Finite density lattice QCD
at low temperatures
Keitaro Nagata
Atsushi Nakamura
RIISE, Hiroshima University & A02

KN
arXiv:1204.6480
Low-T
KN, et al PTEP01A103(2012)
Canonial, Lee-Yang zero,
low T
KN, AN,JHEP1204,092,(2012) EoS near Tc, Taylor,
Reweighting
KN, AN,
PRD83,114507(2011)
Phase boundary,imaginary
KN,
AN,
PRD82,094027(2010)
Reduction
Formula for
Quarksto Universein Computational Science, Nara,
December13-16,

1Introduction

Quarks to Universe in Computational Science, Nara, December13-16,

2Introduction• LQCD at finite T and μ (quark chemicalpotential) are desired to understand origin& properties of matter.
• LQCD has the sign problem at nonzero μ.
• There are many progress in finite density
LQCD for high T and small μ.
• Are those methods applicable to low-T andnonzero μ region ? => two difficulties
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Introduction – small S/N ratio• Example of features of QCD at low-T and finite μ• S/N ratio issmall at low T.

 e.g., 50-100K was
not enough forT/Tc=0.5 withimaginary
approach [Motoki, et.
al. PoS Lat2012].

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4Introduction - Silver Blaze of QCD-Expectatio
nbaryonnumber density iszero for μB<mN (μ<mN/3 )at
T=0.

– quark number starts to increase at μ=mπ/2(’90s)]Blaze [Cohen, PRL91,222001(‘03)]– [Glasgow,isospin Silver- (No silver blaze in MDP, strong coupling)
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5Introduction - Scope of this talk
• An idea is to use an analytic expression
about μ-dependence with the help of areduction formula.
– μ-dependence of det Δ at low T– spectral property and physical meaningof a reduced matrix– low-T limit of det Δ– application to small μ region

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Reduction Formula

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Reduction formula• Fermion determinant det Δ– Locality allows to perform the temporal partof det Δ– Q and C0 are independent of μ– chemical potential and gauge fields areGibbs ('86). Hasenfratz &Toussaint('92). Adams(’04),
separated
Borici('04). KN&AN(‘10), Alexandru &Wenger(’10)

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Reduction formula• Reduction of the rank of det Δ– Δ : 4 Nc N x N y N z N t– Q : 4 N c Nx Ny Nz• With the eigenvalues,– det Δ is an analytic function of μ
– values of det Δ for any value of μ.
Barbour et al. NPB557,327(‘99) etc, Fodor, Katz, JHEP0203, 014(‘0de Forcrand, Kratochvila, PRD73, 114512(‘06), etc,
Quarks to Universe in Computational Science, Nara, December13-16,

9Fermion Property at Low T

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Simulation setup• Action (gauge: RG-improved, fermion: cloverimproved Wilson with Nf=2 )• Mass: mps/mV=0.8• Lattice sizes : 8^3x4, 8^4 (spatial volume fixed)• Temperatures : T/Tpc=0.45~3(beta =1.5~2.4,Nt=4 and 8)• Configurations :10K configurations at μ=0• Eigenvalue: 400 confs. (LAPACK)

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Chemical Potential Dependence at LowT• det Δ is insensitive to μ for μa < 0.5.
• μ-dependence appears at μa = 0.5.
– This value is close to mπ/2 in the presentsetup.
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Fermion Property at Low T

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Meaning of Reduced matrix Q• Relation to energies of quarks– comparison of Nt=4 &8– Q and Polyakov loop

– zeta-regularization method [e.g. Adams, PRD70,045002(’04)]

Quarks to Universe in Computational Science, Nara, December13-16,

14Meaning of Reduced matrix Q• Quark number density

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Spectral property• Eigenvalues form a pair(γ5-hermiticity)– |λ|< 1 for quarks– |λ|> 1 for antiquarks• The Nt-scaling law of the eigenvalues
XQCDJ, PTEP01A103(’12)• Ev’s are related to the pion mass (lightestmass).
Gibbs(‘86),
PLB172,53(’86)
Fodor, Szabo, Toth,
JHEP0708,092(’07)

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Low Temperature LimitEv
distribution
is boundedby pion
mass

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Meaning of Reduced matrix Q• Quark number density

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Free Energy in RF• Free energy• cf

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Summary• We have studied QCD at low T & finite density.
– approach to low-T region with reduction
formula– quark number density is zero for μ<mπ /2 at
T=0 for any configuration, assuming fixedlattice size
– quark number density starts to increase at
μ=mπ /2 for each configuration.
– confirmation: Nt-scaling law, volume
dependence, quark mass dependence, etc

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Summary• To obtain EoS, we have studied/will study• for low T & small μ (small S/N ratio)– expression of free energy in reduction formula• for large μ,– Improvement of overlap : DoS, high density limit,imaginary, isospin, etc, multi-ensemble reweighting– Silver Blaze : finite size effect, increase statistics etc– (MDP, strong coupling ...)

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Buckup Slides

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Free energy

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Spectral property - spectral density

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Spectral property - Symmetry• γ5-hermiticity• Eigenvalues form apair• Small & large eigenvalues correspond to– |λ|< 1 for quarks– |λ|> 1 for anti-quarks

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Spectral property - Gap• Eigenvalues form agap

• Ev’s are related to the pion mass (lightestmass).
Gibbs(‘86),
PLB172,53(’86)

Fodor, Szabo, Toth,JHEP0708,092(’07)
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Spectral property - Nt scaling law• The Nt-scaling law of the eigenvalues

KN et.al. (XQCD-J), (‘12)
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Free Energy in RF• Free energy• This form may be compared toQuarks to Universe in Computational Science, Nara, December13-16,

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Average phase factor vs μ

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Spectral property - Volume• Volume dependence

Quarks to Universe in Computational Science, Nara, December13-16,30Taylor coefficients of EoS– slow convergence of the Taylor series ofEoS– small S/N ratio of chemical potential
dependence
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Fermion properties at low T• Fermion determinants are less sensitive to mu at
lower temperatures.

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Quark mass dependence
mps / mv = 0.6 (red), 0.8 (blue)
Histograms : |ev| (Left), arg(ev) (Right)
confinement (top), deconfinement(bottom)

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Gap is related to pion mass
1ln max | λn |2amπ = −
Nt |λn |<1
Gibbs('86). Eigenvalues and mpi
See also, Fodor, Szabo, Toth ('06). Eigenvalues and hadron spectrum• At low T, mpi/T is well fitted with a/T, a = 4 Tpc (mq heavy)• At high T, mpi approaches to a constant
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Low Temperature Limit• Eigenvalues form apair– gamma5hermiticity– |λ|< 1 for quarks– |λ|> 1 for antiquarks
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Low Temperature Limit• The Nt-scaling law of the eigenvalues• Condition of low-T & small mu limit

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Low Temperature Limit• Condition & pion mass

• Low T and small m limit.

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Low-T limita. small µa. large µ Quarks move on a fixed time slice, there is
no propagation in t-direction(t-linkvanishes) Z3 invariant
Quarksto Universe
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arein notneeded.
 ev’s38

Buckup Slides

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Introduction-LQCD with chemical
potential• High T, small μ(μ/T<1)?

– sign fluctuation is mild
– μ-dependence is• LowT, finite μmeasurable– S/N ratio is small• Calculation of quantities at low T & finite μrequires large statistics.

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Approach to low temperature• Chemical potential is included in the fermion
determinant,• Calculation of det D seems to be inevitable at
low temperatures, e.g.
– slow convergence of Taylor series– small S/N ratio• A reduction formula provides– a way to evaluate det D for large Nt– physical implications & applications
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