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Lattice QCD workshop at KEK, 2014/01/20 -/22
Lattice QCD at low temperature and finite density���- early onset/Silver Blaze problem
Keitaro Nagata KEK, Theory Center
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ü KN, arXiv:1204.6480 ü KN, Motoki, Nakagawa, Nakamura, Saito, PTEP01A103(’12). ü KN, Nakamura, PRD82, 094027(‘10)
Lattice QCD workshop at KEK, 2014/01/20 -/22
Outline
• Introduction • Framework : reduction formula, reduced matrix • Zero temperature limit of the fermion determinant • mπ/2 ? or MN/3 ?
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Introduction
3
Lattice QCD workshop at KEK, 2014/01/20 -/22
Introduction
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εF
• Hadronic phase at low-T - d.o.f is nucleons and pions
ε>εF
• In order to add one particle, energy larger than fermi energy is needed.
T=0
Lattice QCD workshop at KEK, 2014/01/20 -/22
Early onset problem
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• Hadronic phase at low-T : d.o.f is nucleons and pions • µ for the onset of quark number at T=0 – expectation : µ=MN/3 – lattice calculations : µ=mπ/2
Glasgow method - propagator matrix - reweigh5ng
[Barbour et. al. PRD56, 7063 (1997)] 6^4, staggered, mq=0.1
Lattice QCD workshop at KEK, 2014/01/20 -/22
Silver Blaze
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• The effect of µ does not appear, although the action explicitly depends on µ. - well-known property of fermion at T=0. - But it is unclear how it is realized in field theory).
• Cohen(‘04) considered isospin chemical potential case and showed. - fermion determinant of QCD is independent of µ at T=0 for µ
less then half the pion mass. • (baryon Silver Blaze (µ<MN/3) has not been solved)
Lattice QCD workshop at KEK, 2014/01/20 -/22
Motivation
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• In previous studies by Glasgow method, it was unclear what is the origin of the early onset problem.
• Isospin Silver Blaze by Cohen provides an explanation of the early onset observed in Glasgow method.
• Motivation 1. lattice study is important to confirm it non-perturbatively. 2. It is still unclear why it is half the pion mass, not one-third of
the nucleon mass.
• we mostly concentrate on - µ<mπ/2 - property of fermion determinant at configuration level
Lattice QCD workshop at KEK, 2014/01/20 -/22
Motivation
• Question – Why does the quark number start at µ=mπ/2 at T=0 ?
• We discuss the early onset problem in taking the zero temperature limit of the fermion determinant
• relation between fermion determinant and pion mass
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Idea and outline of talk
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µ-independence of fermion determinant
meson mass (meson propagator)
zero T (large Nt) limit
µ-independent lightest meson (pion)
?
? propagation of quarks
Lattice QCD workshop at KEK, 2014/01/20 -/22
Framework
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion matrix
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• Fermion path integral and fermion determinant
- Δ : matrix with the rank 4 Nc Nx Ny Nz Nt - it consists of fermion loops - chemical poten5al comes from temporal loops
Lattice QCD workshop at KEK, 2014/01/20 -/22
Reduction formula
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[Gibbs ('86). Hasenfratz, Toussaint('92). Adams(’03, ‘04), Borici('04). KN&AN(‘10), Alexandru &Wenger(’10)]
← det Δ can be calculcated analy5cally.
• A formula to perform t-‐part of the fermion determinant. - A fermion matrix in t-‐t matrix rep.
Lattice QCD workshop at KEK, 2014/01/20 -/22
Reduction formula
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• det Δ is reduced to
⇠ = e�µ/T
det� = ⇠�Nred/2C0 det(⇠ +Q)
- C0 and Q are func5ons of gauge fields. - chemical poten5al and gauge fields are separated - Q is a matrix with rank Nred=12 Ns
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Reduction formula
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• det Δ is reduced to
⇠ = e�µ/T
det� = ⇠�Nred/2C0 det(⇠ +Q)
• Using eigenvalues of Q, det Δ is given by
⇠ = e�µ/T
• eigenvalues of Q control the µ-‐dependence of det Δ
Lattice QCD workshop at KEK, 2014/01/20 -/22
Reduced matrix (Wilson fermions)
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• It can be applied to e.g. - winding number expansions (Q, QQ, QQQ...) - hadron masses (QQ^+, QQQ ...) [Gibbs(‘86), Fodor, Szabo,
Toth(‘07)]
spa5al temporal r+-‐=(1+-‐gamma4)/2
Lattice QCD workshop at KEK, 2014/01/20 -/22
Spectral properties of reduced matrix 1
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-800-600-400-200
0 200 400 600 800
-600-400-200 0 200 400 600 800
Im[h
]
Re[h]
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[h
]
Re[h]
10-310-210-1100101102103
0 1 2 3 4 5 6n/1000
|hn|
T/Tc=1.08 • Eigenvalues λ of Q - pair (γ5-‐hermi5cy)
�n $ 1/�⇤n
Eigenvalue distribu5on L: large scale R: small scale
independent
Lattice QCD workshop at KEK, 2014/01/20 -/22
10-6
10-4
10-2
100
102
104
106
0 1 2 3 4 5 6n/1000
|h| , |h|1/Nt
Nt=4Nt=8Nt=4 (scaled)Nt=8 (scaled)
Spectral properties of reduced matrix 2
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• Q follows Nt-‐scaling - Q is temporal quark line,
analogous to the Polyakov loop
Nt=4 original
Nt=8 original
Nt=4, 8 scaled
Lattice QCD workshop at KEK, 2014/01/20 -/22
10-6
10-4
10-2
100
102
104
106
0 1 2 3 4 5 6n/1000
|h| , |h|1/Nt
Nt=4Nt=8Nt=4 (scaled)Nt=8 (scaled)
Spectral properties of reduced matrix 3
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• The gap gives the minimum value of epsilon
• Lajce setup β=1.86(same value of a) 8^3xNt, mps/mV=0.8, RG-‐improved gauge and clover-‐Wilson fermion with Nf=2
Lattice QCD workshop at KEK, 2014/01/20 -/22
zero temperature limit
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion determinant • Let us rewrite det Δ, using the proper5es of λ
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion determinant • Let us rewrite det Δ, using the proper5es of λ
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�n $ 1/�⇤n
Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion determinant • Let us rewrite det Δ, using the proper5es of λ
22
�n $ 1/�⇤n
analy5c in µ and T -‐>possible to take T=0 limit
Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion determinant
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• Consider zero temperature limit
Lattice QCD workshop at KEK, 2014/01/20 -/22
Fermion determinant
• Inequality
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• Consider zero temperature limit
- If µ < εmin , then all the µ-‐dependence vanishes.
Lattice QCD workshop at KEK, 2014/01/20 -/22
Is εmin mπ/2 or MN/3?
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Lattice QCD workshop at KEK, 2014/01/20 -/22
Hadron masses in reduction formula
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• Correla5on func5on for charged meson
• Small eigenvalues contribute to hadron correlators. • low temperature limit (large Nt) - m goes to the lightest mass. - Q is dominated by the eigenvalue near the gap.
Phase vanishes
Lattice QCD workshop at KEK, 2014/01/20 -/22
Hadron masses in reduction formula
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• Correla5on func5on for baryon
Phase survives
Lattice QCD workshop at KEK, 2014/01/20 -/22
Numerical result
• μ-‐dependence appears at μa = 0.5(= about mπ/2)
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R =
✓det�(µ)
det�(0)
◆2
= |R|ei✓
0.0001 0.001
0.01 0.1
1 10
100 1000
0 0.2 0.4 0.6 0.8 1
<ln
|R|>
µa
T/Tpc=0.465T/Tpc=0.5T/Tpc=0.54T/Tpc=0.76T/Tpc=1T/Tpc=1.35 low T
Lattice QCD workshop at KEK, 2014/01/20 -/22
Numerical result
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10-810-710-610-510-410-310-210-1100101
0.4 0.5 0.6 0.7 0.8 0.9T/Tc
<|Q|2> (exact)<|Q|2> (LME)<Q3> (exact)<Q3> (LME)
Low Mode approxima5on(LME)
Lattice QCD workshop at KEK, 2014/01/20 -/22
Summary • Using the reduc5on formula, we consider the property of the fermion determinant at T=0.
– fermion determinant is µ-‐independent at T=0 – threshold for µ-independence is given by mπ/2
• a lot of things to do – numerical confirma5on – why does the quark number keep to be zero up to 1/3 MN
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