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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.)
Outline
-Motivations
-The AFDMC method
-Test case: nuclei
-Application: EOS of symmetric nuclear
matter
-Conclusions and perspectives
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)).
Nuclear Hamiltonian
Given A nucleons, the non-relativistic nuclear Hamiltonian is:
A
2
i
M
p
( p)
H 
  v p (rij )O (i, j )
i 1 2mi
i  j p 1
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 M=6, so:
O
p 1... 6
 
 
 (1,  i   j , Sij )  (1, i  j )
with Sij tensor operator


 
Sij  3(rˆij   i )( rˆij   j )   i   j
DMC for central potentials
The formal solution of a Schroedinger equation in imaginary time t
is given by:
 ( R, t )  e
 ( H  ET ) t
 ( R,0)
 e ( E0  ET )t c0 0 ( R,0)   e ( En  ET )t cn  n ( R,0)
n0
It converges to the lowest energy eigenstate not orthogonal to (R,0)
The propagator is written explicitly only for short times:
Re
 Ht
R'  G ( R, R' , t )
3A
2
 m 

e
2

 2 t 
 m ( R  R ') 2
2  2 t
e
 v ( R )  v ( R ')


 ET  t
2


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:
A!
A
2
Z !( A  Z )!
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.
Auxiliary Field DMC
The spin-isospin dependent part of NN interaction can be written as:
Vsid
  1 3 3A ˆ 2
1
   i Ai , j  j i  j   S n n
2 i , j
2  1 n1
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
 
ˆ
S n   i  i  n (i)
i
Auxiliary Field DMC
The Hubbard-Stratonovich transformation is applied to the
Green’s function for the spin-isospin dependent part of the
potential:
e
Vsid t
3A
 e
1
 n Sˆn2 t
2
n 1
e
1
 n Sˆn2 t
2
1

2
 dx e
xn2
   n t xn Sˆn
2
n
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).
Auxiliary Field DMC
The trial wavefunction used for the projection has the following
form
T ( R, S )  J ( R)  A[ji (rj , s j )]
where R=(r1…rA), S=(s1…sA) and {ji} is a single-particle base.
Spin-isospin states are written as complex four-spinor components
 ai 
 
 bi 
si     ai p   bi p   ci n   d i n 
ci
 
d 
 i
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 r0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)).
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)):
method
4He
AFDMC
-27.2(1)
-23.6(5)
GFMC
-26.93(1)
-23.6(1)
[MeV]
8He
[MeV]
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*.
* S. Gandolfi, F. Pederiva, S. Fantoni, K.E. Schmidt, to be submitted for publication.
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 r0=0.16 fm-1.
Single-particle functions are plane waves.
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:
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)
With 76 and 108 nucleons results coincide with that obtained with 28 within 3%.
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).
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.
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.
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.
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.
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 …