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

• n 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 )

• uter crust (nuclei + e -)

inner crust (nuclei + n(1S0) + e -) core superfluid (n(3P2) + p(1S0) + 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 C61, 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
• Central Potential Term
• Kinetic 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

• Evolution Operator
• Thermal Observable

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

10 20 30 40 0.08 0.16 0.24 0.32 E/N [MeV] ρ [fm-3] Temperature 50.0 MeV 20.0 MeV 10.0 MeV 5.9 MeV 4.0 MeV

• T. Abe (2003)

• 3. Thermal Properties of Simple Nucleon Matter
• n a Lattice
• 1S0 Superfluidity of Single Species Nucleon Matter on a Lattice

→ 1S0 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 ( neutron : 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

• Spatial Lattice Discretization
• Identity of Pauli Spin Matrices
• Hamiltonian in discrete space
• nly 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 1S0 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/