Outline 　

1. Critical short-time dynamics

― 　 hot start & cold start 　―

2. Application to lattice gauge theory

― extract critical exponents ―

3. Summary

Short-Time Scaling in SU(2) Lattice Gauge Theory

at Finite Temperature

OTOBE, Tsuyoshi (Waseda Univ.)

OKANO, Keisuke (Tokuyama Univ.)

Motivation

Certain initial State Relaxation in Tc

non-equilibrium

critical slowing down

Initial StateRelaxation in Tc

long-time

Usual method

thermalequilibrium

Ising model etc. (statistical system), lattice gauge theory

New method ?

Ex. : calculating the critical exponents

t

M(t)

t

zt t_mic ： O(101 ～ 2)

t_max ： depending on N_space

tte

Critical short-time dynamics (Hot start & Cold start)

x0 : anomalous dimension of m0

Janssen et. al. Z. Phys. B73(1989)539.

),,,(,,, 011

00 mbLbbtbMbmLtM xzkkk

Generalized scaling law

New universal stage !

tmic ttmax

HOT

COLD

zd

z

z

d

ttM

ttM

ttA

ttM

)2()2(

1

0

~ )(

~),(ln

~ )(

~ )(

HOT start

z

x 0

Initial state

new dynamic exponent

z ： dynamic exponent

ν, β ： static exponent

Second moment M(2)

(t)

Autocorrelation A(t)

M(0) = 0 ： high temperature

Magnetization M(t)

M(0) = m0 ： small but non-ze

ro

(θ, z, ν, β)

z

z

z

d

ttM

ttM

ttU

1

0

~ )(

~ ),(ln

~ )(

1))(()()( 2)2( tMtMtU

LbbtbMbLtM zkkk 11 ,,,,

generalized scaling law

Finite size dynamic scaling law in short-time

m0=1

z ： dynamic exponent

ν, β ： static exponent

Binder cumulant ：

COLD start

Initial state For U(t), M(t) M(0) = 1 ： completely ordered

(z, ν, β)

d ： spatial dim. of the system

・・・

Practical simulation t0 O(102 ～ 3)

…

tM tM

t t

samples

Initial states with the condition

z

~)(

ttM

・・・

・・・

sample averaging

(2+1)-dim. SU(2) lattice gauge theory

at finite temperature

1 , 1

s

t

tt N

N

NaNT

～

Wilson action p

pUUS

0

),( 00x

ii xxUTrW : Polyakov line

i

iWN

M2

1

asymptotic scaling

―Analysis of deconfinement phase transition ―

Heat Bath algorithm

Order parameter

Physical temperatureof the system

Relaxation dynamics

Universality

In relaxation process

2-dim.Ising model2-dim.Ising model (2+1)-dim. (2+1)-dim. SU(2) LGTSU(2) LGT

same universalitysame universality　　 classclass

Same values for （ ν ， β ）

In equilibrium

Same values for （ θ， z ） ?

(1) Christensen & Damgaard ( NP B348 (1991) 226 )

N=64, N0 =2 : 4/gc2 = 3.39, β= 0.120 (8)

(2) Teper ( PL B313 (1993) 417 )

N=64, N0 =2 : 4/gc2 = 3.47, ν= 0.98 (4)

β= 0.125(exact)

ν= 1(exact)

HOT start　・ includes a new phenomena related to θ ( the anomalous dimension of m0 )

　・ difficult to prepare the clean Initial state ( with m0≠0 )

　 Finally one has to extrapolate the result to m0 → 0

⇒ technical difficulty and complexity

　・ less convergent compared to the COLD start

COLD start 　・ very simple to prepare initial state

　・ good convergence compared to the HOT start

　・ needs a relatively bigger lattice

zz

zz

tatb

tFttM

1

1

1

),(

～　

～at τ= 0 → 　 pure power law

for τ≠0 → 　 some modification

Determination of βc from the short-time scaling law (COLD Start)

β ＝ βc

β ＞ βc

β ＜ βc

β

Inclination parameter

①②③

①②③

①②③

①②

③

t

M(t)

τ ： reduced temperature

βc = 3.4505

Fit-range dependence of Inclination parameter

[200,800]

[250,800]

[300,800]

[350,800]

[400,800]

Magnetization (COLD start)

M(t)

zttM

~ )(

t

βc = 3.4505

Lattice size : 1282×2

Sample : 20000

tmic

Cumulant (COLD start)

zttU2

~

βc = 3.4505

Lattice size : 1282×2

Sample : 20000

t

U(t)

tmic

Summary of Results (COLD start)

①① ②② ③③

2/z2/z 1/νz1/νz β/νzβ/νz

SU(2)SU(2) 0.917(4)0.917(4) 0.496(17)0.496(17) 0.0633(2)0.0633(2)

IsingIsing 0.9280.928 0.4640.464 0.05800.0580

zz νν ββ

SU(2)SU(2) 2.182(10)2.182(10) 0.925(40)0.925(40) 0.128(5)0.128(5)

IsingIsing 2.16(2)2.16(2)** 1.01.0**** 0.1250.125****

① cumulant tau-difference of magnetization magnetization ② ③

From K. Okano, L. Schulke and B. Zheng, Nucl. Phys. B485 (1997) 727

** exact** 2.155(03)2.155(03)

* From several literature

Magnetization (HOT start)

tM )( ttM ～

Linear increase of the magnetization

M(t)

t

βc = 3.4505

Lattice size : 1282×2

Sample : 20000 tmic

)()2( tM

tA

Auto-correlation & second moment of magnetization

(HOT start)

t

βc = 3.4505

Lattice size : 1282×2

Sample : 30000

tmic

z

d

ttA

~ )(

zdttM

)2()2( ~ )(

θθ zz νν ββ

hothot 0.1943(4)0.1943(4) 2.108(35)2.108(35) 1.08(9)1.08(9) 0.117(27)0.117(27)

coldcold 2.182(10)2.182(10) 0.925(40)0.925(40) 0.128(5)0.128(5)

IsingIsing

(hot)(hot)0.191(1)0.191(1) 2.16(2)2.16(2) 11 1/81/8

Summary for HOT and COLD simulation

(includes preliminary result)

The 2+1 dim. SU(2) Lattice gauge theory 2-dim. Ising model

Universal (including dynamics)

Summary(1) Short time scaling behavior is observed near the critical point.

(2) It is possible to determine the Critical point βc.

(3) Static critical exponents (ν ， β) (SU(2) LGT) ⇔ (Ising Model) consistent

⇒⇒ 　　 Validity of short-time critical dynamics

(4) Dynamic critical exponents (z , θ) are also obtained. (Also consistent)

(5) 2+1 dim. SU(2) Lattice gauge theory 　⇔　 2-dim. Ising model　　　　　　　　　　　　　　　　　　　　　　　 universal 　　　　　　　　　　　　　　　　（ including the relaxation

dynamics ）

We can obtain Much information from non-equilibrium(short-time).