Superfluidity of Neutron and Nuclear Superfluidity of Neutron and Nuclear MatterMatter

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.)

Why is it of interest?Why is it of interest?

• ““Superfluidity” of nucleiSuperfluidity” of nuclei as been long known. Attractive components of the NN force induce a pairing among nucleons. A few outcomes of it are the even-odd staggering of binding energies or anomalies on the momentum of inertia.

• More recently superfluidity of bulk nuclear matter has been recognized to play a role in the cooling process of cooling process of neutron stars.neutron stars.

Superfulidity and cooling of neutron Superfulidity and cooling of neutron starsstars

Superfluidity of nucleons has essentially three effects on neutrino emission in neutron stars (see e.g. Yakovlev, 2002):

1.1. Suppresses neutrino processes involving nucleons Suppresses neutrino processes involving nucleons (e.g. direct URCA process)(e.g. direct URCA process)

2.2. Initiates a specific mechanism of neutrino emission Initiates a specific mechanism of neutrino emission associated with Cooper pairing of nucleons associated with Cooper pairing of nucleons

3.3. Changes the nucleon heat capacityChanges the nucleon heat capacity

Superfulidity and cooling of neutron Superfulidity and cooling of neutron starsstars

The equations of thermal evolution of a NS are due to Thorne (assuming the internal structure independent on the temperature):

Ter

erc

GmrL

tT

ecQLe

rrcGm

er

r

vr

22

2222

214

2141

Changes occur when T=Tc in a given pairing channel.

It is therefore necessary to know the critical temperature Tc as a function of the density of the nuclear matter.

Superfluid gapSuperfluid gap

The easier way to estimate Tc is through the evaluation of the pairing gap. For instance, in the BCS model we have

76.1

cT

The pairing gap has been estimated by various theories and in different channels (mainly 1S0 and 3P2-3F2).

WE USE AFDMC to estimate the pairing gap as a function WE USE AFDMC to estimate the pairing gap as a function of density.of density.

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

EVEN AT LOW DENSITIES THE DETAIL OF THE EVEN AT LOW DENSITIES THE DETAIL OF THE INTERACTION STILL HAS IMPORTANT INTERACTION STILL HAS IMPORTANT EFFECTS (see Gezerlis, Carlson 2008)EFFECTS (see Gezerlis, Carlson 2008)

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)

Auxiliary Fields DMCAuxiliary Fields DMC

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

expπ21 2Δτλ

21

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

11SS00 gap in neutron matter gap 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

Coefficients from CBF calculations! (Illarionov)

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

Gandolfi S., Illarionov A., Fantoni S., P.F., Schmidt K., PRL 101, 132501 (2008)

13

0101.2105.2

Pair correlation functionsPair correlation functions

Gap in asymmetric matterGap in asymmetric matter

ConclusionsConclusions

• AFDMC can be successfully applied to the study of superfluid gaps in asymmetric nuclear matter and pure neutron matter. Results depend only on the choice of the nn interaction.• Calculations show a maximum of the gap of about 2MeV at about kF=0.6 fm-1

•Large asymmetries seem to increase the value of the gap at the peak•A more systematic analysis is in progress.