1. Superfluidity of Neutron and Nuclear 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.)

2. Why is it of interest? “Superfluidity” of nuclei “Superfluidity” 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. cooling process of neutron stars. More recently superfluidity of bulk nuclear matter has been recognized to play a role in the cooling process of neutron stars.

3. Superfulidity and cooling of neutron stars Superfluidity of nucleons has essentially three effects on neutrino emission in neutron stars (see e.g. Yakovlev, 2002): 1.Suppresses neutrino processes involving nucleons (e.g. direct URCA process) 2.Initiates a specific mechanism of neutrino emission associated with Cooper pairing of nucleons 3.Changes the nucleon heat capacity

4. Superfulidity and cooling of neutron stars The equations of thermal evolution of a NS are due to Thorne (assuming the internal structure independent on the temperature): Changes occur when T=T c in a given pairing channel. It is therefore necessary to know the critical temperature T c as a function of the density of the nuclear matter.

5. Superfluid gap The easier way to estimate Tc is through the evaluation of the pairing gap. For instance, in the BCS model we have The pairing gap  has been estimated by various theories and in different channels (mainly 1 S 0 and 3 P 2 - 3 F 2 ). WE USE AFDMC to estimate the pairing gap as a function of density.

6. 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).

7. Nuclear Hamiltonian The interaction used in this study is AV 8 ’ cut to the first six operators. where EVEN AT LOW DENSITIES THE DETAIL OF THE INTERACTION STILL HAS IMPORTANT EFFECTS (see Gezerlis, Carlson 2008)

8. DMC for central potentials The formal solution lowest energy eigenstatenot orthogonal to (R,0) converges to the lowest energy eigenstate not orthogonal to (R,0)

9. Auxiliary 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)) quadraticlinear 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.

10. 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

11. 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.

12. 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 r 0 =0.16 fm -1. The antisymmetric product A is a Slater determinant of plane waves. Wave Function many-nucleon wave function Jastrow factorantisymmetric mean field wave function The many-nucleon wave function is written as the product of a Jastrow factor and an antisymmetric mean field wave function:

13. 1 S 0 gap in neutron matter AFDMC should allow for an accurate estimate of the gap in superfluid neutron matter. INGREDIENT NEEDED: A “SUPERFLUID” WAVEFUNCTION. Jastrow-BCS 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 Coefficients from CBF calculations ! (Illarionov)

14. 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.

15. Gap in Neutron Matter Gandolfi S., Illarionov A., Fantoni S., P.F., Schmidt K., PRL 101, 132501 (2008)

16. Pair correlation functions

17. Gap in asymmetric matter

18. Conclusions 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 k F =0.6 fm -1 Large asymmetries seem to increase the value of the gap at the peak A more systematic analysis is in progress.