1. Auxiliary Field Diffusion Monte Carlo study of symmetric nuclear matter S. Gandolfi Dipartimento di Fisica and INFN, Università di Trento I-38050 Povo, Trento, Italy Coworkers F. Pederiva (Trento) S. Fantoni (SISSA) K.E. Schmidt (Arizona S.U.)

2. Outline -Motivations -The AFDMC method -Test case: nuclei -Application: EOS of symmetric nuclear matter -Conclusions and perspectives

3. Motivations - EOS of asymmetric nuclear matter is relevant for astronuclear physic (evolution of neutron stars). - Theoretical uncertainties on the symmetric EOS derive both from the approximations introduced in the many-body methods and from using model interactions (maybe…). - Properties of nuclei are well described by realistic NN and TNI interactions but limited to A=12 with GFMC technique (S.C. Pieper, Nucl. Phys. A 751 (2005)).

4. Nuclear Hamiltonian Given A nucleons, the non-relativistic nuclear Hamiltonian is: where i and j label nucleons and O (p) are operators including spin, isospin, tensor and others. M is the maximum number of operators (18 for the Argonne v 18 potential). In this study M=6, so: with S ij tensor operator

5. DMC for central potentials Schroedinger equation in imaginary time t The formal solution of a Schroedinger equation in imaginary time t is given by: lowest energy eigenstate It converges to the lowest energy eigenstate not orthogonal to The propagator is written explicitly only for short times:

6. DMC and nuclear Hamiltonians The DMC technique is easy to apply when the interaction is purely central. quadratic spin and isospin operatorssummation A is limited to 12 For realistic NN potentials, the presence of quadratic spin and isospin operators in the propagator imposes the summation over all the possible good spin-isospin single-particle states. This is the standard approach of the GFMC of Pieper, Carlson et al. With this explicitely summation A is limited to 12, because the huge number of possible states:

7. Auxiliary Field DMC The basic idea of AFDMC is to sample spin-isospin states instead of explicitely summing over all the possible configurations. The application to pure neutron systems is due to Fantoni and Schmidt (K.E. Schmidt and S. Fantoni, Phys. Lett. 445, 99 (1999)), but it is never had employed for nuclear matter or nuclei. The method consists in using the Hubbard-Stratonovich transformation in order to reduce the spin-isospin operators in the Green’s function from quadratic to linear.

8. Auxiliary Field DMC The spin-isospin dependent part of NN interaction can be written as: where A is a matrix containing the interaction between nucleons, are the eigenvalues of A, and S are operators written in terms of eigenvectors of A

9. Auxiliary Field DMC The Hubbard-Stratonovich transformation is applied to the Green’s function for the spin-isospin dependent part of the potential: The x n are auxiliary variables to be sampled. The effect of the S n is a rotation of the four-component spinors of each particle (written in the proton-neutron up-down basis).

10. Auxiliary Field DMC The trial wavefunction used for the projection has the following form where R=(r 1 …r A ), S=(s 1 …s A ) and {  i } is a single-particle base. Spin-isospin states are written as complex four-spinor components

11. Light nuclei For nuclei, the Jastrow factor  J is a product of two-body factors, which are taken as the scalar components of the FHNC/SOC correlation operator which minimizes the energy per particle of nuclear matter at equlibrium density    0.16 fm -1. The single-particle base is obtained from a radial part coupled to spherical harmonics; the antisymmetric wavefunction is buit to be an eigenstate of total angular momentum J. Radial functions are computed by Hartree-Fock with Skyrme force fitted to light nuclei (X. Bai and J. Hu, Phys. Rev. C 56, 1410 (1997)).

12. Light nuclei With the Argonne v 6 ’ interaction our results for alpha particle and 8 He are in agreement of about 1% with those given by GFMC (R.B. Wiringa and Steven C. Pieper, PRL 89, 18 (2002)): We also computed the ground state energy of 16 O with Argonne v 14 ’ cutted to v 6 ’ and our results are lower of about 10% respect other variational results. Preliminary results for 40 Ca are also available*. method 4 He [MeV] 8 He [MeV] AFDMC-27.2(1)-23.6(5) GFMC-26.93(1)-23.6(1) * S. Gandolfi, F. Pederiva, S. Fantoni, K.E. Schmidt, to be submitted for publication.

13. Nuclear matter For nuclear matter the Jastrow factor  J is taken as the scalar component of the FHNC/SOC correlation operator which minimizes the energy per particle for each density. Calculations were performed with A=28 nucleons in a periodic box in a range of densities from 0.5 to 3 times the experimental equlibrium density of heavy nuclei   =0.16 fm -1. Single-particle functions are plane waves.

14. Nuclear matter: finte-size effects To avoid large finite-size effects, the calculation of two-body interaction is performed with a summation over the first shell of periodic replicas of the simulation cell. However, to test the accuracy of this method, we have done several simulations at the highest and at the lowest density with different numbers of particles: With 76 and 108 nucleons results coincide with that obtained with 28 within 3%.   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)

15. Nuclear matter We computed the energy of 28 nucleons interacting with Argonne v 8 ’ cutted to v 6 ’ for several densities*, and we compare our results with those given by FHNC/SOC and BHF calculations**: Our EOS differs from both EOS computed with different methods. * S. Gandolfi, F. Pederiva, S. Fantoni, K.E. Schmidt, to be published in PRL. **I. Bombaci, A. Fabrocini, A. Polls, I. Vidaña, Phys. Lett. B 609, 232 (2005).

16. Nuclear matter FHNC leads to an overbinding at high density. FHNC/SOC contains two intrinsic approximations violating the variational principle: 1)the absence of contributions from the elementary diagrams. 2)the absence of contributions due to the non-commutativity of correlation operators entering in the variational wavefunction.

17. Nuclear matter S. Fantoni et al. computed the lowest order of elementary diagrams, showing that they are not negligible and give an important contribution to the energy: With the addition of this class of diagrams, FHNC/SOC results are much closer to the AFDMC ones. However the effect of higher order elementary diagrams and commutators is unknown.

18. Nuclear matter BHF predicts an EOS with a shallower binding that the AFDMC one. It has been shown that for Argonne v 18 and v 14 interactions, the contribution from three hole-line diagrams in the BHF calculations add a contribution up to 3 MeV at density below  , and  decrease the energy at higher (Song et al., PRL 81, 1584 (1998)). Maybe for the v 8 ’ interaction such corrections would be similar.

19. Conclusions - AFDMC is an efficient and fast projection algorithm for the computation the ground state energy of nuclei and nuclear matter at zero temperature. - We showed that AFDMC works efficiently with NN interactions containing tensor force, and our results are in agreement with other methods (both for nuclei and nuclear matter). - The number of nucleons in the Hamiltonian has practically no limitatons.

20. Perspectives - Addition of missing terms in the Hamiltonian, such spin-orbit and three-body interactions. - Calculation of EOS of asymmetric nuclear matter, particularly important for prediction of properties of neutron stars. - Calculation of binding energy of heavy nuclei to predict coefficients in the Weizsacker formula to be compared with experimental data to test NN and TNI interactions. - Other …