Presentazione di PowerPoint

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
  A12 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 v18 potential). In
this study M6, so with Sij tensor operator
5
DMC for central potentials
The formal solution of a Schroedinger equation in
imaginary time t is given by 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. 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
Greens 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 Greens function for the
spin-isospin dependent part of the
potential The xn are auxiliary variables
to be sampled. The effect of the Sn 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(r1rA),
S(s1sA) and ji is a single-particle
base. Spin-isospin states are written as complex
four-spinor components
11
Light nuclei
For nuclei, the Jastrow factor fJ 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 r00.16
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 v6 interaction our results for
alpha particle and 8He 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 16O with
Argonne v14 cutted to v6 and our results are
lower of about 10 respect other variational
results. Preliminary results for 40Ca are also
available.
method 4He MeV 8He 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 fJ 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 A28 nucleons in a periodic box in
a range of densities from 0.5 to 3 times the
experimental equlibrium density of heavy nuclei
r00.16 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.
r/r0 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 v8 cutted to v6 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
• the absence of contributions from the elementary
  diagrams.
• 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 v18 and v14
interactions, the contribution from three
hole-line diagrams in the BHF calculations add a
contribution up to 3 MeV at density below r0, and
decrease the energy at higher (Song et al., PRL
81, 1584 (1998)). Maybe for the v8 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