Superfluid Gap in Neutron Matter from a Microscopic Effective Interaction

Transcription

1 Superfluid Gap in Neutron Matter from a Microscopic Effective Interaction Omar Benhar INFN and Department of Physics, Sapienza University I Roma, Italy Based on work done in collaboration with Giulia De Rosi (University of Trento), Alessandro Lovato (ANL), and Angela Mecca (University of Lisbon) Superfluidity and Pairing Phenomena: from Cold Atomic Gases to Neutron Stars ECT*, March 20-25, 2017

2 OUTLINE Motivation Derivation of a microscopic effective interaction from the correlated basis function (CBF) formalism, using the cluster expansion technique Perturbative calculation of equilibrium and non equilibrium properties of nuclear matter Calculation of the 1 S 0 superfluid gap in pure neutron matter Summary & Outlook 1 / 24

3 MOTIVATION Besides being highly valuable in its own right, the study of superfluidity in neutron matter is relevant to the description of many astrophysical processes, such as neutron star cooling and the onset of the gravitational-wave driven CFS instability in rotating relativistic stars Astrophysical applications require theoretical approaches capable to provide a consistent description of properties of neutron star matter other than the equation of state, ranging from the neutrino emission and absorption rates, to the transport coefficients and the superfluid gaps Effective interactions obtained from a microscopic nuclear dynamics strongly constrained by phenomenology combine the flexibility of the effective interaction approach with the ability to provide a realistic description of a variety of equilibrium and non equilibrium properties. 2 / 24

4 Not a new idea. Pioneering studies of the properties of neutron star matter based on microscopic effective interactions date back to the mid 1980s Nuclear Physics A437 (1985) North-Holland Publishing Company SUPERFLUIDITY OF NEUTRON MAlTER (I). Singlet pairing L. AMUNDSEN and E. 0STGAARD Fysisk Institutt, AVH, Universitetet i Trondheim, 7055 Dragvoll, Norway Received 15 June 1984 (Revised 2 November 1984) NUCLEAR PHYSICS A NuclearPhysicsA555 (1993) Abstract: The possibility of neutron or proton singlet pairing and superfluidity in neutron star matter is North-Holland investigated, and the energy gap and corresponding critical temperature is calculated or estimated as a function of Fermi momentum or density. The calculations are performed for three different potentials: a one-pion-exchange gaussian (OPEC) potential, an effective OPEC potential, and an effective Reid soft-core potential obtained by a method of lowest-order constrained variation. The results indicate that neutron superfluidity, corresponding specifically to &-state pairing, Quasiparticle interactions in neutron matter for applications in neutron stars+ may exist in a low-density shell in the nuclear-matter region in neutron stars, i.e. for densities 4.6 x IO g/cm3 < p < 1.6 x loi g/cm3, and the maximum self-consistent energy gap is A( k;) = 2-5 MeV for a neutron Fermi momentum k;= 0.7-l.O fm-. Superfluidity or superconductivity corresponding to $,-state pairing for the proton subsystem is quite likely at higher densities, i.e. for 2.4 x loi g/cm3 < p (7.8 x loi g/cm3, and the maximum energy gap for the OPEC potential is A( kfl) f J. MeV Warnbach, for a proton T.L. Fermi Ainsworth momentum kf and D. fm-. Pines The estimated critical temperatures seem to be higher than expected temperatures inside neutron stars. Department of Physics, University of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, II 61801, USA Received 1. Introduction 4 May 1992 (Revised 16 July 1992) Migdal ) first suggested the possibility of superfluid states in neutron star matter. The effective interaction between two neutrons (or protons) is a combination of Abstract: A microscopic model for the quasiparticle interaction in neutron matter is presented. Both 3 / 24

5 MODELING NUCLEAR DYNAMICS ab initio (bottom-up) approach H = i p 2 i 2m + v ij + v ijk j>i k>j>i v ij provides a very accurate descritpion of the observed properties of the two-nucleon system, in both bound and scettering states, and reduces to Yukawa s one-pion-exchange potential at large distances inclusion of v ijk needed to explain the ground-state energies of the three-nucleon systems v ij is spin and isospin dependent, non spherically symmetric, and strongly repulsive at short distance nuclear interactions can not be treated in perturbation theory in the basis of eigenstates of the non interacting system 4 / 24

6 CORRELATED BASIS FUNCTIONS (CBF) Replace the basis states of the non-interacting system (Fermi gas states in the case of uniform nuclear matter) with a set of correlated states n F G n = F n F G n F G F F n F G = 1 F n 1/2 F G Nn F = S j>i f ij the structure of the two-nucleon correlation operator reflects the complexity of nuclear dynamics f ij = [f T S (r ij ) + δ S1 f tt (r ij )S ij ]P ST P ST S,T =0,1 ( r spin isospin projector operator, S ij = σi α σ β α ij r β ij j δ αβ) shapes of f T S (r ij ) and f tt (r ij ) determined form minimization of the ground-state energy r 2 ij 5 / 24

7 NN POTENTIAL AND CORRELATION FUNCTIONS ANL v 6 potential 6 / 24

8 CBF EFFECTIVE INTERACTION In principle, the complete set of correlated states can be employed to carry out perturbative calculations, using the bare nuclear Hamiltonian However, correlated states are non orthogonal. Owing to this feature, perturbation theory in the correlated basis involves serious additional difficulties Alternatively, the formalism of correlated basis functions can be exploited to obtain a well behaved effective interaction, suitable for perturbation theory in the basis of eigenstates of the non-interacting system 7 / 24

9 Cluster expansion of the Hamiltonian expectation value in the CBF ground state of nuclear matter at density ρ = νk 3 F /6π2 (ν is the degeneracy of the momentum eigenstates) k 2 F H = 0 H 0 = 3 5 2m + ( E[f T S, f tt ]) n n 2 The shapes of the f T S, f tt are determined from minimisation of H computed using the FHNC/SOC integral equations, which allow to sum the relevant cluster terms at all orders The CBF effective interaction is defined adjusting the correlation functions in such a way as to satisfy the relation H FHNC/SOC = 3 k 2 nmax F 5 2m + k 2 F n=2 = 3 5 2m + 0 F G j>i ( E[ f T S, f ) tt ] n v eff ij 0 F G 8 / 24

10 Neglecting three-nucleon forces, one may set n max = 2, and obtain ( f ) 2 ij + fij v ij fij v eff ij = 1 m Note that the correlation function f ij depends on density, and so does the effective interaction Adding three-body cluster terms allows to take into account the leading contributions arising from the three-nucleon potential, the inclusion of which is essential to obtain saturation in isospin-symmetric nuclear matter 9 / 24

11 CBF effective interaction in the T = 1 channel at nuclear matter equilibrium density, obtained from the Argonne v 6 + UIX nuclear Hamiltonian v(r) [MeV] (a) v eff S=0,T =1 (r) v bare S=0,T =1 (r) v(r) [MeV] (b) v eff S=1,T =1 (r) v bare S=1,T =1 (r) r [fm 1 ] v(r) [MeV] 80 (c) r [fm 1 ] v eff t,t =1 (r) v bare t,t =1 (r) r [fm 1 ] 10 / 24

12 DENSITY DEPENDENCE OF THE EFFECTIVE INTERACTION 1 S 0 channel 11 / 24

13 GROUND STATE ENERGY AND SINGLE-PARTICLE SPECTRUM The ground state energy per baryon can be computed at first order in the effective interaction that is, in Hartree Fock approximation for fixed baryon density and arbitrary proton fraction and polarizartions E = N B kλ k 2 2m n λ(k) kλ,k λ kλ k λ v eff kλ k λ A n λ (k)n λ (k ) where λ = 1, 2, 3, 4 corresponds to p, p, n, n, and n λ (k) = θ(k Fλ k ), k Fλ = (3π 2 ρ λ ) 1/3 The same approximation can be employed to obtain the single-nucleon spectrum and the effective masses e λ (k) = k2 2m + k λ kλ k λ v eff kλ k λ A n λ (k), 1 m = 1 de λ (k) k d k 12 / 24

14 Density dependence of the ground state energy per nucleon of unpolarized pure neutron matter (PNM) and isopspin-symmetric nuclear matter (SNM) obtained from the Argonne v 6 + UIX nuclear Hamiltonian Note that the v 6 + UIX Hamiltonian, while predicting saturation at ρ ρ 0 = 0.16 fm 3, underestimates the equilibrium energy of SNM by 5 MeV 13 / 24

15 SINGLE-NUCLEON SPECTRUM Kinetic energy (dashed line) and Hartree-Fock (solid line) spectrum in PNM at nuclear matter equilibrium density 14 / 24

16 EFFECTIVE MASS AND CHEMICAL POTENTIAL left: m (k F )/m, right: µ = e(k F ) 15 / 24

17 Energy of unpolarized nuclear matter as a function of baryon density and proton fraction 0 x p 0.5 E/A [MeV] x u ρ [fm 3 ] / 24

18 EXTENSION TO T > 0 Assuminhg that thermal effect do not significantly affect the dynamics of strong interactions, the effective interaciotns can be used to obtain the properties of nuclear matter at T > 0 Replace θ(k F k) {1 + exp[e(k) µ]/t } 1 17 / 24

19 CALCULATION OF THE 1 S 0 GAP IN PNM Gap equation for l = 0 neutron pairs (k) = 1 π 0 v(k, k ) (k ) [ξ 2 (k ) + 2 (k )] 1/2 k 2 dk where v(k, k ) = 0 j 0 (k) v eff ( 1 S 0 ) j 0 (k ) x 2 dx, j 0 (z) = sin z z and ξ 2 (k) = [e(k) µ] 2 All elements of the calculations can be consistently obtained from the effective interaction 18 / 24

20 The numerical solution of the gap equation is performed using the algorithm developed by Khodel, Khodel and Clark, based on the separation ansatz where v(k, k ) = v F φ(k)φ(k ) + W (k, k ) v F = v(k F, k F ), φ(k) = v(k, k F ) v F φ(k F ) = 1 The gap equation is rewritten in terms of χ(k) = (k)/ F, with F = (k F ) χ(k) = φ(k) 1 π 0 W (k, k )χ(k ) [ξ 2 (k ) + F χ 2 (k )] 1/2 k 2 dk and solved iteratively using matrix inversions 19 / 24

21 DENSITY DEPENDENCE OF THE REDUCED GAP Recall: χ(k) = (k)/ F 20 / 24

22 DEPENDENCE ON THE SINGLE-NUCLEON SPECTRUM Gap function obtained using kinetic energy (dashed line) or Hartree-Fock (solid line) spectrum 21 / 24

23 DENSITY DEPENDENCE OF F Gap function obtained using the bare v 6 potential (dashed line) with kinetic energy spectrum (dashed line) and the CBF effective interaction with Hartee-Fock spectrum (solid line) 22 / 24

24 COMPARISON WITH DIFFERENT APPROACHES Comparison with the gap function obtained using different theoretical approaches. Diamonds: Gandolfi et al (AFDMC); dashed line: Ding et al (CBF); dot-dash line: Cao et al (SFGF); squares: Cao et al (G-matrix) 23 / 24

25 SUMMARY & OUTLOOK The CBF formalism provides a consistent theoretical framework for the unified description of equilibrium and non equilibrium properties of nuclear matter The results of the exploratory studies of zero-temperature PNM and SNM support the validity of the assumptions underlying the CBF effective interaction approach The studies of the superfluid gap will be extended to the 3 P 2 3 F 2 channel, whose understanding is needed for the analysis of the CFS instability Calculations of a variety of properties of nuclear matter at temperatures < 50 MeV are on their way. Early results of these studies are being used in simulations of GW emission from protoneutron stars. 24 / 24

26 Backup slides 25 / 24

27 INTRODUCTION CBF EFFECTIVE INTERACTION EQUILIBRIUM HS TRANSPORT HS SUMMARY &PROSPECTS THE HARD-SPHERE MODEL The Fermi hard-sphere model: point-like spin one-half particles 1 r<a v(r) = 0 r>a? Valuable model to study properties of nuclear matter.? Purely repulsive potential to prevent the possibility of Cooper pairs formation.? A simple many-body system to investigate the validity and robustness of the assumptions of CBF effective interaction approach. 26 / 24

28 INTRODUCTION CBF EFFECTIVE INTERACTION EQUILIBRIUM HS TRANSPORT HS SUMMARY &PROSPECT THE GROUND-STATE ENERGY E 0 = 3k2 F (1 + ) 10m I The accuracy of the variational results depends on the quality of the trial wave function. I Long-range statistical correlations effects in f(r) much larger for =2 than for =4. I DMC overcomes the limitations of the variational approach by using a projection technique on the trial wave function. 9 / 27 / 24

29 INTRODUCTION CBF EFFECTIVE INTERACTION EQUILIBRIUM HS TRANSPORT HS SUMMARY &PROSPECTS MOMENTUM DISTRIBUTION =4 In comparison with non orthogonal CBF perturbation theory Momentum distribution of HS c k F a =0.55 corresponds to n(k) of nuclear matter NM =0.16fm 3 k F =1.33fm 1 S. Fantoni and V. R. Pandharipande, Nucl. Phys. A 427(1984) Nucleons in nuclear matter HS of radius a =0.55/ fm. Virtual scattering processes between strongly correlated particles are mainly driven by the short-range repulsive core of the nucleon-nucleon interaction. 14 / / 24

30 PRESSURE OF SNM AND SYMMETRY ENERGY 29 / 24