Quantum Monte Carlo calculations of the equation of state of neutron matter with chiral EFT interactions

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1 Quantum Monte Carlo calculations of the equation of state of neutron matter with chiral EFT interactions 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

2 Outline 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 N 2 LO e.g. Epelbaum et al., PPNP (2006) and RMP (2009) Gezerlis, IT, et al., PRL & PRC (2013, 2014, 2016) ØResults for neutron matter, light nuclei, and n-alpha scattering Ø S and L constraints from lower bound of neutron matter Apr. 4, 2017 Ingo Tews, ICNT program 2

3 Motivation n n n " %& To obtain the equation of state we need: q A theory for the strong interactions among nucleons n " #$ n Phenomenological forces or Chiral EFT n q An ab initio method to solve the many-body Schrödinger equation n " '() n Many-body Pert. Theory (MBPT), Quantum Monte Carlo (QMC), Coupled Cluster, Feb. 8, 2017 Ingo Tews, Frontiers Meeting 3

Motivation E/N [MeV] 20 15 10 this work LS 180 LS 220 LS 375 FSU2.

15 n [fm -3 ] IT, Krüger, Hebeler, Schwenk (2013) Put constraints on symmetry energy and its density

4 Motivation E/N [MeV] this work LS 180 LS 220 LS 375 FSU2.1 NL3 TM1 DD2 SFHo SFHx 5 MBPT, Fourth order in chiral EFT n [fm -3 ] IT, Krüger, Hebeler, Schwenk (2013) Put constraints on symmetry energy and its density dependence L: Ø * + = MeV Ø 4 = MeV Good agreement with experimental constraints Lattimer, Lim, ApJ (2013) Apr. 4, 2017 Ingo Tews, ICNT program 4

5 Motivation E/N [MeV] this work LS 180 LS 220 LS 375 FSU2.1 NL3 TM1 DD2 SFHo SFHx 5 MBPT, Fourth order in chiral EFT n [fm -3 ] IT, Krüger, Hebeler, Schwenk (2013) Quantum Monte Carlo, Phenomenological forces Credit: Stefano Gandolfi 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 Ø QMC calculations with local chiral EFT interactions Apr. 4, 2017 Ingo Tews, ICNT program 4

6 Chiral effective field theory for nuclear forces 2 LECs 7 LECs 15 LECs \ 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... > Apr. 4, 2017 Ingo Tews, ICNT program 5

7 Chiral effective field theory for nuclear forces 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 manybody sector Apr. 4, 2017 Ingo Tews, ICNT program 5

8 Quantum Monte Carlo method 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) Apr. 4, 2017 Ingo Tews, ICNT program 6

9 Quantum Monte Carlo method Particle in a 1D box, solution: B C D = 2 sin IJD, L C = I$ J $ 2 Basic steps: Ø Choose parabolic trial wavefunction which overlaps with the ground state Animation by Joel Lynn, TU Darmstadt Apr. 4, 2017 Ingo Tews, ICNT program 7

10 Quantum Monte Carlo method Particle in a 1D box, solution: B C D = 2 sin IJD, L C = I$ J $ 2 Ø Make consecutive small timesteps, A = 1.4 Animation by Joel Lynn, TU Darmstadt # M NOP Apr. 4, 2017 Ingo Tews, ICNT program 7

11 Quantum Monte Carlo method He E 0 E( ) (Exp) (GFMC) E(!1) (GFMC) E (MeV) (MeV 1 ) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. Apr. 4, 2017 Ingo Tews, ICNT program 8

12 Quantum Monte Carlo method S. Pieper and R. Wiringa Apr. 4, 2017 Ingo Tews, ICNT program 9

13 Local chiral interactions To evaluate the propagator for small timesteps ΔA we need local potentials: Chiral Effective Field Theory interactions generally nonlocal: Ø Momentum transfer S T U T Ø Momentum transfer in the exchange channel V = W X (T + TU ) Ø Fourier transformation: S \, V Derivatives Sources of nonlocalities: ØUsual regulator in relative momenta Solutions: Ø Choose local regulators: Øk-dependent contact operators Ø Use Fierz freedom to choose local set of contact operators Apr. 4, 2017 Ingo Tews, ICNT program 10

14 Local chiral interactions Ø Leading order " (]) = "^_`a ] + " bcd Ø Pion exchange local local regulator e f_`g (h) = 1 exp( h & /m ] & ) Ø Contact potential: Only two independent (Pauli principle) e no_pa (h) = q exp( h & /m ] & ) Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer... Apr. 4, 2017 Ingo Tews, ICNT program 11

15 Local chiral interactions Ø Choose local set of short-range operators at NLO (7 out of 14) v v v Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer... Apr. 4, 2017 Ingo Tews, ICNT program 12

Local chiral interactions Ø Choose local set of short-range operators at NLO (7 out of 14) Ø Pion exchanges up to N 2 LO are local Ø This freedom can be used to remove all nonlocal operators up to

16 Local chiral interactions Ø Choose local set of short-range operators at NLO (7 out of 14) Ø Pion exchanges up to N 2 LO are local Ø This freedom can be used to remove all nonlocal operators up to N²LO Gezerlis, IT, Epelbaum, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRC (2014) Ø LECs fit to phase shifts Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer... Apr. 4, 2017 Ingo Tews, ICNT program 13

17 QMC results for NN forces Q Q Q NN-only calculation: Ø QMC: Statistical uncertainty of points negligible Ø Bands include NN cutoff variation m ] = fm Ø Order-by-order convergence up to saturation density Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) and PRC (2014) Apr. 4, 2017 Ingo Tews, ICNT program 14

18 Benchmark of MBPT 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 Apr. 4, 2017 Ingo Tews, ICNT program 16

19 Local chiral interactions Inclusion of leading 3N forces: " y " u " M Three topologies: Ø Two-pion exchange " y Ø One-pion-exchange contact " u Ø Three-nucleon contact " M Only two new couplings: t u and c d. Fit to uncorrelated observables: Ø Probe properties of light nuclei: 4 He E B Ø Probe T=3/2 physics: n-q scattering Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer... Apr. 4, 2017 Ingo Tews, ICNT program 17

20 QMC with chiral 3N forces Usually Two-pion-exchange most important in PNM: t # term: Tucson-Melbourn S-wave interaction t %,& term: Fujita-Miyazawa interaction Usually " u and " M vanish in neutron matter: t u due to spin-isospin structure, t M due to Pauli principle see also Hebeler, Schwenk, PRC (2010) Only true for regulator symmetric in particle labels like commonly used nonlocal regulators, not for local regulators Apr. 4, 2017 Ingo Tews, ICNT program 18

21 QMC results with 3N TPE E/N [MeV] Local N 2 LO (this work) MBPT EGM N 2 LO pp ladder EM N 2 LO CC N 2 LO opt SCGF N 2 LO opt Ø Only three-nucleon two-pion exchange t # and t % Ø Auxiliary-field diffusion Monte Carlo: Ø NN + 3N TPE forces Ø m ] = fm Ø m %z = fm Ø 3N cutoff dependence small 5 Ø TPE 3N contributions 1 2 MeV at I ] Ø smaller than for nonlocal regulators n [fm -3 ] IT, Gandolfi, Gezerlis, Schwenk, PRC (2016) Apr. 4, 2017 Ingo Tews, ICNT program 19

22 Fits of 3N LECs Ø Fit t M and t u to 4 He binding energy and n-q scattering Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016) Apr. 4, 2017 Ingo Tews, ICNT program 20

23 Results N 2 LO (D2,E1) N 2 LO (D2,EP) N 2 LO (D2,E ) Local N 2 LO (this work) MBPT EGM N 2 LO pp ladder EM N 2 LO CC N 2 LO opt SCGF N 2 LO opt E/A (MeV) 10 8 E/N [MeV] n (fm 3 ) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016) n [fm -3 ] 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) Apr. 4, 2017 Ingo Tews, ICNT program 21

24 Results N 2 LO (D2,E1) N 2 LO (D2,EP) N 2 LO (D2,E ) Deuteron: R 0 =1.0 fm R 0 =1.2 fm Exp E/A (MeV) n (fm 3 ) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016) E (MeV) LO NLO NLO + pert. (2) NLO + pert. (3) N 2 LO Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. Ø Chiral interactions at N $ LO simultaneously reproduce the properties of A 5 systems and of neutron matter (uncertainty estimate as in E. Epelbaum et al, EPJ (2015) ) Ø Commonly used phenomenological 3N interactions fail for neutron matter Sarsa, Fantoni, Schmidt, Pederiva, PRC (2003) Apr. 4, 2017 Ingo Tews, ICNT program 22

25 Results E (MeV) He E (MeV) He E (MeV) H q hr 2 pti (fm) LO NLO N 2 LO LO NLO N 2 LO LO NLO N 2 LO NLO N 2 LO (D2,E ) Exp. 3 H 3 He 4 He Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. Ø Chiral interactions at N $ LO simultaneously reproduce the properties of A 5 systems and of neutron matter (uncertainty estimate as in E. Epelbaum et al, EPJ (2015) ) Ø Commonly used phenomenological 3N interactions fail for neutron matter Sarsa, Fantoni, Schmidt, Pederiva, PRC (2003) Apr. 4, 2017 Ingo Tews, ICNT program 22

26 Results 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 N 2 LO and includes also hard interactions IT, Krüger, Hebeler, Schwenk, PRL (2013) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, PRL (2016) Ø Improve local chiral interactions: Ø Develop N 3 LO potentials Apr. 4, 2017 Ingo Tews, ICNT program 23

27 Next step: N 3 LO Improve local chiral interactions: Ø Develop maximally local N 3 LO potentials Ø Inclusion of Delta degree of freedom Ø Problem: only 8 out of 30 possible operators local Ø But: work in progress! Apr. 4, 2017 Ingo Tews, ICNT program 24

28 Now: Constraints on S and L from lower bound of neutron matter energy Kolomeitsev, Lattimer, Ohnishi, IT, arxiv: Apr. 4, 2017 Ingo Tews, ICNT program 25

29 S and L constraints from lower bound of neutron matter energy E PNM [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) n [fm -3 ] 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 Kolomeitsev, Lattimer, Ohnishi, IT, arxiv: Empirical observation: Unitary gas energy seems to be lower bound to neutron-matter energy Ø Constraints on S and L Apr. 4, 2017 Ingo Tews, ICNT program 26

30 S and L constraints from lower bound of neutron matter energy E PNM [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) S(u) (MeV) 50 Allowed u t = LB u 30 t =1 (S 0,L0 ) 20 S LB (u) 10 Excluded u=n/n n [fm -3 ] Kolomeitsev, Lattimer, Ohnishi, IT, arxiv: Empirical observation: Unitary gas energy seems to be lower bound to neutron-matter energy Ø Constraints on S and L Apr. 4, 2017 Ingo Tews, ICNT program 26

31 S and L constraints from lower bound of neutron matter energy L (MeV) Excluded TMA NLρδ NL3 STOS,TM1 NLρ LS220 u t =1/2 KVOR FSUgold DBHF TKHS KVR DD2, IUFSU DD,D 3 C,DD-F SFHo GCR HS LB (S 0,L0 ) MKVOR u t =1 SFHx Allowed S 0 (MeV) Kolomeitsev, Lattimer, Ohnishi, IT, arxiv: Put constraints on symmetry energy S and its density dependence L. Apr. 4, 2017 Ingo Tews, ICNT program 27

32 Summary 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. 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) 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. Kolomeitsev, Lattimer, Ohnishi, IT, arxiv: Apr. 4, 2017 Ingo Tews, ICNT program 28 L (MeV) Excluded TMA NLρδ NL3 STOS,TM1 NLρ LS220 u t =1/2 KVOR FSUgold DBHF TKHS KVR DD2, IUFSU DD,D 3 C,DD-F SFHo GCR LB (S 0,L0 ) u t =1 HS MKVOR SFHx Allowed S 0 (MeV)

Gandolfi Ø University of Guelph: A. Gezerlis Ø Forschungszentrum Ju lich: A.

33 Thanks Thanks to my collaborators: Ø Technische Universita t Darmstadt: K. Hebeler, J. Lynn, A. Schwenk Ø Universita t Bochum: E. Epelbaum Ø Los Alamos National Laboratory: J. Carlson, S. Gandolfi Ø University of Guelph: A. Gezerlis Ø Forschungszentrum Ju lich: A. Nogga Ø Matej Bel University: Evgeni Kolomeitsev Ø Stony Brook: Jim Lattimer Ø Yukawa Institute Kyoto: Akira Ohnishi Thank you for your attention. Apr. 4, 2017 Ingo Tews, ICNT program 29

Neutron matter from chiral effective field theory interactions

Neutron matter from chiral effective field theory interactions Ingo Tews, In collaboration with K. Hebeler, T. Krüger, A. Schwenk, JINA Neutron Stars, May 26, 2016, Athens, OH Chiral effective field theory