Thermal Properties of Simple Nucleon Matter on a Lattice Takashi - PowerPoint PPT Presentation

Spin and Quantum Structure in Hadrons, Nuclei and Atoms (SQS04) Thermal Properties of Simple Nucleon Matter on a Lattice Takashi Abe Department of Physics Tokyo Institute of Technology In Collaboration with R. Seki & A.N. Kocharian (CSUN) 2/19/04 @ Tokyo Inst. Tech.

Contents 1. Introduction: Neutron Matter 2. Nuclear Matter on a Lattice 3. Thermal Properties of Simple Nucleon Matter on a Lattice 4. Summary & Future

1. Introduction: Neutron Matter • Neutron Matter → supposed to exist inside the neutron star rich phase structure @ finite T & ρ • Interior of Neutron Star Neutron Matter ( neutron : spin-up , : spin-down ) outer crust (nuclei + e - ) inner crust (nuclei + n( 1 S 0 ) + e - ) core superfluid (n( 3 P 2 ) + p( 1 S 0 ) + e - ) pion (kaon) condensates? quark matter?

Some Early Studies • Numerous studies of grand state properties of nuclear & neutron matters [ R.B. Wiringa, V. Fiks, and A. Fabrocini, PR C38 , 1010 (1988); A. Akmal and V.R. Pandharipande, PR C56 , 2261 (1997); etc. ] First lattice study of nuclear matter • (quantum hadrodynamics on momentum lattice) [ R. Brockmann and J. Frank, PRL 68 , 1830, (1992) ] • First study on spatial lattice @ finite temperature [ H.-M. Müller, S.E. Koonin, R. Seki, and U. van Kolck, PR C 61 , 044320 (2000) ]

2. Nuclear Matter on a Lattice • Nuclear Lattice Simulation with Simple Form of Interactions Monte Carlo Methods for Nuclear & Neutron Matter [ H.-M. Müller, S.E. Koonin, R. Seki, and U. van Kolck, PR C61, 044320 (2000) ] Simple form of interactions (central + spin-exchange) to reproduce the saturation density and the binding energy of nuclear matter → Phase transition @ T ≈ 15 MeV & ρ≈ 0.32 fm -3 Nuclear Matter on a Lattice ( neutron : spin-up , : spin-down ) ( proton : spin-up , : spin-down )

Hamiltonian of the System • Initial Hamiltonian in Continuum Space Potential (Central & Spin-exchange Interactions) • Skyrm-like On-site & Next-neighbor Interactions •

Lattice Discretization 1 • Spatial Lattice Discretization • Kinetic Term • Central Potential Term

Lattice Discretization 2 Spin-exchange Potential Term •

Thermal Formalism • Grand Canonical Partition Function • Trotter-Suzuki Approximation • Hubbard-Stratonovitch Transformation For simplicity, only considering the transformation for the on-site part of central potential at one particular site ↓ exponential of a one-body operator and an integration over the auxiliary field χ

Monte Carlo Methods Thermal Observable Evolution Operator • •

Parameters @ Nuclear Lattice Simulation • On-site & Next-neighbor Potential Parameters @ Nuclear Lattice Simulation determined to reproduce the saturation property of nuclear matter • Lattice Spacing setting the quarter-filling of lattices to the normal nuclear density • Spatial Lattices: 4 x 4 x 4 • Temporal Lattices: 2 ~ 25 ( T = 50.0 ~ 4.0 MeV) 4 x 4 x 4 spatial lattices

E/N of Neutron Matter 40 30 E/N [MeV] 20 Temperature 50.0 MeV 20.0 MeV 10 10.0 MeV 5.9 MeV 4.0 MeV 0 0 0.08 0.16 0.24 0.32 ρ [fm -3 ] T. Abe (2003)

3. Thermal Properties of Simple Nucleon Matter on a Lattice 1 S 0 Superfluidity of Single Species Nucleon Matter on a Lattice • → 1 S 0 Superfluidity of Neutron Matter @ Low-density [ T. Abe, R. Seki, and A.N. Kocharian, nucl-th/0312125 ] Mean Field calculations based on BCS formalism with Hartree-Fock Bogoliubov (HFB) approximation to understand the qualitative features in a lattice formulation of Neutron Matter → Support the Nuclear Lattice Simulation Neutron Matter on a Lattice : spin-up , : spin-down ) ( neutron

Hamiltonian of the System • Initial Hamiltonian in Continuum Space Potential (Central & Spin-exchange Interactions) • Skyrm-like On-site & Next-neighbor Interactions •

Lattice Discretization • Spatial Lattice Discretization • Identity of Pauli Spin Matrices • Hamiltonian in discrete space only keeping the single species of nucleons (such as neutron matter) → Extended Attractive Hubbard Model

Mean Field Approach • Hartree-Fock Bogoliubov Approximation On-site (U-term) Next-neighbor (V-term) and assuming both terms as

Hamiltonian in Momentum Space • Fourier Transformation • Bogoliubov-Valatin Transformation Quasi-particle Hamiltonian in momentum space •

Gap Equations from Extended Attractive Hubbard Model • Free Energy of the System Equilibrium Conditions Gap Equations @ T ≠ 0 Gap Equations @ T = 0 • • From these coupled equations, Δ and μ are determined.

Parameters @ Mean Field Calculations • On-site & Next-neighbor Potential Parameters @ Extended Hubbard Model • Lattice Spacing same value appeared in Monte Carlo lattice simulation of nuclear matter Thermodynamic Limit ( N → ∞ ) •

Order parameter as a function of temperature T. Abe, R. Seki, and A.N. Kocharian, nucl-th/0312125

Phase Diagram T. Abe, R. Seki, and A.N. Kocharian, nucl-th/0312125

4. Summary & Future • Summary Monte Carlo simulation shows the phase transition of neutron matter around T ≈ 4.0 MeV & ρ ≈ 0.16 fm -3 . It corresponds to the 1 S 0 superfluid phase transition of single species nucleon matter within a mean field approximation. • Future Superfluid phase transition of double-species nucleon matter have to be investigated within a mean field approximation in order to clarify whether the phenomenological potential parameters are really appropriate or not. Phase shift equivalent potential regularized on a lattice should be determined. □ Nuclear Lattice Collaboration HP □ http://www.csun.edu/~rseki/collaboration/