有限密度格子QCDの研究-高密度への挑戦K. Nagata,Hiroshima Univ., RIISE
S. Motoki (KEK), A. Nakamura (Hiroshima)
in XQCD-J collaboration

Talk given at
Nemu no Sato, Shima, Mie, 2011/12/04

Contents• Introduction• Reduction formula for Wilson fermion determinant• Application of reduction formula
1. Taylor expansion
2. Lee-Yang zeros
Imaginary chemical potential• SummaryQCD at finite temperature and density• nature of matter in our universe– ordinary : hadron, nuclei, nuclear matter– dense : compact stars– hot : HIC, creation of matter in early universe• QCD is non-perturbative.
• Finite density LQCD : sign problem

To reach at low T & high mu• Several methods.
• Most studies have been done for regions near Tc.
• Each method has its own difficulty• Taylor : truncation error• MPR : overlap & sign problems• Imaginary : uncertainty of functional form
• etc.....
• We have been studied those methods,• Wilson fermion, formula for determinant, real &imaginaryFermion determinant• effects of chemical potential• sign problem• numerical hot spotA formula for determinant• Quark action as t-t matrix– temporal hop accompanies chemical potential– spatial (diag), temporal (n.n + b.c)• Performing temporal determinant by hand– rank reduces to N/Nt (memory & CPU time ~1/Nt^2)– det D is an analytic function of mu.

• Reduction formula enables us– exact evaluation of determinant (useful for large mu)– for arbitrary values of chemical potential– suppress CPU time for low T (T=1/a Nt)

Reduction formula• Q : rank = Nr = 4 Nc Ns^3 matrix (Nr = N/Nt)– function of link variables– independent of chemical potential

Wilson type : KN&AN,
PRD82,094027 ('10),• It can be used for many applications

Taylor expansion• Coefficients are calculated at mu=0 , and functions of T.

Taylor coefficients

Data of "WHOT" are taken from
WHOT, Ejiri et al.,
PRD82, 014508 (2010),
arXiv:0909.2121

clover-Wilson + RG-gauge(Nf=2)
Volume : 8^3x4quark mass : mps/mV ~ 0.8
Configurations : HMC at mu=0
11 K steps including ( therm. as 3-5K)

Eigen values : 400 configs.

Taylor coefficients• At high T, almost converge up to O((mu/T)^4).
• Near and below Tc, slow convergence. Higher orderterms are non-negligible.
We are preparing for higher order coefficientsand those at low T

EoS & Phase boundary
Phase boundary can be obtained from convergence radius(in principle)Lee-Yang zeros• Zeros of partition function (Lee-Yang) implies phasetranstions• LY zeros are distributed on the complex fugacity plane.
• LY zeros approach to Re[xi] AXIS as V -> 0.

Lee-Yang zerosln Z

Im[xi]

Re[xi]

Preliminary data :config : clover-Wilson
Measurement : w/o clover-term
statistics 100

Lee-Yang zeros
Im[xi]
Re[xi]

quark : inside unit circle
anti-quark : outside unit circle

Imaginary chemical potential• Sign problem is absent for pure imag. mu– Generating configs. at imaginary mu– approach to real mu• analytic continuation• canonical approachImaginary chemical potential• MC simulations in mu^2<0 region and analytic continuation• This approach may be useful for the study of low-T region.
For properties ofthis phase diagram
e.g. KN&AN,PRD83,114507 ,arXiv:1104.2142.

– wide regions– RW periodicity

Polyakov loop at low T
We found the effect of chemical potential at T/Tc=0.5.

Fit & analytic continuation
RW periodicity -> Fourier series (cos) -> coshConsistency• Each approach has its own difficulty, consistency
check is helpful• Taylor : truncation error• MPR : overlap & sign problems• Imaginary : uncertainty of functional form
• They would be consistent if calculations are donein enough precision.

Consistency• Taylor and MPR are consistent– consistent up to mu/T = 0.6– Taylor : truncation error– MPR : overlap & sign problem

Summary• We are studying several ideas towards low T & high mu– Taylor, Lee-Yang, Imaginary etc– phase boundary, EoS ...

• Further investigations are in progress– larger volume, increasing statistics , low T (T=0.5TcTc)