[PPT] - Qua Quantum um Monte e Carlo calcul ulations ns of the he equa PowerPoint Presentation

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Qua Quantum um Monte e Carlo calcul ulations ns of the he equa equation n of

f state of
f neutron
n matter

r wi with ch chiral EFT inter eract ctions

Ingo Tews

(Institute for Nuclear Theory Seattle)

In collaboration with A.Gezerlis, J. Carlson, S. Gandolfi, J. Lynn, A. Schwenk,

E. Kolomeitsev, J. Lattimer, A. Ohnishi

Extracting Bulk Properties of Neutron-Rich Matter with Transport Models in Bayesian Perspective, April 4th, 2017, FRIB-MSU, East Lansing

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Outl utline ne

Apr. 4, 2017

Ingo Tews, ICNT program 2

Ø Motivation

Ø Chiral effective field theory

Systematic basis for nuclear forces, naturally includes many-body forces
Very successful in calculations of nuclei and nuclear matter

Ø Quantum Monte Carlo method

Very precise for strongly interacting systems
Need of local interactions (depend only on )

ØLocal chiral interactions

Can be constructed up to N2LO

ØResults for neutron matter, light nuclei, and n-alpha scattering Ø S and L constraints from lower bound of neutron matter

e.g. Epelbaum et al., PPNP (2006) and RMP (2009) Gezerlis, IT, et al., PRL & PRC (2013, 2014, 2016)

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Feb. 8, 2017

Ingo Tews, Frontiers Meeting 3

n n n n n n n n To obtain the equation of state we need: q A theory for the strong interactions among nucleons → Phenomenological forces or Chiral EFT q An ab initio method to solve the many-body Schrödinger equation → Many-body Pert. Theory (MBPT), Quantum Monte Carlo (QMC), Coupled Cluster, … "

#$

"

%&

"

'()

Mo Motiv tivatio tion

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Mo Motiv tivatio tion

Apr. 4, 2017

Ingo Tews, ICNT program 4

0.05 0.1 0.15

n [fm-3]

5 10 15 20

E/N [MeV]

this work LS 180 LS 220 LS 375 FSU2.1 NL3 TM1 DD2 SFHo SFHx

MBPT, Fourth order in chiral EFT

IT, Krüger, Hebeler, Schwenk (2013) Lattimer, Lim, ApJ (2013)

Put constraints on symmetry energy and its density dependence L: Ø *+ = 28.9 − 34.9 MeV Ø 4 = 43.0 − 66.6 MeV

Good agreement with experimental constraints

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Mo Motiv tivatio tion

Apr. 4, 2017

Ingo Tews, ICNT program 4

0.05 0.1 0.15

n [fm-3]

5 10 15 20

E/N [MeV]

this work LS 180 LS 220 LS 375 FSU2.1 NL3 TM1 DD2 SFHo SFHx

MBPT, Fourth order in chiral EFT

Credit: Stefano Gandolfi

Quantum Monte Carlo, Phenomenological forces

Status: Ø Sizeable uncertainty for chiral EFT calculations of neutron matter Ø Quantum Monte Carlo: very precise method for strongly interacting systems Ø Phenomenological interactions provide a good description of light nuclei and nuclear matter, but it is not clear how to systematically improve their quality, no uncertainty estimates

IT, Krüger, Hebeler, Schwenk (2013)

Ø QMC calculations with local chiral EFT interactions

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Apr. 4, 2017

Ingo Tews, ICNT program 5

\ Systematic expansion of nuclear forces in low momenta 8 over breakdown scale Λ:: Ø Pions and nucleons as explicit degrees of freedom Ø Long-range physics explicit, short-range physics expanded in general operator basis, couplings (LECs) fit to experiment Ø Separation of scales: Expand in powers of

; <= >

∼

# % >

Ø Power counting scheme Ø Can work to desired accuracy with systematic error estimates

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ...

2 LECs 7 LECs 15 LECs

Chi hiral effective field d the heory for nuc nuclear fo forces

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Chi hiral effective field d the heory for nuc nuclear forces

Apr. 4, 2017

Ingo Tews, ICNT program 5

Many-body forces: Ø Crucial for nuclear physics Ø Natural hierarchy of nuclear forces Ø Fitting: NN forces in NN system (NN phase shifts), 3N forces in 3N/4N system (Binding energies, radii) Ø Consistent interactions: Same couplings for two-nucleon and many- body sector

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Apr. 4, 2017

Ingo Tews, ICNT program 6

Qua Quantum tum Mo Monte Carlo lo me meth thod

d

Solve the many-body Schrödinger equation Basic steps: Ø Choose trial wavefunction which overlaps with the ground state Ø Evaluate propagator for small timestep ΔA, feasible only for local potentials Ø Make consecutive small time steps using Monte Carlo techniques to project out ground state

More details: Carlson, Gandolfi, Pederiva, Pieper, Schiavilla, Schmidt, Wiringa, RMP (2015)

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Apr. 4, 2017

Ingo Tews, ICNT program 7

Particle in a 1D box, solution: BC D = 2

sin IJD , LC = I$J$

2 Basic steps: Ø Choose parabolic trial wavefunction which overlaps with the ground state

Animation by Joel Lynn, TU Darmstadt

Qua Quantum tum Mo Monte Carlo lo me meth thod

d

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Qua Quantum tum Mo Monte Carlo lo me meth thod

d
Apr. 4, 2017

Ingo Tews, ICNT program 7

Particle in a 1D box, solution: BC D = 2

sin IJD , LC = I$J$

2 Ø Make consecutive small timesteps, A = 1.4

# MNOP

Animation by Joel Lynn, TU Darmstadt

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Apr. 4, 2017

Ingo Tews, ICNT program 8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 τ (MeV−1) −29.0 −28.0 −27.0 −26.0 −25.0 −24.0 E (MeV)

E0 (Exp) E(τ) (GFMC) E(τ → ∞) (GFMC)

0.52 0.54 0.56 0.58 0.60 0.62 −28.7 −28.6 −28.5 −28.4 −28.3 −28.2

Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. 4He

Qua Quantum tum Mo Monte Carlo lo me meth thod

d

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Apr. 4, 2017

Ingo Tews, ICNT program 9

Qua Quantum tum Mo Monte Carlo lo me meth thod

d
S. Pieper and R. Wiringa

SLIDE 13

Apr. 4, 2017

Ingo Tews, ICNT program 10

Sources of nonlocalities: ØUsual regulator in relative momenta Øk-dependent contact operators Solutions: Ø Choose local regulators: Ø Use Fierz freedom to choose local set of contact operators

To evaluate the propagator for small timesteps ΔA we need local potentials:

Chiral Effective Field Theory interactions generally nonlocal: Ø Momentum transfer S → TU − T Ø Momentum transfer in the exchange channel V =

W X (T + TU)

Ø Fourier transformation: S → \, V → Derivatives

Lo Local ch chir iral al in interactio actions

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Lo Local ch chir iral al in interactio actions

Apr. 4, 2017

Ingo Tews, ICNT program 11

Ø Leading order "(]) = "

^_`a ] + "bcd

Ø Pion exchange local → local regulator Ø Contact potential:

→ Only two independent (Pauli principle)

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ...

e

f_`g(h) = 1 − exp(−h&/m] &)

e

no_pa(h) = q exp(−h&/m] &)

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Lo Local ch chir iral al in interactio actions

Apr. 4, 2017

Ingo Tews, ICNT program 12

Ø Choose local set of short-range operators at NLO (7 out of 14)

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ...

v v v

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Lo Local ch chir iral al in interactio actions

Apr. 4, 2017

Ingo Tews, ICNT program 13

Ø Choose local set of short-range operators at NLO (7 out of 14) Ø Pion exchanges up to N2LO are local Ø This freedom can be used to remove all nonlocal operators up to N²LO Ø LECs fit to phase shifts

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ... Gezerlis, IT, Epelbaum, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRC (2014)

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QMC QMC re results fo for NN NN fo forces

Apr. 4, 2017

Ingo Tews, ICNT program 14

NN-only calculation: Ø QMC: Statistical uncertainty of points negligible Ø Bands include NN cutoff variation m] = 1.0 − 1.2 fm Ø Order-by-order convergence up to saturation density

Q Q Q

Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) and PRC (2014)

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Be Benchmark of

f MB

MBPT

Apr. 4, 2017

Ingo Tews, ICNT program 16 Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRC (2014)

Many-body perturbation theory: Ø Excellent agreement with QMC for soft potentials (m] = 1.2 fm) Ø Validates perturbative calculations for those interactions

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Lo Local ch chir iral al in interactio actions

Apr. 4, 2017

Ingo Tews, ICNT program 17

Only two new couplings: tu and cd. Fit to uncorrelated observables: Ø Probe properties of light nuclei: 4He EB Ø Probe T=3/2 physics: n-q scattering Inclusion of leading 3N forces: Three topologies: Ø Two-pion exchange "

y

Ø One-pion-exchange contact "

u

Ø Three-nucleon contact "

M

"

u

"

y

"

M

Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ...

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QMC QMC with with ch chir iral al 3N 3N fo forces

Apr. 4, 2017

Ingo Tews, ICNT program 18

Usually "

u and " M vanish in neutron matter:

tu due to spin-isospin structure, tM due to Pauli principle Only true for regulator symmetric in particle labels like commonly used nonlocal regulators, not for local regulators Usually Two-pion-exchange most important in PNM: t# term: Tucson-Melbourn S-wave interaction t%,& term: Fujita-Miyazawa interaction

QMC QMC results sults with with 3N TPE TPE

Apr. 4, 2017

Ingo Tews, ICNT program 19

0.05 0.1 0.15

n [fm

3]

5 10 15 20

E/N [MeV]

Local N2LO (this work) MBPT EGM N2LO pp ladder EM N2LO CC N2LOopt SCGF N2LOopt IT, Gandolfi, Gezerlis, Schwenk, PRC (2016)

Ø Only three-nucleon two-pion exchange ∼ t# and t% Ø Auxiliary-field diffusion Monte Carlo: Ø NN + 3N TPE forces Ø m] = 1.0 − 1.2 fm Ø m%z = 1.0 − 1.2 fm Ø 3N cutoff dependence small Ø TPE 3N contributions ≈ 1 − 2 MeV at I] Ø smaller than for nonlocal regulators

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Fi Fits of 3N LECs Cs

Apr. 4, 2017

Ingo Tews, ICNT program 20 Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016)

Ø Fit tM and tu to 4He binding energy and n-q scattering

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Re Results

Apr. 4, 2017

Ingo Tews, ICNT program 21

Ø Less repulsion from TPE, but additional contributions due to shorter-range 3N forces Ø After inclusion of all contributions we find agreement of various approaches (different way of uncertainty estimate, see EKM, PRC 2015)

0.05 0.1 0.15

n [fm

3]

5 10 15 20

E/N [MeV]

Local N2LO (this work) MBPT EGM N2LO pp ladder EM N2LO CC N2LOopt SCGF N2LOopt

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)

Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016)

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Re Results

Apr. 4, 2017

Ingo Tews, ICNT program 22

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)

LO NLO NLO + pert.

(2) NLO + pert. (3)

N2LO

−2.3 −2.2 −2.1 −2.0 −1.9 −1.8 −1.7 E (MeV) R0 = 1.0 fm R0 = 1.2 fm Exp

Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016)

Ø Chiral interactions at N$LO simultaneously reproduce the properties of A≤5 systems and of neutron matter (uncertainty estimate as in ) Ø Commonly used phenomenological 3N interactions fail for neutron matter

Sarsa, Fantoni, Schmidt, Pederiva, PRC (2003)

E. Epelbaum et al, EPJ (2015)

Deuteron:

Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation.

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Re Results

Apr. 4, 2017

Ingo Tews, ICNT program 22 Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. LO NLO N2LO −75 −65 −55 −45 −35 −25 −15 −5 E (MeV)

4He

LO NLO N2LO −16 −14 −12 −10 −8 −6 −4 E (MeV)

3He

LO NLO N2LO −16 −14 −12 −10 −8 −6 −4 E (MeV)

3H

Ø Chiral interactions at N$LO simultaneously reproduce the properties of A≤5 systems and of neutron matter (uncertainty estimate as in ) Ø Commonly used phenomenological 3N interactions fail for neutron matter

Sarsa, Fantoni, Schmidt, Pederiva, PRC (2003)

E. Epelbaum et al, EPJ (2015)

3H 3He 4He

0.8 1.0 1.2 1.4 1.6 1.8 2.0

q hr2

pti (fm)

NLO N2LO (D2, Eτ) Exp.

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Re Results

Apr. 4, 2017

Ingo Tews, ICNT program 23 IT, Krüger, Hebeler, Schwenk, PRL (2013) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016)

Comparing to N%LO calculation:

Chiral EFT forces with the Quantum Monte Carlo method: Ø Energies agree well with MBPT result within uncertainty bands Ø Many-body uncertainty negligible Ø uncertainties comparable but QMC band only at N2LO and includes also hard interactions

Ø Improve local chiral interactions: Ø Develop N3LO potentials

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Ne Next s step: N3LO

LO

Apr. 4, 2017

Ingo Tews, ICNT program 24

Ø Problem: only 8 out of 30 possible operators local Ø But: work in progress! Improve local chiral interactions: Ø Develop maximally local N3LO potentials Ø Inclusion of Delta degree of freedom

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Now: Constraints on S and L from lower bound of neutron matter energy

Apr. 4, 2017

Ingo Tews, ICNT program 25

Kolomeitsev, Lattimer, Ohnishi, IT, arXiv:1611.07133

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S S and nd L L cons nstraints from lower bo bound und

f ne

neut utron n ma matter er ene nergy gy

0.05 0.1 0.15 0.2

n [fm-3]

5 10 15 20 25

EPNM [MeV]

Lynn et al. (2016) SCGF (2016) Hebeler et al. (2010) TT (2013) GCR (2012) GC (2010) APR (1998) FP (1981) Unitary gas (ξ=0.37)

Apr. 4, 2017

Ingo Tews, ICNT program 26

Empirical observation: Unitary gas energy seems to be lower bound to neutron-matter energy Ø Constraints on S and L

Kolomeitsev, Lattimer, Ohnishi, IT, arXiv:1611.07133

Unitary gas: Ø Gas interacting via two-body interactions with infinite scattering length and vanishing effective range Ø Then, system has no scale except density, and can be described by a dimensionless parameter, (Bertsch parameter) Ø Details of the interaction become irrelevant (universality) Ø Experiment and theory: ≈ 0.37

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S S and nd L L cons nstraints from lower bo bound und

f ne

neut utron n ma matter er ene nergy gy

0.05 0.1 0.15 0.2

n [fm-3]

5 10 15 20 25

EPNM [MeV]

Lynn et al. (2016) SCGF (2016) Hebeler et al. (2010) TT (2013) GCR (2012) GC (2010) APR (1998) FP (1981) Unitary gas (ξ=0.37)

Apr. 4, 2017

Ingo Tews, ICNT program 26

Kolomeitsev, Lattimer, Ohnishi, IT, arXiv:1611.07133

S(u) (MeV) u=n/n0 10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4 Excluded Allowed

ut=0.5-1.1 ut=1 (S0

LB,L0)

SLB(u)

Empirical observation: Unitary gas energy seems to be lower bound to neutron-matter energy Ø Constraints on S and L

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Apr. 4, 2017

Ingo Tews, ICNT program 27

S S and nd L cons nstraints from lower bo bound und

f ne

neut utron n matter ene nergy gy

Kolomeitsev, Lattimer, Ohnishi, IT, arXiv:1611.07133

Put constraints on symmetry energy S and its density dependence L.

L (MeV) S0 (MeV) 20 40 60 80 100 120 24 26 28 30 32 34 36 38 40 Excluded Allowed

(S0

LB,L0)

ut=1

SFHx SFHo IUFSU FSUgold DD2, STOS,TM1 NL3 TMA LS220 NLρ NLρδ DBHF DD,D3C,DD-F KVR KVOR TKHS HS GCR ut=1/2 MKVOR

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Su Summary

Apr. 4, 2017

Ingo Tews, ICNT program 28

Ø QMC calculations of neutron matter, light nuclei, and n-alpha scattering with local chiral potentials up to N²LO including NN and 3N forces can serve as nonperturbative benchmarks. Ø Chiral interactions at N$LO simultaneously reproduce the properties of A≤5 systems and of neutron matter, commonly used phenomenological 3N interactions fail. Ø Further improvements will allow to determine neutron-matter EOS with improved uncertainties (factor of 2). Ø Constraints on symmetry energy and its slope parameter from lower bound of neutron-matter energy.

Gezerlis, IT, Epelbaum, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRC (2014) IT, Gandolfi, Gezerlis, Schwenk, PRC (2016) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016)

L (MeV) S0 (MeV) 20 40 60 80 100 120 24 26 28 30 32 34 36 38 40 Excluded Allowed

(S0

LB,L0)

ut=1

SFHx SFHo IUFSU FSUgold DD2, STOS,TM1 NL3 TMA LS220 NLρ NLρδ DBHF DD,D3C,DD-F KVR KVOR TKHS HS GCR ut=1/2 MKVOR

Kolomeitsev, Lattimer, Ohnishi, IT, arXiv:1611.07133

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Tha Thank nks

Apr. 4, 2017

Ingo Tews, ICNT program 29

Thank you for your attention.

Thanks to my collaborators: Ø Technische Universität Darmstadt:

K. Hebeler, J. Lynn, A. Schwenk

Ø Universität Bochum: E. Epelbaum Ø Los Alamos National Laboratory: J. Carlson, S. Gandolfi Ø University of Guelph: A. Gezerlis Ø Forschungszentrum Jülich: A. Nogga Ø Matej Bel University: Evgeni Kolomeitsev Ø Stony Brook: Jim Lattimer Ø Yukawa Institute Kyoto: Akira Ohnishi