手征极限下带温度和化学势的两味道 Wilson 费米

子 QCD 的相结构

吴良凯 罗向前 （教授）中山大学

罗向前 教授

Outline

• Introduction

• Lattice formulation

• Lattice QCD with Imaginary Chemical Potential with Wilson Quarks

• Conclusion

Satz’s and Aoki’s talks

Phase diagram of QCD at zero-densityI. Introduction

两味道 QCD 在手征极限下的相图

Four fermion model: Alford, Wilczek, et al.,

QGP

2SC

Tricritical pointHadronic phase

II. Lattice Formulation

fN 味夸克的系统的配分函数为（带有化学势）

gf sN eU))[dU]Det(M(

fg ssedddUZ ]][][[

fsgs 为纯规范场作用量， 为夸克作用量

• 纯规范场作用量

with β=6/g2

• 在格点上代换为 Wilson 作用量

利用 Wilson 费米子 , 则费米子矩阵为：

在需要考虑化学势时，代换费米子作用量中时间方向的链，引入化学势。

但是：引入化学势后 , 对 SU(3)

5

5

† †5

† †5

( ) , ( 0)

( ) , ( 0)

M M

M M

费米子矩阵的行列式为复数，使得 Monte Carlo 模拟不能进行。

• 连续的夸克作用量

• 在格点上代换为离散的夸克作用量

M 是离散的费米子矩阵

解决办法a. Improved reweighting

b. Imaginary chemical potential

III. Lattice QCD with Imaginary Chemical Potential With Wilson Quarks

The Phase diagram suggested by Roberge and Weiss

First order

Polyakov loop

Chiral condensate

1

0

( ) [ ( )]tN

tt

P x Tr U x

( )

1

1[ ][ ][ ]

1[ ] ( )( ( ))

G F

f G

S S

N S

t

dU d d eZ

dU M U DetM U eZVN

Some observables considered

Nf=2 of KS fermions

Nf=4 of KS fermions

Deconfinement phase transition

Phase diagran suggested by MC study

Z(3) transition, First order

The results above indicate that at higher T, there is Z(3) first order phase transition for QCD with Wilson quarks at imaginary chemical potential.

First Results from two flavors of Wilson fermions

Second scan in this direction, deconfinement transition

First scanThis directionZ(3) tranition

History and histogram at a I =0.262

The phase of Polyakov loop changes with imaginary chemical potential at different coupling at kappa=0.1

6

The determination of chiral limit

• Determine the chiral limit through the axial vector Ward-Takahashi identity

Y.Iwasaki, K.Kanaya,et al, Phys.Rev.Lett.67,1494(1991)

On the lattice

with

The average number of iteration for the fermionic matrix inversio invN

invN Is enormously large at chiral limit in the confining phase

invN Is of order several hundreds in the deconfining phase

invN From hot start cold start

Results from the scanning along the temperature axis, i.e. beta axis.

Critical beta as a function of imaginary chemical potential

)())(27(649.0)1(203.5 442IIc aOa

To obtain critical beta as a function of real chemical potential, replace I by Ii

)())(27(649.0)1(203.5 442IIc aOa

Using the renormalization group equation

and

obtain

The finite size scaling of chiral condensate

The history and histogran of chiral condensate

IV. Conclusion

Second order

Crossover First orderL.G.Yaffe, B.Svetisky, Phys.Rev.D26,963(1982)

QCD Phase Diagram on the (T,μ) plane

from lattice QCD

Multi-dimensional reweighting: Fodor and Katz, …

Hamiltonian lattice QCD with Wilson quarks

X.Q. Luo, Phys. Rev. D70 （ 2004 ） 091504 　 (Rapid Commun.)

X.L. Yu, X.Q. Luo, hep-lat/0508032

Hamiltonian lattice QCD: Greogry, Guo, Kroger, X.Q. Luo, Phys. Rev. D62 (2000) 054508.

Y. Fang, X.Q. Luo, Phys. Rev. D69 (2004) 114501.

Lagrangian Lattice QCD from Imaginary chemical potential method: de Forcrand, Lombardo, H. Chen, X.Q. Luo, Phys. Rev. D72 (2005) 034504

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