Complex phase in QCD
with finite chemical potential
arXiv:1111.6363

Shinji Takeda
Kanazawa U.
in collaboration withY. Kuramashi & A. Ukawa

QCD phase structureDifficulties• The complex phase ruins an applicability of
the Monte Carlo technique• Reweighting• Large phase fluctuation

Sign problem

Phase (of determinant) study
So far•We try to do here

Lattice, Taylor expansion
Allton (2005)•We want to obtain/understandthe phase more precisely &irrespective of the location of
the phase diagram•

Especially we are interested inlow T & high μEjiri (2008)– Not reliable when μ/T>1•

Effective theory/model
Splittorff & Verbaarschot (2008)
Han & Stephanov (2008)– Not reliable in region neartransition/systematic error

Plan• Winding number expansion– Better convergence in low temperature region• Property of the phase– Understanding the sign problem– Is there a parameter region where the signproblem can be controlled?

• Reweighting– 4 flavor QCD phase structureWinding number expansion (WNE)
Danzer & Gattringer PRD78.114506 (2008)

Winding number

μ-independent & real

q=0

Gauge loops

Phase

q=1q=2

Phase in terms of WNE

NT• converges if• better convergence range
than Taylor expansion

Convergence is better for larger!!! Really?

Numerical check of the convergence
Kentucky group PRD82.054502(2010):

On phase quenched configurations O(1000)

Convergence is better for larger!!!

An upper bound for the phase
WNE
Hopping parameter expansion (heavy mass)

The phase gets exponentially suppressed for larger!!!

Distribution of the exact phaseOn the same configurations as before

Fluctuation of the phase gets small for large NT

Reweighting factor
On the phase quenched configurations

easy

Sign problem

severePhase reweighting

By using reduction technique we can computephase, Quark Number & Susceptibility exactlyQuark Number and its susceptibility
transition point at T=150MeV (NT=4)
determined by the canonicalapproach where Tc=160MeV (μ=0)
Kentucky group (2010)

Polyakov loop and its susceptibility

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Histogram of plaquette value onphase-quenched configuration

Technical remarks• Reduction technique [Danzer et al., (2008)] +Original further reduction technique– Inverse matrix (rank ∝ NL3) & matrix-matrix
product ➠ LAPACK– Cost ∝(NL3)3×NT– Memory size ∝ (NL3)2 independent of NT• Machine: T2K-Tsukuba

SummaryOn the phase• Winding number expansion
is a good approximation forlarger NT• The phase can be controlled
for large NT
We roughly observe suchbehavior numerically
In heavy mass region

Reweighting• Nf =4 at μ/T=0.8(T=150MeV), we observe
1st order like behavior• We plan Nf =3 or 2+1 withfinite size scaling(Y.Nakamura and X-Y. Jin)

Other use of the WNE?

Far future planobservableweight• Real part=> HMC hamiltonian=> WNE to get analytic form of MD force=> approximation can be corrected by computing determinant once per HMC
trajectory• Observable=> sign problem=> For large N_T, fluctuation should be small

Constant physics with rescaling cut off
Constant physics

Rescaling by factor b

Exact vs. WNE for reweighting factor

WNE for physical quantities