Recent progress in Auxiliary-Field Diffusion Monte Carlo computation of EOS of nuclear and neutron matter

woundcallousΗμιαγωγοί

1 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

72 εμφανίσεις

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.)

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

Our general goal


SOLVE THE NUCLEAR NON
-
RELATIVISTIC PROBLEM WITH “NO”
APPROXIMATIONS BY DMC (~GFMC).

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).

Nuclear Hamiltonian

The interaction used in this study is A
V
8

cut to

the first six operators
.

where

Inclusion of spin
-
orbit and three body forces is possible
(already done for pure neutron systems).

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
t

=
it

is the imaginary time.

DMC for central potentials

The formal solution

converges to the
lowest energy eigenstate

not
orthogonal to

(R,0)

DMC for central potentials

We can write explicitly the propagator only for
short times:

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
.

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

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, PRL
74
, 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))

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
.

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

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.

Nuclear matter

The functions
f
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

The
many
-
nucleon wave function

is written as the product
of a
Jastrow factor

and an
antisymmetric mean field
wave function
:

Nuclear matter

Simulations

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.



Some checks against simulations with a
larger number
of nucleons (N=76,
108
)

were performed at the extrema
of the density interval considered.

Nuclear matter

Finite size effects

r/r
0

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%!

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

r
s

(expected)

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

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)

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)

---

4x
E
(
4
He) =
-
108.52 MeV:
UNSTABLE!!

10x
E
(
4
He) =
-
271.3 MeV:
BARELY STABLE!!

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.

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
.

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

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

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

Gap in neutron matter

AFDMC should allow for an accurate estimate of the gap in
superfluid
neutron matter.

INGREDIENT NEEDED: A “SUPERFLUID” 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

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.

Gap in Neutron Matter

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.

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).