T. Abe (CNS, U. of Tokyo)T. Abe (CNS, U. of Tokyo)

in collaboration within collaboration with

R. Seki (CSUN & KRL, Caltech)R. Seki (CSUN & KRL, Caltech)

Thermal properties of neutron matter Thermal properties of neutron matter by lattice calculation with NN effective field theory by lattice calculation with NN effective field theory

at the next-to-leading orderat the next-to-leading order

高エネルギー加速器研究機構高エネルギー加速器研究機構素粒子原子核研究所素粒子原子核研究所

Nov. 20, 2007Nov. 20, 2007

KEKKEK　研究会　「原子核・ハドロン物理：横断研究会」　研究会　「原子核・ハドロン物理：横断研究会」　　 RCNP 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」研究会　「核子と中間子の多体問題の統一的描像に向けて」

大阪大学核物理研究センタ大阪大学核物理研究センターー

Dec. 15, 2007Dec. 15, 2007

OutlineOutline

1.1. MotivationMotivation

2.2. Formulation: NN EFT on the LatticeFormulation: NN EFT on the Lattice

3.3. LO calc. (cLO calc. (c00 only): only):

a.a. 11SS00 Pairing Gap @ T ~ 0 in Thermodynamic & Continuum Limits Pairing Gap @ T ~ 0 in Thermodynamic & Continuum Limits

b.b. Phase Diagram of Low-Density Neutron Matter in Thermodynamic & Phase Diagram of Low-Density Neutron Matter in Thermodynamic & Continuum Limits Continuum Limits

4.4. NLO calc. (cNLO calc. (c00 & c & c22): Preliminary Results & Comparisons w/ LO ): Preliminary Results & Comparisons w/ LO calc.calc.

a.a. 11SS00 Pairing Gap @ T ~ 0 in Ns = 4 Pairing Gap @ T ~ 0 in Ns = 43 3 & n = 1/4& n = 1/4

b.b. Phase Diagram of Neutron Matter in Ns = 4Phase Diagram of Neutron Matter in Ns = 433 & n = 1/4 & n = 1/4

5.5. Summary & OutlookSummary & Outlook

• D. J. Dean & M. Hjorth-Jensen, Rev. Mod. Phys. 75, 607 (2003)

kF ~ 1.68 fm-1 (ρ ~ 0.16 fm-3)

for neutron matter

Polarization effects

BCS calc

BCS gap equation

• 11SS00 Pairing gap Pairing gap △ △ (Neutron Matter)(Neutron Matter)1.1. MotivationMotivation

1.1. MotivationMotivation

• Thermal PropertiesThermal Properties (Low-density Neutron Matter) (Low-density Neutron Matter)

- - 11SS00 Pairing Gap Pairing Gap

- Phase Diagram- Phase Diagram

Normal-to-Superfluid Phase Transition

Tc(Tc(ρρ))

△△(T~0)(T~0)

• Calculation MethodCalculation Method

Lattice FrameworkLattice Framework

Quantum Monte Carlo Quantum Monte Carlo (QMC)(QMC)

Hybrid Monte Carlo (HMC)Hybrid Monte Carlo (HMC)

Nucleon-Nucleon Nucleon-Nucleon

Effective Field TheoryEffective Field Theory

(NN EFT)(NN EFT)

++

2.2. Formulation: NN EFT on the LatticeFormulation: NN EFT on the Lattice - Effective Field Theory (EFT) - Effective Field Theory (EFT)

low-energy physics long-distance dynamics

- Nucleon-Nucleon Effective Field Theory (NN EFT)- Nucleon-Nucleon Effective Field Theory (NN EFT)

Symmetries of underlying theory (QCD)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- 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)connection to the underlying theory (QCD)• systematic improvement of the calculationssystematic improvement of the calculations

• Non-relativistic HamiltonianNon-relativistic Hamiltonian

w/

• Non-relativistic Lattice HamiltonianNon-relativistic Lattice Hamiltonian

c.f.) Attractive Hubbard Modelc.f.) Attractive Hubbard Model

Extended Attractive Hubbard Model Extended Attractive Hubbard Model

2.2. Formulation Formulation

cc00 (LO) (LO)cc00 & c & c22 (NLO) (NLO)

• K (reaction) MatrixK (reaction) Matrix

• Potential TermsPotential Terms

Effective Range Expansion on the LatticeEffective Range Expansion on the Lattice

Luscher’s method ~ K matrix with asymptotically standing-wave boundary conditionLuscher’s method ~ K matrix with asymptotically standing-wave boundary condition

• R. Seki, & U. van Kolck, Phys. Rev. C 73, 044006 (2006)

• R. Seki, & U. van Kolck, Phys. Rev. C 73, 044006 (2006).

Observables(a0, r0)

Coupling Constants & Regularization Scale(c0, c2, …, Λ(~π/a))

where

3.3. LO calc. (cLO calc. (c00 only) only)

• Calculation MethodsCalculation Methods

- Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique- Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique - Hybrid Monte Carlo (HMC)- Hybrid Monte Carlo (HMC)

Set upSet up

• Parameter set upParameter set up

- k- kFF = 15, 30, 60 MeV = 15, 30, 60 MeV

- N- Ntt = 2 – 128 = 2 – 128 (for observing the phase transition)(for observing the phase transition)

- N- Nss = 4 = 433, 6, 633, 8, 83 3 (DQMC), (DQMC), && 10 103 3 (HMC) (HMC) (for taking the thermodynamic limit)(for taking the thermodynamic limit)

- n = 1/16, 1/8, 3/16 1/4, 3/8, - n = 1/16, 1/8, 3/16 1/4, 3/8, && 1/2 1/2 (for taking the continuum limit)(for taking the continuum limit)

sample # ~ 2,000 – 10,000 with 50 – 100 thermalization stepssample # ~ 2,000 – 10,000 with 50 – 100 thermalization steps

Performed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAMEPerformed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAME

3.3. Results & DiscussionsResults & Discussions

• S-wave Pair Correlation FunctionS-wave Pair Correlation Function

w/ S-wave pair field S-wave pair field & # of spatial lattice sites # of spatial lattice sites

• Estimation of Estimation of ΔΔ

a. a. 11SS00 Pairing Gap @ T ~ 0 Pairing Gap @ T ~ 0kF = 0.15 fm-1(30 MeV)

a = 12.82 fm

t = 0.1261 MeV

Ns = 83

• M. Guerrero, G. Ortiz, & E. Gubernatis, Phys. Rev. B M. Guerrero, G. Ortiz, & E. Gubernatis, Phys. Rev. B 6262, 600 (2000), 600 (2000)

• Matrix-decomposition Matrix-decomposition

Stabilization MethodStabilization Method

kF = 0.3 fm-1(60 MeV)

Ns = 63

continuum limitcontinuum limit

n -> 0 (a -> 0)n -> 0 (a -> 0)

thermodynamic limitthermodynamic limit

N -> ∞N -> ∞

11SS00 pairing gap △ in thermodynamic & continuum limits pairing gap △ in thermodynamic & continuum limits

• Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen & E. Osnes, Nucl. Phys. A 604, 446 (1996)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)J. Wambach, T. L. Ainsworth & D. Pines, Nucl. Phys. A 555, 128 (1993)

BCS calcBCS calc

w/ polarization effectsw/ polarization effects

11SS00 pairing gap △ in thermodynamic & continuum limits pairing gap △ in thermodynamic & continuum limits

• T. Abe & R. Seki, arXiv:07082523 T. Abe & R. Seki, arXiv:07082523

• 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)T. Abe & R. Seki, arXiv:0708.2523 (2007)

Lattice EFTLattice EFT

• the size of △the size of △

no evidence of significant suppression of △no evidence of significant suppression of △

kkFF [MeV] [MeV] ρρ △△MCMC [MeV] [MeV] △△MFMF [MeV] [MeV] △△MCMC /△ /△MF MF

15 ~ 9 x 10-5 ρρ0 0.0207(9) 0.029680.02968 0.70(2)0.70(2)

30 ~ 7 x 10-4 ρρ0 0.127(5)0.127(5) 0.19380.1938 0.66(2)0.66(2)

60 ~ 6 x 10-3 ρρ0 0.62(2)0.62(2) 0.8750.875 0.71(3)0.71(3)

(ρ0 = 0.16 fm-

3)

• ratio of ratio of △△MCMC to △ to △MFMF

noticeable reduction of noticeable reduction of △△MCMC fromfrom △ △MFMF by ~ 30 %by ~ 30 %

importance of pairing correlation induced by many-body importance of pairing correlation induced by many-body effects effects

even at low density even at low density

ρ ~ 10-4 ρ0 – 10-2 ρ0

Discussions about Discussions about 11SS00 pairing gap △ pairing gap △

TcTc

kF = 0.15 fm-1(30 MeV)

a = 12.82 fm

t = 0.1261 MeV

N = 83

b. Phase Diagram: b. Phase Diagram: 11SS00 Superfluid Phase Superfluid Phase TransitionTransition

• S-wave Pair Correlation FunctionS-wave Pair Correlation Function

w/w/

• Determination of TcDetermination of Tc

TcTc is given by the inflexion point of

3.3. Results & DiscussionsResults & Discussions

3.3. Results & DiscussionsResults & Discussions

• Pauli Spin SusceptibilityPauli Spin Susceptibility

b. Phase Diagram: Pseudo Gapb. Phase Diagram: Pseudo Gap

• Determination of T*Determination of T*

T*T* is identified with the maximum position of

T*T*

(BCS limit)(BCS limit) (BEC limit)(BEC limit)

kF = 0.15 fm-1(30 MeV)

a = 12.82 fm

t = 0.1261 MeV

N = 83

• A. Sewer, X. Zotos & H. Beck, Phys. Rev. B66, 140504 (2002)

BECBECBCSBCS

BCS-BEC CrossoverBCS-BEC Crossover

(BCS limit)(BCS limit)

(BEC limit)(BEC limit)

A. Sewer, X. Zotos & H. Beck, Phys. Rev. B66, 140504 (2002)

|c|c00|/(a|/(a33t) = 0, 2, 4, 6, 8, 10, 12 t) = 0, 2, 4, 6, 8, 10, 12

(from top to bottom)(from top to bottom)

• E. Burovski, N. Prokofev, B. Svistunov, & M. Troyer, Phys. Rev. Lett. 96, 160402 (2006)

thermodynamic limitthermodynamic limit

continuum limitcontinuum limit

kF = 60 MeV

kF = 30 MeVkF = 15 MeV

Finite-size Scaling for Tc & T*Finite-size Scaling for Tc & T*

Phase Diagram in thermodynamic & continuum limits Phase Diagram in thermodynamic & continuum limits

normalnormalpseudo gappseudo gap

11SS00 superfluid superfluid

T*T*

TcTc

• T. Abe & R. Seki, arXiv:07082523 T. Abe & R. Seki, arXiv:07082523

• △△/T/T

- approach the BCS value (△- approach the BCS value (△MFMF/Tc ~ 1.76) as the density decreases/Tc ~ 1.76) as the density decreases

evidence of the deviation from BCS weak-coupling approx.evidence of the deviation from BCS weak-coupling approx. even at low density rangingeven at low density ranging ρρ ~ 10 ~ 10-4-4 ρρ00 – 10 – 10-2-2 ρρ00

• phase diagram of low-density neutron matterphase diagram of low-density neutron matter - - drawn for the first time in a sense of ab initio calculationdrawn for the first time in a sense of ab initio calculation - existence of pseudo gap phase - existence of pseudo gap phase induced by the strong short-range correlationinduced by the strong short-range correlation

kkFF [MeV] [MeV] ρρ △△ [MeV][MeV] Tc Tc [MeV][MeV] T*T* [MeV] [MeV] △△/Tc/Tc △△/T*/T*

15 ~ 10-4 ρρ0 0.0207(9) 0.0124(6)0.0124(6) 0.013(1)0.013(1) 1.7(2)1.7(2) 1.6(2)1.6(2)

30 ~ 10-3 ρρ0 0.127(5)0.127(5) 0.066(3)0.066(3) 0.081(9)0.081(9) 1.9(2)1.9(2) 1.6(2)1.6(2)

60 ~ 10-2 ρρ0 0.62(2)0.62(2) 0.30(1)0.30(1) 0.49(5)0.49(5) 2.1(2)2.1(2) 1.3(2)1.3(2)

Discussions about Phase DiagramDiscussions about Phase Diagram

4.4. NLO calc. (cNLO calc. (c00 & c & c22))

• Lattice HamiltonianLattice Hamiltonian Determinantal Quantum Monte Carlo (DQMC) MethodDeterminantal Quantum Monte Carlo (DQMC) Method

cc00 (LO) (LO)

cc00 & c & c22 (NLO) (NLO)

• Calculation MethodsCalculation Methods

- Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique- Detarminantal Quantum Monte Carlo (DQMC) w/ MDS technique

All orders in cAll orders in c00 included, c included, c22 treated perturbatively treated perturbatively

Set upSet up

• Parameter set upParameter set up

- - kkFF = 60, 90, 120 MeV = 60, 90, 120 MeV

- - Temporal lattice: NTemporal lattice: Ntt = 4 – 128 = 4 – 128 (for observing the phase transition)(for observing the phase transition)

- Spatial lattice: N- Spatial lattice: Nss = 4 = 433

- Lattice filling: n = 1/4- Lattice filling: n = 1/4

sample # ~ 1,000 – 10,000 with 10 – 100 thermalization stepssample # ~ 1,000 – 10,000 with 10 – 100 thermalization steps

Performed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAMEPerformed @ NERSC Seaborg, Bassi & Titech GRID, TSUBAME Comparison w/ one-parameter calc. @ Ns = 4Comparison w/ one-parameter calc. @ Ns = 433 & k & kFF = 60, 90, 120 MeV = 60, 90, 120 MeV

BCS calcBCS calc

w/ polarization effectsw/ polarization effects

LO (cLO (c00 only) only)NLO (cNLO (c00 & c & c22))

• Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen & E. Osnes, Nuclear Phys. A 604, 446 (1996)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)J. Wambach, T. L. Ainsworth & D. Pines, Nuclear Phys. A 555, 128 (1993)

4. Preliminary Result & Comparison: △ @ Ns = 44. Preliminary Result & Comparison: △ @ Ns = 433 & n = 1/4 & n = 1/4 (w/o taking thermodynamic & continuum limits)(w/o taking thermodynamic & continuum limits)

ρ ~ 0.05ρ0

ρ ~ 0.02ρ0

11SS00 superfluid superfluid

pseudo gappseudo gapnormalnormal

T*T*

TcTc

4. Preliminary Result & Comparison: Phase Diagram @ Ns = 44. Preliminary Result & Comparison: Phase Diagram @ Ns = 43 3 & & n = 1/4 n = 1/4 (w/o taking thermodynamic & continuum limits)(w/o taking thermodynamic & continuum limits)

4.4. Summary Summary

• LO calc. @ T≠0 in Ns = 4LO calc. @ T≠0 in Ns = 433, 6, 633, 8, 833 (DQMC), 10 (DQMC), 103 3 (HMC) & Nt =2 – 128 (HMC) & Nt =2 – 128 • NLO calc. @ T≠0 in Ns = 4NLO calc. @ T≠0 in Ns = 433 & Nt =4 – 128 (DQMC) & Nt =4 – 128 (DQMC)

11SS00 pairing gap @ T ~ 0 pairing gap @ T ~ 0

- Reduction of △ by - Reduction of △ by ~ 30 %~ 30 % from BCS weak-coupling approx. from BCS weak-coupling approx.

Phase diagramPhase diagram

- - Existence ofExistence of Pseudo gap Pseudo gap

Importance of neutron-neutron pairing correlation even at low densityImportance of neutron-neutron pairing correlation even at low density

11SS00 pairing gap @ T ~ 0 & phase diagram pairing gap @ T ~ 0 & phase diagram

@ Ns = 4@ Ns = 433, n = 1/4 & k, n = 1/4 & kFF = 60, 90, 120 MeV = 60, 90, 120 MeV

- △ decreased- △ decreased - Tc & T* unaltered btw LO & NLO calc. -> thermodynamics controlled by LO ??- Tc & T* unaltered btw LO & NLO calc. -> thermodynamics controlled by LO ??

Feasible approach for the consistent calculation w/ NN EFT Feasible approach for the consistent calculation w/ NN EFT

up to ~ the pion mass (at least @ T~ 0) up to ~ the pion mass (at least @ T~ 0)

LO calc. (cLO calc. (c00 only) only)

NLO calc. (cNLO calc. (c00 & c & c22)) preliminarypreliminary

4.4. Outlook Outlook

• Completion of NLO calc. in thermodynamic & continuum limitsCompletion of NLO calc. in thermodynamic & continuum limits• Calculation @ higher density Calculation @ higher density by including pionspions, …

• Other partial wavesOther partial waves 3P-F2, … <- astrophysical interest from internal structure of NS • Nuclear matter Nuclear matter enhancement of △ by polarization effects??, …enhancement of △ by polarization effects??, …• Application to the finite nucleiApplication to the finite nuclei di-neutron correlation in halo nuclei (dimer), neutron droplets, …

ρρ kkFF [MeV] [MeV] Potential termPotential term

~ 0.006ρρ0 0 - 60 1 term (a1 term (a00))

~ 0.07ρρ0 0 - 1400 - 140 2 terms (a2 terms (a00, r, r00))

~ 0.5ρρ0 0 - 2600 - 260 2 terms + 2 terms + ππ

ρ0 = 0.16 fm-3