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

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

Outl utline ne Ø Motivation Ø Chiral effective field theory e.g. Epelbaum et al. , PPNP (2006) and RMP (2009) • 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 ) Ø Local chiral interactions Gezerlis, IT, et al., PRL & PRC (2013, 2014, 2016) • Can be constructed up to N 2 LO Ø Results 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

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

Mo Motiv tivatio tion this work 20 LS 180 LS 220 LS 375 FSU2.1 NL3 15 TM1 DD2 E/N [MeV] SFHo SFHx 10 5 MBPT, Fourth order in chiral EFT 0 0 0.05 0.1 0.15 n [fm -3 ] IT, Krüger, Hebeler, Schwenk (2013) Put constraints on symmetry energy and its density dependence L: Ø * + = 28.9 − 34.9 MeV Ø 4 = 43.0 − 66.6 MeV Lattimer, Lim, ApJ (2013) Good agreement with experimental constraints Apr. 4, 2017 Ingo Tews, ICNT program 4

Mo Motiv tivatio tion this work 20 LS 180 LS 220 LS 375 FSU2.1 NL3 15 TM1 DD2 E/N [MeV] SFHo SFHx 10 5 MBPT, Quantum Monte Carlo, Fourth order in chiral EFT Phenomenological forces 0 0 0.05 0.1 0.15 n [fm -3 ] Credit: Stefano Gandolfi IT, Krüger, Hebeler, Schwenk (2013) 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

Chi hiral effective field d the heory for nuclear fo nuc forces Systematic expansion of nuclear forces in low momenta 8 over breakdown scale Λ : : Ø Pions and nucleons as explicit 2 LECs degrees of freedom Ø Long-range physics explicit, short-range physics expanded in general operator basis, couplings (LECs) fit to experiment 7 LECs Ø Separation of scales: \ > > ; # Expand in powers of ∼ < = % Ø Power counting scheme Ø Can work to desired accuracy with systematic error estimates 15 LECs Weinberg, van Kolck, Kaplan, Savage, Wise, Epelbaum, Kaiser, Machleidt, Meißner, Hammer ... Apr. 4, 2017 Ingo Tews, ICNT program 5

Chi hiral effective field d the heory for nuc 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 many- body sector Apr. 4, 2017 Ingo Tews, ICNT program 5

Qua Quantum tum Mo Monte Carlo lo me meth thod od 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

� Qua Quantum tum Mo Monte Carlo lo me meth thod od Particle in a 1D box, solution: sin IJD , L C = I $ J $ B C D = 2 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

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

Qua Quantum tum Mo Monte Carlo lo me meth thod od − 24 . 0 (Exp) E 0 4 He E ( τ ) (GFMC) E ( τ → ∞ ) (GFMC) − 25 . 0 − 28 . 2 − 28 . 3 − 26 . 0 E (MeV) − 28 . 4 − 28 . 5 − 27 . 0 − 28 . 6 − 28 . 7 0 . 52 0 . 54 0 . 56 0 . 58 0 . 60 0 . 62 − 28 . 0 − 29 . 0 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 τ (MeV − 1 ) Lynn, IT, Carlson, Gandolfi, Gezerlis, Schmidt, Schwenk, in preparation. Apr. 4, 2017 Ingo Tews, ICNT program 8

Qua Quantum tum Mo Monte Carlo lo me meth thod od S. Pieper and R. Wiringa Apr. 4, 2017 Ingo Tews, ICNT program 9

Local ch Lo chir iral al in interactio actions 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 W X (T + T U ) Ø Momentum transfer in the exchange channel V = Ø Fourier transformation: S → \, V → Derivatives Solutions: Sources of nonlocalities: Ø Choose local regulators: Ø Usual regulator in relative momenta Ø k-dependent contact operators Ø Use Fierz freedom to choose local set of contact operators Apr. 4, 2017 Ingo Tews, ICNT program 10

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

Local ch Lo chir iral al in interactio actions 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

Lo Local ch chir iral al in interactio actions 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

QMC QMC re results fo for NN NN fo forces NN-only calculation: Q Q Ø QMC: Q 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 Gezerlis, IT, Epelbaum, Freunek, Gandolfi, Hebeler, Nogga, Schwenk, PRL (2013) and PRC (2014) Apr. 4, 2017 Ingo Tews, ICNT program 14

Be Benchmark of of MB 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

Lo Local ch chir iral al in interactio actions 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

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