 400 500 600 Elab (MeV)

  • 40
  • 20

20 40 60 δ (deg) Argonne V8’ SAID 1S0 100 200 300 400 500 600 Elab (MeV)

  • 20

20 40 60 80 100 120 δ (deg) Argonne V8’ SAID 3S1 100 200 300 400 500 600 Elab (MeV)

  • 40
  • 20

δ (deg) Argonne V8’ SAID 1P1 100 200 300 400 500 600 Elab (MeV)

  • 40
  • 20

20 δ (deg) Argonne V8’ SAID 3P0 100 200 300 400 500 600 Elab (MeV)

  • 50
  • 40
  • 30
  • 20
  • 10

δ (deg) Argonne V8’ SAID 3P1 100 200 300 400 Elab (MeV)

  • 10

10 20 30 δ (deg) Argonne V8’ SAID 3P2 100 200 300 400 500 600 Elab (MeV)

  • 40
  • 30
  • 20
  • 10

δ (deg) Argonne V8’ SAID 3D1 100 200 300 400 500 600 Elab (MeV) 10 20 30 40 50 60 δ (deg) Argonne V8’ SAID 3D2 100 200 300 400 500 600 Elab (MeV) 2 4 6 8 δ (deg) Argonne V8’ SAID

ε1

Difference AV8′-AV18 less than 0.2 MeV per nucleon up to A=12.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 3 / 27

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SLIDE 21

Nuclear Hamiltonian

Chiral EFT interactions Short range operators need to be regulated → cutoff dependency!

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 4 / 27

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SLIDE 22

Three-body forces

Urbana–Illinois Vijk models processes like

π π ∆ π π π π ∆ π π π ∆ π ∆

+ short-range correlations (spin/isospin independent). Chiral forces at N2LO:

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 5 / 27

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SLIDE 23

Nuclear Hamiltonians

Advantages: Argonne interactions fit phase shifts up to high energies: accurate up to high densities. Provide a very good description of several

  • bservables in light nuclei.

Interactions derived from chiral EFT can be systematically improved. Changing the cutoff probes the physics and energy scales entering into observables. They are generally softer, and make most of the calculations easier to converge. Disadvantages: Phenomenological interactions are phenomenological, not clear how to improve their quality. Systematic uncertainties hard to quantify. Chiral interactions describe low-energy (momentum) physics: bad for high densities. How do they work at large momenta, (i.e. e and ν scattering)? Important to consider both and compare predictions

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 6 / 27

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SLIDE 24

Scattering data and neutron matter

The energy of scattering data included in the fit gives an idea of the validity of the interaction in dense matter. Two neutrons have k ≈

  • Elab m/2 ,

→ kF that correspond to kF → ρ ≈ (Elab m/2)3/2/2π2 . Elab=150 MeV corresponds to about 0.12 fm−3. Elab=350 MeV to 0.44 fm−3. Argonne potentials useful for dense matter well above ρ0=0.16 fm−3 Recent chiral forces fit 30 < Elab < 200 MeV.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 7 / 27

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SLIDE 25

Nuclear Hamiltonian

Phase shifts, LO, NLO and N2LO with R0=1.0 and 1.2 fm:

50 100 150 200 250

  • Lab. Energy [MeV]

10 20 30 40 50 60 70

Phase Shift [deg]

LO NLO N2LO

50 100 150 200 250

  • Lab. Energy [MeV]

50 100 150 200 50 100 150 200 250

  • Lab. Energy [MeV]
  • 25
  • 20
  • 15
  • 10
  • 5

50 100 150 200 250

  • Lab. Energy [MeV]

1 2 3 4 1S0 3S1 3D1

ε1

50 100 150 200 250

  • Lab. Energy [MeV]
  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]
  • 10
  • 5

5 10 15 20 25 30 35 40 50 100 150 200 250

  • Lab. Energy [MeV]
  • 30
  • 25
  • 20
  • 15
  • 10
  • 5

1P1 3P0 3P1

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 8 / 27

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SLIDE 26

Nuclear Hamiltonian

Phase shifts, LO, NLO and N2LO with R0=1.0 and 1.2 fm:

50 100 150 200 250

  • Lab. Energy [MeV]

5 10 15 20

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]

0.5 1 1.5 2

50 100 150 200 250

  • Lab. Energy [MeV]
  • 4
  • 3.5
  • 3
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

3P2 3F2

ε2

50 100 150 200 250

  • Lab. Energy [MeV]

5 10 15

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]

10 20 30 40 50 50 100 150 200 250

  • Lab. Energy [MeV]
  • 4
  • 3
  • 2
  • 1

1D2 3D2 1F3 Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 9 / 27

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SLIDE 27

Nuclear Hamiltonian

Phase shifts, LO, NLO and N2LO with R0=1.0 and 1.2 fm:

50 100 150 200 250

  • Lab. Energy [MeV]
  • 3
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]

0.2 0.4 0.6 0.8 1 1.2 50 100 150 200 250

  • Lab. Energy [MeV]

1 2 3 4 5 6 3F3 1G4 3G4 50 100 150 200 250

  • Lab. Energy [MeV]

1 2 3 4 5

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]
  • 3
  • 2.5
  • 2
  • 1.5
  • 1
  • 0.5

50 100 150 200 250

  • Lab. Energy [MeV]

1 2 3 4 5 6 7 3D3 3G3

ε3

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 10 / 27

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SLIDE 28

Nuclear Hamiltonian

Phase shifts, LO, NLO and N2LO with R0=1.0 and 1.2 fm:

50 100 150 200 250

  • Lab. Energy [MeV]

0.5 1 1.5 2

Phase Shift [deg]

50 100 150 200 250

  • Lab. Energy [MeV]

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 50 100 150 200 250

  • Lab. Energy [MeV]
  • 2
  • 1.5
  • 1
  • 0.5

3F4 3H4

ε4

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 11 / 27

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SLIDE 29

4He energy with chiral two-body interactions. Binding energy of 4He with only two-body interactions:

  • 48
  • 46
  • 44
  • 42

LO NLO N2LO v8

′

Eb (MeV) Chiral Order

  • 30
  • 28
  • 26
  • 24
  • 22
  • 20
  • 18
  • 16

Exp. R0 = 1.0 fm R0 = 1.1 fm R0 = 1.2 fm

Lynn, Carlson, Epelbaum, Gandolfi, Gezerlis, Schwenk, PRL (2014).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 12 / 27

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SLIDE 30

Neutron matter

Equation of state of neutron matter using NN chiral forces:

0.05 0.1 0.15

n [fm-3]

5 10 15

E/N [MeV]

AFDMC LO AFDMC NLO AFDMC N2LO R0=1.0 fm R0=1.2 fm

Gezerlis, Tews, et al., PRL (2013), PRC (2014)

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 13 / 27

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SLIDE 31

Chiral three-body forces

Coefficients cD and cE fit to reproduce the binding energy of 4He and neutron-4He scattering. → more information on T=3/2 part of three-body interaction.

1 2 3 4 5 30 60 90 120 150 180 Ec.m. (MeV) δJL (degrees)

1 2

+

1 2

  • AV18

AV18+UIX AV18+IL2 R-Matrix

3 2

  • GFMC neutron-4He results

using Argonne Hamiltonians. Nollett, Pieper, Wiringa, Carlson, Hale, PRL (2007).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 14 / 27

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SLIDE 32

4He binding energy and p-wave n-4He scattering Regulator: δ(r) =

1 πΓ(3/4)R3

0 exp[−(r/R0)4]

Cutoff R0 taken consistently with the two-body interaction.

−1.0 0.0 1.0 2.0 3.0 4.0 cD −1.5 −1.0 −0.5 0.0 0.5 1.0 cE R0 = 1.0 fm R0 = 1.2 fm R0 = 1.0 fm

(a)

N2LO (D1, Eτ) N2LO (D2, Eτ) N2LO (D2, E1) N2LO (D2, EP) 1 2 3 4 Ecm (MeV) 20 40 60 80 100 120 140 δ (deg.) 3 2 − 1 2 −

(b)

NLO N2LO (D2, Eτ) N2LO (D2, EP) R−matrix

No fit to both observables can be obtained for R0 = 1.2 fm and VD1 Lynn, Tews, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk PRL (2016).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 15 / 27

slide-33
SLIDE 33

A=3, 4 nuclei at N2LO

−9 −8 −7 −6 −5 E (MeV)

3H 3He 4He

−31 −25 −19 −13

3H 3He 4He

0.8 1.0 1.2 1.4 1.6 1.8 2.0

  • r2

pt (fm)

NLO N2LO (D2, Eτ) Exp.

Error quantification: define Q = max

  • p

Λb , mπ Λb

  • and calculate:

∆(N2LO) = max

  • Q4| ˆ

OLO|, Q2| ˆ OLO − ˆ ONLO|, Q| ˆ ONLO − ˆ ON2LO

  • Epelbaum, Krebs, Meissner (2014).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 16 / 27

slide-34
SLIDE 34

Neutron matter at N2LO

EOS of pure neutron matter at N2LO, R0=1.0 fm. Error quantification estimated as previously.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 n (fm−3) 2 4 6 8 10 12 14 16 18 E/A (MeV) N2LO (D2, E1) N2LO (D2, EP) N2LO (D2, Eτ)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Neutron Density (fm

  • 3)

2 4 6 8 10 12 14 16 18

Energy per Neutron (MeV)

AV8’+UIX AV8’

Lynn, Tews, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk PRL (2016).

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 17 / 27

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SLIDE 35

Quantum Monte Carlo

Projection in imaginary-time t: H ψ( r1 . . . rN) = E ψ( r1 . . . rN) ψ(t) = e−(H−ET )tψ(0) Ground-state extracted in the limit of t → ∞. Propagation performed by ψ(R, t) = R|ψ(t) =

  • dR′G(R, R′, t)ψ(R′, 0)

dR′ → dR1 dR2 . . . , G(R, R′, t) → G(R1, R2, δt) G(R2, R3, δt) . . . Importance sampling: G(R, R′, δt) → G(R, R′, δt) ΨI(R′)/ΨI(R) Constrained-path approximation to control the sign problem. Unconstrained calculation possible in several cases (exact). Ground–state obtained in a non-perturbative way. Systematic uncertainties within 1-2 %.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 18 / 27

slide-36
SLIDE 36

Overview

Recall: propagation in imaginary-time e−(T+V )∆τψ ≈ e−T∆τe−V ∆τψ Kinetic energy is sampled as a diffusion of particles: e−∇2∆τψ(R) = e−(R−R′)2/2∆τψ(R) = ψ(R′) The (scalar) potential gives the weight of the configuration: e−V (R)∆τψ(R) = wψ(R) Algorithm for each time-step: do the diffusion: R′ = R + ξ compute the weight w compute observables using the configuration R′ weighted using w

  • ver a trial wave function ψT.

For spin-dependent potentials things are much worse!

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 19 / 27

slide-37
SLIDE 37

Branching

The configuration weight w is efficiently sampled using the branching technique: Configurations are replicated or destroyed with probability int[w + ξ] Note: the re-balancing is the bottleneck limiting the parallel efficiency.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 20 / 27

slide-38
SLIDE 38

GFMC and AFDMC

Because the Hamiltonian is state dependent, all spin/isospin states of nucleons must be included in the wave-function. Example: spin for 3 neutrons (radial parts also needed in real life): GFMC wave-function:

ψ =            a↑↑↑ a↑↑↓ a↑↓↑ a↑↓↓ a↓↑↑ a↓↑↓ a↓↓↑ a↓↓↓            A correlation like 1 + f (r)σ1 · σ2 can be used, and the variational wave function can be very good. Any operator accurately computed.

AFDMC wave-function:

ψ = A

  • ξs1

a1 b1

  • ξs2

a2 b2

  • ξs3

a3 b3

  • We must change the propagator by using

the Hubbard-Stratonovich transformation: e

1 2 ∆tO2 =

1 √ 2π

  • dxe− x2

2 +x

√ ∆tO

Auxiliary fields x must also be sampled. The wave-function is pretty bad, but we can simulate larger systems (up to A ≈ 100). Operators (except the energy) are very hard to be computed, but in some case there is some trick!

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 21 / 27

slide-39
SLIDE 39

Propagator

We first rewrite the potential as: V =

  • i<j

[vσ(rij) σi · σj + vt(rij)(3 σi · ˆ rij σj · ˆ rij − σi · σj)] = =

  • i,j

σiαAiα;jβσjβ = 1 2

3N

  • n=1

O2

nλn

where the new operators are On =

  • jβ

σjβψn,jβ Now we can use the HS transformation to do the propagation: e−∆τ 1

2

  • n λO2

nψ =

  • n

1 √ 2π

  • dxe− x2

2 +√−λ∆τxOnψ

Computational cost ≈ (3N)3.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 22 / 27

slide-40
SLIDE 40

Three-body forces

Three-body forces, Urbana, Illinois, and local chiral N2LO can be exactly included in the case of neutrons. For example: O2π =

  • cyc
  • {Xij, Xjk} {τi · τj, τj · τk} + 1

4 [Xij, Xjk] [τi · τj, τj · τk]

  • =

2

  • cyc

{Xij, Xjk} = σiσkf (ri, rj, rk) The above form can be included in the AFDMC propagator.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 23 / 27

slide-41
SLIDE 41

Variational wave function

E0 ≤ E = ψ|H|ψ ψ|ψ =

  • dr1 . . . drN ψ∗(r1 . . . rN)Hψ∗(r1 . . . rN)
  • dr1 . . . drN ψ∗(r1 . . . rN)ψ∗(r1 . . . rN)

→ Monte Carlo integration. Variational wave function: |ΨT =  

i<j

fc(rij)    

i<j<k

fc(rijk)    1 +

  • i<j,p
  • k

uijkfp(rij)Op

ij

  |Φ where Op are spin/isospin operators, fc, uijk and fp are obtained by minimizing the energy. About 30 parameters to optimize. |Φ is a mean-field component, usually HF. Sum of many Slater determinants needed for open-shell configurations. BCS correlations can be included using a Pfaffian.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 24 / 27

slide-42
SLIDE 42

The Sign problem in one slide

Evolution in imaginary-time: ψI(R′)Ψ(R′, t + dt) =

  • dR G(R, R′, dt)ψI(R′)

ψI(R) ψI(R)Ψ(R, t) note: Ψ(R, t) must be positive to be ”Monte Carlo” meaningful. Fixed-node approximation: solve the problem in a restricted space where Ψ > 0 (Bosonic problem) ⇒ upperbound. If Ψ is complex: |ψI(R′)||Ψ(R′, t + dt)| =

  • dR G(R, R′, dt)
  • ψI(R′)

ψI(R)

  • |ψI(R)||Ψ(R, t)|

Constrained-path approximation: project the wave-function to the real

  • axis. Extra weight given by cos ∆θ (phase of Ψ(R′)

Ψ(R) ), Re{Ψ} > 0 ⇒ not

necessarily an upperbound.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 25 / 27

slide-43
SLIDE 43

Unconstrained-path

GFMC unconstrained-path propagation:

10 20 30 40

  • 93
  • 92
  • 91

nu 〈H〉 (MeV)

12C(0+) − AV18 & AV18+IL7 with various corrs. − 〈H〉 − 27 Feb 2010

g.s., IL7, 6 state 18

Changing the trial wave function gives same results.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 26 / 27

slide-44
SLIDE 44

Unconstrained-path

AFDMC unconstrained-path propagation:

0.0005 0.001 0.0015 0.002 0.0025 τ (MeV

  • 1)
  • 110
  • 108
  • 106
  • 104
  • 102
  • 100
  • 98
  • 96
  • 94
  • 92
  • 90

E (MeV) 16O, AV6’+Coulomb 0.00025 0.0005 0.00075 0.001 τ (MeV

  • 1)
  • 100
  • 95
  • 90
  • 85
  • 80

E (MeV) 16O, AV7’+Coulomb

The difference between CP and UP results is mainly due to the presence

  • f LS terms in the Hamiltonian. Same for heavier systems.

Work in progress to improve Ψ to improve the constrained-path.

Stefano Gandolfi (LANL) - stefano@lanl.gov Momentum distributions and pair distribution functions 27 / 27