Recent progress in Auxiliary-Field Recent progress in Auxiliary-Field Diffusion Monte Carlo computation of Diffusion Monte Carlo computation of

EOS of nuclear and neutron matterEOS of nuclear and neutron matter

F. PederivaDipartimento di Fisica

Università di Trento I-38050 Povo, Trento, ItalyCNR/INFM-DEMOCRITOS

National Simulation Center, Trieste, Italy

CoworkersCoworkers

S. Gandolfi (SISSA)A. Illarionov (SISSA)S. Fantoni (SISSA)K.E. Schmidt (Arizona S.U.)

PunchlinesPunchlines

High quality (=benchmark) Diffusion High quality (=benchmark) Diffusion Monte Carlo calculations are available Monte Carlo calculations are available now for pure neutron matter EOS with AV* now for pure neutron matter EOS with AV* and U*-IL* potentials. Can we trust and U*-IL* potentials. Can we trust presently available results?presently available results?

We have an accurate estimate of the gap We have an accurate estimate of the gap in superfluid NMin superfluid NM

Our general goalOur general goal

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

Nuclear HamiltonianNuclear Hamiltonian

The interaction between N nucleons can be written in terms of an Hamiltonian of the form:

ji

M

p

pijp

N

i i

i VjiOrvmpH

13

)(

1

2

),()(2

where i and j label the nucleons, rij 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 v18 potential).

Nuclear HamiltonianNuclear Hamiltonian

The interaction used in this study is AV8’ cut to the first six operators.

)(),,(6...1jijji

piSO ττσσ1

wherejijijiijijS σσσrσr ))((3

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

DMC for central potentialsDMC for central potentials

Important fact:The Schroedinger equation in imaginary time is a diffusion equation:

τ),(τ

τ),()(2

22

RRERVm T

where R represent the coordinates of the nucleons, and = it is the imaginary time.

DMC for central potentialsDMC for central potentials

The formal solution

0

τ)(00

τ)(

τ)(

)0,()0,(

)0,(τ),(

0

nnn

EEEE

EH

RceRce

ReR

TnT

T

converges to the lowest energy eigenstatelowest energy eigenstate not not orthogonal to orthogonal to (R,0)(R,0)

DMC for central potentialsDMC for central potentials

We can write explicitly the propagator only for short times:

Δτ2

4σ

πσ21

Δτ),',('||

22

Δτ2

)'()(σ2

)'(2

3

2

Δτ

2

2

m

ee

RRGReR

TERVRVRRA

H

DMC and Nuclear HamiltoniansDMC 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 Sz. For realistic potentials the presence of quadratic spin presence of quadratic spin and isospin operatorsand isospin operators imposes the summation over all summation over all the possible good the possible good SSzz and and TTzz states states..

A

ZAZA 4

)!(!!

The huge number of states limits

present calculations to A14

Auxiliary FieldsAuxiliary 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))

The method consists of using the Hubbard-Stratonovich transformation in order to reduce the spin operators appearing in the Green’s function from quadraticquadratic to linearlinear.

Auxiliary FieldsAuxiliary Fields

For N nucleons the NN interaction can be re-written as

ji

jjiisisdsi sAsVVVV;

ββα;

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:

pdncpbna iiiii

THE INCLUSION OF TENSOR-ISOSPIN TERMS HAS BEEN THE MOST RELEVANT DIFFICULTY IN THE APPLICATION

OF AFDMC SO FAR

Auxiliary FieldsAuxiliary Fields

We can apply the Hubbard-Stratonovich transformation to the Green’s function for the spin-dependent part of the potential:

nnnn

n

O

N

n

OV

Oxxdxe

ee

nn

nnsd

Δτλ2

expπ2

1 2Δτλ21

3

1

Δτλ21

Δτ

2

2 Commutators neglected

The xn are auxiliary variables to be sampled. The effect of the On is a rotation of the spinors of each particle.

Nuclear matter 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 r0=0.16 fm-1. The antisymmetric product A is a Slater determinant of plane waves.

Wave FunctionThe many-nucleon wave functionmany-nucleon wave function is written as the product of a Jastrow factorJastrow factor and an antisymmetric mean field antisymmetric mean field wave functionwave function:

)...;...;...()()...;...;...( 111111 NNNji

ijJNNN Ar ττσσrrττσσrr

Nuclear matterNuclear matter

SimulationsMost simulations were performed in a periodic boxperiodic box containing 28 nucleons28 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 larger number of nucleons (N=76,of nucleons (N=76,108108)) were performed at the extrema of the density interval considered.

Nuclear matterNuclear matter

Finite size effects

E/A(28) [MeV]E/A(28) [MeV] E/A (76) [MeV]E/A (76) [MeV] E/A (108) [MeV]E/A (108) [MeV]

0.50.5 -7.64(3)-7.64(3) -7.7(1)-7.7(1) -7.45(2)-7.45(2)

3.03.0 -10.6(1)-10.6(1) -10.7(6)-10.7(6) -10.8(1)-10.8(1)

CORRECTIONS ARE LESS THAN 3%!

Nuclear matterNuclear matter

We computed the energy of 28 nucleons interacting with Argonne AV8’ 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)

NucleiNucleiNuclei can be treated the same way as nuclear matter. The main technical difference is in the construction of wavefunctions with the correct correct symmetrysymmetry for a given total angular momentum J. At present we confine ourselves to closed-shell closed-shell nucleinuclei (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:

),()( icmii

A

ji

cij sRAfR

r

R: collective coordinate (space, spin, isospin), s: spin, isospin, Rcm: Center of mass coordinate

NucleiNucleiWe performed calculations for 4He, 8He, 16O, 40Ca with a AV6’ interaction and without inclusion of the Coulomb potential.

E(E(44He)He) (MeV)(MeV)

E(E(88He)He) (MeV)(MeV)

AFDMCAFDMC -27.13(10)-27.13(10) -23.6(5)-23.6(5)

GFMCGFMC11 -26.93(1)-26.93(1) -23.6(1)-23.6(1)

EIHHEIHH22 -26.85(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 OPEN SHELL!! (only 1P1P3/23/2 filled, degenerate filled, degenerate with 1Pwith 1P1/21/2 w/o spin- w/o spin-orbit)orbit)

NucleiNuclei

EE (MeV) (MeV) E/AE/A (MeV) (MeV) EEexpexp/A/A (MeV) (MeV)

44HeHe -27.13(10)-27.13(10) -6.78-6.78 -7.07-7.07

88HeHe -23.6(5)-23.6(5) -2.95-2.95 -3.93-3.93

1616OO -100.7(4)-100.7(4) -6.29-6.29 -7.98-7.98

4040CaCa -272(2)-272(2) -6.80-6.80 -8.55-8.55

Periodic Periodic (A=28)(A=28) ------ -12.8(1)-12.8(1) ------

4xE(4He) = -108.52 MeV: UNSTABLE!!

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

Neutron MatterNeutron MatterWe 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 nucleonenergy per nucleon computed with the AV8’ potential in PNM with A=14 neutrons in a periodic box is now 17.586(6)17.586(6) MeV, which compares very well with the GFMC-UC calculations of J. Carlson et al. which give 17.00(27)17.00(27) MeV. The previous published AFDMC result was 20.32(6)20.32(6) MeV.

Neutron MatterNeutron 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 Ravenhall1.

1. A. Akmal, V.R. Pandharipande, and D.G. Ravenhall, PRC 58, 1804 (1998)

+ UIX

Neutron StarsNeutron 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 StarsNeutron 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 matterGap in neutron matterAFDMC 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 wavefunctionJastrow-BCS wavefunction

),()()( SRrfR BCSji

ijJT

where the BCS part is a Pfaffian of orbitals of the form

a

jii

k

kjiij sse

uv

ss ij

a

a ),(),,( rkr

Gap in neutron matterGap in neutron matter

The gap is estimated by the even-odd energy difference at fixed density:

)1()1(21)()( NENENEN

•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 MatterGap in Neutron Matter

13

0101.2105.2

ConclusionsConclusions

• 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 nextWhat’s next Add three-body forces and spin-orbit in the Add three-body forces and spin-orbit in the

nuclear matter propagator (explicit nuclear matter propagator (explicit or or fake nucleons).fake nucleons).

Asymmetric nuclear matter (easy with Asymmetric nuclear matter (easy with some redefinition of the boundary some redefinition of the boundary conditions of the problem)conditions of the problem)

Explicit inclusion of pion (and delta) fields: Explicit inclusion of pion (and delta) fields: development of an EFT-DMC (with P. development of an EFT-DMC (with P. Faccioli and P. Armani, Trento).Faccioli and P. Armani, Trento).