1. Recent progress in Auxiliary-Field Diffusion Monte Carlo computation of EOS of nuclear and neutron matter F. Pederiva Dipartimento di Fisica Università di Trento I-38050 Povo, Trento, Italy CNR/INFM-DEMOCRITOS National Simulation Center, Trieste, Italy Coworkers S. Gandolfi (SISSA) A. Illarionov (SISSA) S. Fantoni (SISSA) K.E. Schmidt (Arizona S.U.)

2. Punchlines  High quality (=benchmark) Diffusion Monte Carlo calculations are available now for pure neutron matter EOS with AV* and U*-IL* potentials. Can we trust presently available results?  We have an accurate estimate of the gap in superfluid NM

3. Our general goal  SOLVE THE NUCLEAR NON- RELATIVISTIC PROBLEM WITH “NO” APPROXIMATIONS BY DMC (~GFMC).

4. Nuclear Hamiltonian The interaction between N nucleons can be written in terms of an Hamiltonian of the form: where i and j label the nucleons, r ij is the distance between the nucleons and the O (p) are operators including spin, isospin, and spin-orbit operators. M is the maximum number of operators (M=18 for the Argonne v 18 potential).

5. Nuclear Hamiltonian The interaction used in this study is AV 8 ’ cut to the first six operators. where Inclusion of spin-orbit and three body forces is possible (already done for pure neutron systems).

6. DMC for central potentials Important fact: The Schroedinger equation in imaginary time is a diffusion equation: where R represent the coordinates of the nucleons, and  = it is the imaginary time.

7. DMC for central potentials The formal solution lowest energy eigenstatenot orthogonal to (R,0) converges to the lowest energy eigenstate not orthogonal to (R,0)

8. DMC for central potentials We can write explicitly the propagator only for short times:

9. DMC and Nuclear Hamiltonians The standard QMC techniques are easy to apply whenever the interaction is purely central, or whenever the wavefunction can be written as a product of eigenfunctions of S z. presence of quadratic spin and isospin operatorssummation over all the possible good S z and T z states. For realistic potentials the presence of quadratic spin and isospin operators imposes the summation over all the possible good S z and T z states. The huge number of states limits present calculations to A14

10. Auxiliary Fields The use of auxiliary fields and constrained paths is originally due to S. Zhang for condensed matter problems (S.Zhang, J. Carlson, and J.Gubernatis, PRL74, 3653 (1995), Phys. Rev. B55. 7464 (1997)) Application to the Nuclear Hamiltonian is due to S.Fantoni and K.E. Schmidt (K.E. Schmidt and S. Fantoni, Phys. Lett. 445, 99 (1999)) quadraticlinear The method consists of using the Hubbard- Stratonovich transformation in order to reduce the spin operators appearing in the Green’s function from quadratic to linear.

11. Auxiliary Fields For N nucleons the NN interaction can be re-written as where the 3Nx3N matrix A is a combination of the various v(p) appearing in the interaction. The s include both spin and isospin operators, and act on 4-component spinors: THE INCLUSION OF TENSOR-ISOSPIN TERMS HAS BEEN THE MOST RELEVANT DIFFICULTY IN THE APPLICATION OF AFDMC SO FAR

12. Auxiliary Fields We can apply the Hubbard-Stratonovich transformation to the Green’s function for the spin-dependent part of the potential: Commutators neglected The x n are auxiliary variables to be sampled. The effect of the O n is a rotation of the spinors of each particle.

13. Nuclear matter The functions  J in the Jastrow factor are taken as the scalar components of the FHNC/SOC correlation operator which minimizes the energy per particle of SNM at saturation density r 0 =0.16 fm -1. The antisymmetric product A is a Slater determinant of plane waves. Wave Function many-nucleon wave function Jastrow factorantisymmetric mean field wave function The many-nucleon wave function is written as the product of a Jastrow factor and an antisymmetric mean field wave function:

14. Nuclear matter Simulations periodic box 28 nucleons Most simulations were performed in a periodic box containing 28 nucleons (14 p and 14 n). The density was changed varying the size of the simulation box. Particular attention must be paid to finite size effects. At the higher densities we performed a summation over the first shell of periodic replicas of the simulation cell. larger number of nucleons (N=76,108) Some checks against simulations with a larger number of nucleons (N=76,108) were performed at the extrema of the density interval considered.

15. Nuclear matter Finite size effects   E/A(28) [MeV] E/A (76) [MeV] E/A (108) [MeV] 0.5-7.64(3)-7.7(1)-7.45(2) 3.0-10.6(1)-10.7(6)-10.8(1) CORRECTIONS ARE LESS THAN 3%!

16. Nuclear matter We computed the energy of 28 nucleons interacting with Argonne AV 8 ’ cut to six operators for several densities*, and we compare our results with those given by FHNC/SOC and BHF calculations**: AFDMC EOS differs from all other previous estimates! S. Gandolfi, F. Pederiva, S. Fantoni, K.E. Schmidt, PRL 98, 102503 (2007) **I. Bombaci, A. Fabrocini, A. Polls, I. Vidaña, Phys. Lett. B 609, 232 (2005). Wrong prediction of  s (expected)

17. Nuclei correct symmetry closed-shell nuclei Nuclei can be treated the same way as nuclear matter. The main technical difference is in the construction of wavefunctions with the correct symmetry for a given total angular momentum J. At present we confine ourselves to closed-shell nuclei (J=0) for which the many-body wavefunction is expected to have full spherical symmetry (J=0). In this case it is easy to write the wavefunction as: R: collective coordinate (space, spin, isospin), s: spin, isospin, R cm : Center of mass coordinate

18. Nuclei We performed calculations for 4 He, 8 He, 16 O, 40 Ca with a AV6’ interaction and without inclusion of the Coulomb potential. E( 4 He) (MeV) E( 8 He) (MeV) AFDMC-27.13(10)-23.6(5) GFMC 1 -26.93(1)-23.6(1) EIHH 2 -26.85(2)--- 1. R.B. Wiringa, S.C. Pieper, PRL 89, 182501 (2002) 2. G. Orlandini, W. Leidemann, private comm. OPEN SHELL!! (only 1P 3/2 filled, degenerate with 1P 1/2 w/o spin- orbit)

19. Nuclei E (MeV) E/A (MeV) E exp /A (MeV) 4 He -27.13(10)-6.78-7.07 8 He -23.6(5)-2.95-3.93 16 O -100.7(4)-6.29-7.98 40 Ca -272(2)-6.80-8.55 Periodic (A=28) ----12.8(1)--- 4xE( 4 He) = -108.52 MeV: UNSTABLE!! 10xE( 4 He) = -271.3 MeV: BARELY STABLE!!

20. Neutron Matter We revised the computations made on Neutron Matter to check the effect of the use of the fixed-phase approximation. Results are more stable, and some of the issues that were not cleared in the previous AFDMC work are now under control. energy per nucleon 17.586(6) 17.00(27) 20.32(6) In particular the energy per nucleon computed with the AV8’ potential in PNM with A=14 neutrons in a periodic box is now 17.586(6) MeV, which compares very well with the GFMC-UC calculations of J. Carlson et al. which give 17.00(27) MeV. The previous published AFDMC result was 20.32(6) MeV.

21. Neutron Matter Equation of state of PNM modeled with the AV8’ potential with and without the inclusion of the three-body UIX potential, compared with the results of Akmal, Pandharipande and Ravenhall 1. 1. A. Akmal, V.R. Pandharipande, and D.G. Ravenhall, PRC 58, 1804 (1998) + UIX

22. Neutron Stars Mass-radius relation in a neutron star obtained solving the Tolman Oppenheimer Volkov (TOV) equation using the EOS of pure neutron matter from AFDMC and variational calculations. Mass in is units of M ., radius in Km

23. Neutron Stars Mass-core density relation in a neutron star obtained solving the Tolman Oppenheimer Volkov (TOV) equation using the EOS of pure neutron matter from AFDMC and variational calculations. Mass in is units of M ., core density in fm -3

24. Gap in neutron matter AFDMC should allow for an accurate estimate of the gap in superfluid neutron matter. INGREDIENT NEEDED: A “SUPERFLUID” WAVEFUNCTION. Jastrow-BCS wavefunction Nodes and phase in the superfluid are better described by a Jastrow-BCS wavefunction where the BCS part is a Pfaffian of orbitals of the form

25. Gap in neutron matter The gap is estimated by the even-odd energy difference at fixed density: For our calculations we used N=12-18 and N=62-68. The gap slightly decreases by increasing the number of particles. The parameters in the pair wavefunctions have been taken by CBF calculatons.

26. Gap in Neutron Matter

27. Conclusions AFDMC can be successfully applied to the study of symmetric nuclear matter and pure neutron matter. Results depend only on the choice of the nn interaction. The algorithm has been successfully applied to nuclei The estimates of the EOS computed with the same potential and other methods are quite different. Pure neutron matter has been revised. The AP EOS underestimates the hardness when a pure two body potential is considered We have estimates of the gap within range of other DMC and recent BHF calculations.

28. What’s next  Add three-body forces and spin-orbit in the nuclear matter propagator (explicit  or fake nucleons).  Asymmetric nuclear matter (easy with some redefinition of the boundary conditions of the problem)  Explicit inclusion of pion (and delta) fields: development of an EFT-DMC (with P. Faccioli and P. Armani, Trento).