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高エネルギー加速器研究機構 素粒子原子核研究所 Nov. 20, 2007

RCNP 研究会　「核子と中間子の多体問題の統一的描像に向けて」. KEK 　研究会　「原子核・ハドロン物理：横断研究会」. Thermal properties of neutron matter by lattice calculation with NN effective field theory at the next-to-leading order. T. Abe (CNS, U. of Tokyo) in collaboration with R. Seki (CSUN & KRL, Caltech). 高エネルギー加速器研究機構 素粒子原子核研究所

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高エネルギー加速器研究機構 素粒子原子核研究所 Nov. 20, 2007

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  1. RCNP 研究会　「核子と中間子の多体問題の統一的描像に向けて」 KEK　研究会　「原子核・ハドロン物理：横断研究会」 Thermal properties of neutron matter by lattice calculation with NN effective field theory at the next-to-leading order T. Abe (CNS, U. of Tokyo) in collaboration with R. Seki (CSUN & KRL, Caltech) 高エネルギー加速器研究機構 素粒子原子核研究所 Nov. 20, 2007 大阪大学核物理研究センター Dec. 15, 2007

  2. Outline • Motivation • Formulation: NN EFT on the Lattice • LO calc. (c0 only): • 1S0 Pairing Gap @ T ~ 0 in Thermodynamic & Continuum Limits • Phase Diagram of Low-Density Neutron Matter in Thermodynamic & Continuum Limits • NLO calc. (c0 & c2): Preliminary Results & Comparisons w/ LO calc. • 1S0 Pairing Gap @ T ~ 0 in Ns = 43 & n = 1/4 • Phase Diagram of Neutron Matter in Ns = 43 & n = 1/4 • Summary & Outlook

  3. 1S0 Pairing gap △ (Neutron Matter) Motivation BCS gap equation BCS calc kF ~ 1.68 fm-1 (ρ ~ 0.16 fm-3) for neutron matter Polarization effects • D. J. Dean & M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003)

  4. Motivation • Thermal Properties (Low-density Neutron Matter) - 1S0 Pairing Gap - Phase Diagram Normal-to-Superfluid Phase Transition △(T~0) Tc(ρ) • Calculation Method + Nucleon-Nucleon Effective Field Theory (NN EFT) Lattice Framework Quantum Monte Carlo (QMC) Hybrid Monte Carlo (HMC)

  5. Formulation: NN EFT on the Lattice - Effective Field Theory (EFT) low-energy physics long-distance dynamics - Nucleon-Nucleon Effective Field Theory (NN EFT) Symmetries of underlying theory (QCD) Low-energy theory with the relevant degrees of freedom (N, π, etc.) based on the relevant symmetries of the underlying theory (QCD) in low-energy physics (Lorentz, parity, time-reversal etc.) - Power counting Systematic expansion in powers of p / Q (p: long-distance scale, Q: short-distance scale) Coupling constants Experimental data (phase shift …) connection to the underlying theory (QCD) systematic improvement of the calculations

  6. Formulation • Non-relativistic Hamiltonian w/ • Non-relativistic Lattice Hamiltonian c.f.) Attractive Hubbard Model Extended Attractive Hubbard Model c0 (LO) c0 & c2 (NLO)

  7. R. Seki, & U. van Kolck, Phys. Rev. C 73, 044006 (2006) Effective Range Expansion on the Lattice • Potential Terms • K (reaction) Matrix Luscher’s method ~ K matrix with asymptotically standing-wave boundary condition

  8. Observables (a0, r0) Coupling Constants & Regularization Scale (c0, c2, …, Λ(~π/a)) where R. Seki, & U. van Kolck, Phys. Rev. C 73, 044006 (2006).

  9. LO calc. (c0 only)

  10. Set up • Calculation Methods - Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique - Hybrid Monte Carlo (HMC) • Parameter set up - kF = 15, 30, 60 MeV - Nt = 2 – 128 (for observing the phase transition) - Ns = 43, 63, 83 (DQMC), & 103 (HMC) (for taking the thermodynamic limit) - n = 1/16, 1/8, 3/16 1/4, 3/8, & 1/2 (for taking the continuum limit) sample # ~ 2,000 – 10,000 with 50 – 100 thermalization steps Performed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAME

  11. Results & Discussions a. 1S0 Pairing Gap @ T ~ 0 kF = 0.15 fm-1(30 MeV) a = 12.82 fm t = 0.1261 MeV Ns = 83 • S-wave Pair Correlation Function w/ S-wave pair field & # of spatial lattice sites • Estimation of Δ • Matrix-decomposition Stabilization Method M. Guerrero, G. Ortiz, & E. Gubernatis, Phys. Rev. B 62, 600 (2000)

  12. 1S0 pairing gap △ in thermodynamic & continuum limits thermodynamic limit continuum limit N -> ∞ n -> 0 (a -> 0) kF = 0.3 fm-1(60 MeV) Ns = 63

  13. 1S0 pairing gap △ in thermodynamic & continuum limits T. Abe & R. Seki, arXiv:07082523 BCS calc w/ polarization effects Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen & E. Osnes, Nucl. Phys. A 604, 446 (1996) J. Wambach, T. L. Ainsworth & D. Pines, Nucl. Phys. A 555, 128 (1993)

  14. Lattice EFT A. Fabrocini, S. Fantoni, A. Y. Illarionov, & K. E. Schmidt, PRL 95, 192501 (2005) J. Carlson, Nucl. Phys. A 787, 516c-523c (2007); A. Gezerlis & J. Carlson, arXiv:0711.3006 (2007) T. Abe & R. Seki, arXiv:0708.2523 (2007)

  15. Discussions about 1S0 pairing gap △ • the size of △ no evidence of significant suppression of △ • ratio of △MC to △MF noticeable reduction of △MCfrom △MFby ~ 30 % • importance of pairing correlation induced by many-body effects even at low density ρ ~ 10-4ρ0– 10-2ρ0 (ρ0 = 0.16 fm-3)

  16. b. Phase Diagram: 1S0 Superfluid Phase Transition • Results & Discussions • S-wave Pair Correlation Function w/ Tc kF = 0.15 fm-1(30 MeV) a = 12.82 fm t = 0.1261 MeV N = 83 • Determination of Tc Tc is given by the inflexion point of

  17. Results & Discussions b. Phase Diagram: Pseudo Gap kF = 0.15 fm-1(30 MeV) a = 12.82 fm t = 0.1261 MeV N = 83 • Pauli Spin Susceptibility T* • Determination of T* T*is identified with the maximum position of (BCS limit) (BEC limit) • A. Sewer, X. Zotos & H. Beck, Phys. Rev. B66,140504 (2002)

  18. BCS-BEC Crossover BEC BCS (BCS limit) (BEC limit) |c0|/(a3t) = 0, 2, 4, 6, 8, 10, 12 (from top to bottom) A. Sewer, X. Zotos & H. Beck, Phys. Rev. B66,140504 (2002)

  19. Finite-size Scaling for Tc & T* continuum limit kF = 60 MeV kF = 30 MeV kF = 15 MeV thermodynamic limit E. Burovski, N. Prokofev, B. Svistunov, & M. Troyer, Phys. Rev. Lett. 96, 160402 (2006)

  20. Phase Diagram in thermodynamic & continuum limits T. Abe & R. Seki, arXiv:07082523 T* normal pseudo gap Tc 1S0 superfluid

  21. Discussions about Phase Diagram • phase diagram of low-density neutron matter - drawn for the first time in a sense of ab initio calculation - existence of pseudo gap phase induced by the strong short-range correlation • △/T - approach the BCS value (△MF/Tc ~ 1.76) as the density decreases • evidence of the deviation from BCS weak-coupling approx. even at low density rangingρ ~ 10-4ρ0 – 10-2ρ0

  22. NLO calc. (c0 & c2)

  23. Determinantal Quantum Monte Carlo (DQMC) Method • Lattice Hamiltonian c0 (LO) c0 & c2 (NLO)

  24. Set up • Calculation Methods - Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique All orders in c0 included, c2 treated perturbatively • Parameter set up - kF = 60, 90, 120 MeV - Temporal lattice: Nt = 4 – 128 (for observing the phase transition) - Spatial lattice: Ns = 43 - Lattice filling: n = 1/4 sample # ~ 1,000 – 10,000 with 10 – 100 thermalization steps Performed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAME • Comparison w/ one-parameter calc. @ Ns = 43 & kF = 60, 90, 120 MeV

  25. 4. Preliminary Result & Comparison: △ @ Ns = 43 & n = 1/4 (w/o taking thermodynamic & continuum limits) BCS calc preliminary LO (c0 only) NLO (c0 & c2) w/ polarization effects ρ ~ 0.05ρ0 ρ ~ 0.02ρ0 Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen & E. Osnes, Nuclear Phys. A 604, 446 (1996) J. Wambach, T. L. Ainsworth & D. Pines, Nuclear Phys. A 555, 128 (1993)

  26. 4. Preliminary Result & Comparison: Phase Diagram @ Ns = 43 & n = 1/4 (w/o taking thermodynamic & continuum limits) T* preliminary Tc normal pseudo gap 1S0 superfluid

  27. Summary LO calc. (c0 only) • LO calc. @ T≠0 in Ns = 43, 63, 83 (DQMC), 103 (HMC) & Nt =2 – 128 • NLO calc. @ T≠0 in Ns = 43 & Nt =4 – 128 (DQMC) • 1S0 pairing gap @ T ~ 0 - Reduction of △ by ~ 30 % from BCS weak-coupling approx. • Phase diagram - Existence of Pseudo gap • Importance of neutron-neutron pairing correlation even at low density NLO calc. (c0 & c2)preliminary • 1S0 pairing gap @ T ~ 0 & phase diagram @ Ns = 43, n = 1/4 & kF = 60, 90, 120 MeV - △ decreased - Tc & T* unaltered btw LO & NLO calc. -> thermodynamics controlled by LO ?? • Feasible approach for the consistent calculation w/ NN EFT up to ~ the pion mass (at least @ T~ 0)

  28. Outlook • Completion of NLO calc. in thermodynamic & continuum limits • Calculation @ higher density by includingpions, … • Other partial waves 3P-F2, … <- astrophysical interest from internal structure of NS • Nuclear matter enhancement of △ by polarization effects??, … • Application to the finite nuclei di-neutron correlation in halo nuclei (dimer), neutron droplets, … ρ0 = 0.16 fm-3

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