AFDMC Method for Nuclear Physics and Nuclear Astrophysics
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1 AFDMC Method for Nuclear Physics and Nuclear Astrophysics Thanks to INFN and to F. Pederiva (Trento)
2 Outline Motivations: NN scattering data few body theory. Few-body many body experiments/observations? The AFDMC method Test case: nuclei and neutron drops Neutron rich isotopes Equation of state (EOS) of nuclear and neutron matter Superfluid low-density neutron matter Conclusions
3 Motivations From scattering data and very light nuclei it is possible to fit very accurate realistic Hamiltonians, whose ground state is very hard (or impossible) to be solved. Properties of nuclei are well described by realistic NN and TNI interactions but limited to A=12 with GFMC technique (or less with other accurate few-body methods). EOS of nuclear and neutron matter relevant for nuclear astrophysics (evolution of neutron stars). Theoretical uncertainties on the calculation of symmetric EOS derive both from the approximations introduced in the many-body methods and/or from using model interactions. We address to the first problem.
4 Hamiltonian We consider A non-relativistic nucleons with an effective NN and TNI forces that model the pion-exchange between nucleons: H = 2 2m NN is usually written as v ij = A i=1 2 i + i<j v ij + i<j<k V ijk (1) M v p (r ij )O (p) (i, j) (2) p=1 where O (p) are operators including spin, isospin, tensor and others. The TNI models 2- and 3-π exchange between nucleons with also some excited state. The general form is V ijk = A PW 2π O 2π,PW ijk + A SW 2π O 2π,SW ijk + A R 3π O 3π, R ijk + A R Oijk R. (3)
5 NN and TNI interactions The main contribution to NN interaction is given by OPE, but also other processes are included. The most important operators are O p=1,8 ij = (1, σ i σ j, S ij, L ij S ij ) (1, τ i τ j ), (4) where L ij is the relative angular momentum, S ij is the total spin, and The TNI models following processes: S ij = 3( σ i ˆr ij )( σ j ˆr ij ) σ i σ j. (5) For example the Fujita-Miyazawa diagram gives: O 2π,PW ijk = cyc [ {X ij, X jk } {τ i τ j, τ j τ k } + 1 ] 4 [X ij, X jk ] [τ i τ j, τ j τ k ], (6)
6 DMC for central potentials The formal solution of a Schrodinger equation in imaginary time t is given by: ψ(r, t) = e (H E T )t ψ(r, 0) = = e (E 0 E T )t c 0 ψ 0 (R, 0) + n 0 e (En E T )t c n φ n (R, 0) In the limit of t it converges to the lowest energy eigenstate not orthogonal to ψ(r, 0).
7 DMC for central potentials The propagation is performed by means of the integral equation ψ(r, t) = R ψ(t) = dr G(R, R, t)ψ(r, 0) (7) The propagator is written explicitly only for short times: G(R, R, t) = R e H t R = = ( m ) 3A 2 2π 2 t h 2 2 t e V (R)+V (R ) 2 E T i t e m(r R ) 2 Then we need to iterate many times the above integral equation in the small time-step limit.
8 DMC and nuclear Hamiltonians The DMC technique is easy to apply when the interaction is purely central (spin isospin independent). For realistic NN potentials, the presence of quadratic spin and isospin operators in the propagator imposes the summation over all the possible good spin-isospin single-particle states because ( σ 1 σ 2 ) = (8) This is the approach of the GFMC of Pieper et al., including a huge number of states in the wave function: # A! Z!(A Z)! 2A (9)
9 Auxiliary Field DMC The basic idea of AFDMC is to sample spin-isospin states instead of the explicit summation. The application to pure neutron systems is due to Schmidt and Fantoni 1. Unfortunately the extension to proton-neutron systems interacting with some tensorial force was unexpectedly difficult (due to the bad constraint used to control the Fermion sign problem). 1 K.E. Schmidt and S. Fantoni, Phys. Lett. 445, 99 (1999)
10 Auxiliary Field DMC The method consists in using the Hubbard-Stratonovich transformation in order to reduce the spin-isospin operators in the Green s function from quadratic to linear: e 1 2 to2 = 1 2π dxe x2 2 +x to (10) We need to sample the x auxiliary fields.
11 Auxiliary Field DMC The trial wave function used for the projection has the following form: ψ T (R, S) = Φ J (R) A[φ i ( r j, s j )] (11) where R = ( r 1... r A ), S = (s 1...s A ) and {φ i } is a single-particle base. Φ J (R) is a Jastrow (scalar) factor: Φ J (R) = i<j f (r ij ) (12) Spin-isospin states are written as complex four-spinor components: a i s i b i c i = a i p + b i p + c i n + d i n, d i
12 Nuclei and neutron drops Nuclei and neutron drops: ideal tests to compare the ground state energy and the Spin-Orbit splitting (SOS) with other accurate methods. The Jastrow factor Φ J is a product of two-body factors related to the scalar component of the NN interaction The single-particle base is obtained from a radial part coupled to spherical harmonics; the wave function is an eigenstate of total angular momentum J For neutron drops the Hamiltonian includes an external well to confine neutrons: A H = T i + v ij + V ijk + i=1 i<j i<j<k i V ext (r i ) (13) The estimate of SOS is obtained as the difference between two excited states.
13 Light nuclei With the Argonne AV6 interaction our results for alpha particle and 8 He are in agreement of about 1% with those given by GFMC 2 and EIHH: method 4 He [MeV] 8 He [MeV] AFDMC -27.2(1) 23.6(5) GFMC (1) -23.6(1) EIHH (2) We also computed the ground state energy of 16 O with Argonne AV14 cut to v6 and our results are lower of about 8% with respect other variational estimates: method 16 O [MeV] AFDMC -90.8(1) VMC FHNC/SOC R.B. Wiringa and Steven C. Pieper, Phys. Rev. Lett. 89, 18 (2002)
14 Light nuclei We extended the calculation to the 40 Ca and nuclear matter using the Argonne AV6 3 : nucleus E AFDMC /A E exp /A 4 He He O Ca (1) -16 This simple NN interaction cannot give the experimental ground-state, but the resulting surface coefficient in the Weizsacker formula BE(A, Z) A 1 = a v a s A a Z(Z 1) (A 2Z) 2 1/3 c a A 4/3 sym A 2 + δ A, (14) is 20.5 MeV, not too far from the experimental value of MeV. 3 SG et al., Phys. Rev. Lett. 99, (2007)
15 SOS in Neutron drops In the neutron drop case the NN interaction is the Argonne AV8 and different form of TNI were considered. The AFDMC results for SOS are very close to that of GFMC 4 : TNI AFDMC-SOS GFMC-SOS UIX 1.55(5) 1.5(1) IL1 2.70(5) 2.8(3) IL2 2.93(5) 2.8(3) IL3 3.39(6) 3.6(4) IL4 2.47(6) 2.4(4) The previously constrained-path calculation using AV8 +UIX Hamiltonian estimated SOS=0.8 MeV 5. 4 S.C. Pieper et al., Phys. Rev. C 64, (2001) 5 F. Pederiva et al., Nucl. Phys. A742, 255 (2004)
16 Neutron rich isotopes Neutron rich isotopes 18 O 22 O and 42 Ca 48 Ca modeled by considering only valence neutrons in an external well, and interacting with realistic NN and TNI forces. Very good agreement with measured energies 6. 6 SG et al., Phys. Rev. C 73, (2006), SG et al., Eur. Phys. J. A 35, 207 (2008)
17 Nuclear Matter Nuclear and neutron matter For nuclear and neutron matter the Jastrow factor is essentially the same of that used for nuclei. Calculations were performed with nucleons in a periodic box for several densities. Single-particle orbitals are plane waves fitting the box.
18 Nuclear Matter We computed the energy of 28 nucleons interacting with Argonne AV8 cut to v6 for several densities 7, and we compare our results with those given by FHNC/SOC and BHF calculations 8 : Wrong prediction of equilibrium density ρ 0 =0.16 fm 3 (expected for the absence of TNI). 7 SG et al., Phys. Rev. Lett. 98, (2007) 8 I. Bombaci et al., Phys. Lett. B 609, 232 (2005)
19 Neutron matter The same calculation performed with GFMC 9 was repeated. Using the Argonne AV8 in the Hamiltonian, the energy of 14 neutrons in a periodic box is: ρ [fm 3 ] FP-AFDMC CP-GFMC UC-GFMC CP-AFDMC (2) 6.43(01) 6.32(03) (8) 10.02(02) 9.591(06) (6) 18.54(04) 17.00(27) 20.32(6) (9) 30.04(04) 28.35(50) The fixed-phase approximation improves the agreement between AFDMC and GFMC. 9 J. Carlson et al., Phys. Rev. C 68, (2003)
20 Neutron matter We use the realistic nuclear Hamiltonian AV8 +UIX and 66 neutrons in a periodic box. The AFDMC EOS is compared with the VCS one of Akmal, Pandharipande and Ravenhall 10. The VCS seems to overestimate the TNI contribution. 10 A. Akmal et al., Phys. Rev. C 58, 1804 (1998)
21 Neutron star structure We solved the TOV equation to compare the structure of the star predicted by AFDMC-EOS and the APR one. The less hardness predicted by APR essentially does not change the structure of the star.
22 Superfluid Neutron Matter Superfluid neutron matter The value of the superfluid gap is crucial to study the cooling process of neutron stars. It can be computed from even-odd staggering of estimated energies E(N) of neutron matter. The antisymmetric part of the trial wave function has a BCS structure with a projected number N of neutrons: Φ BCS = A[φ( r 1, s 1, r 2, s 2 )...φ( r i, s i, r j, s j )ψ k ( r l )] (15) The pairing orbitals have the form φ( r ij, s i, s j ) = α c α e i K α r ij ξ S (s i, s j ) (16) and the c α parameters are determined with a CBF calculation 11. The unpaired orbitals, needed for odd number of neutrons, are plane waves. 11 A. Fabrocini et al., Phys. Rev. Lett. 95, (2005)
23 Superfluid Neutron Matter We computed the superfluid gap considering N = and N = neutrons 12. The most recent results predict a larger value of than the previous estimates. The AFDMC calculation is the only one using the full Hamiltonian rather than an effective one. 12 SG et al., Phys. Rev. Lett. in press
24 Conclusions AFDMC useful to study properties of neutron matter with a realistic Hamiltonian, and nuclear matter with a semi-realistic Hamiltonian. The ground state energy of nuclei and neutron drops is in a very good agreement with other accurate methods. The energy of neutron rich isotopes in a well agreement with experimental data. We revisited the EOS of neutron matter and the structure of a neutron star. The 1 S 0 superfluid gap has been accurately computed including the full Hamiltonian rather then some effective interaction.
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