1. 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.)

2. Motivation Certain initial State Relaxation in Tc non-equilibrium critical slowing down Initial State Relaxation in Tc long-time Usual method thermal equilibrium Ising model etc. (statistical system), lattice gauge theory New method ? Ex. : calculating the critical exponents

3. M(t)M(t) t t_mic ： O(10 1 ～ 2 ) t_max ： depending on N_space Critical short-time dynamics (Hot start & Cold start) x 0 : anomalous dimension of m 0 Janssen et. al. Z. Phys. B73(1989)539. Generalized scaling law New universal stage ! t mic t tmax HOT COLD

4. HOT start Initial state new dynamic exponent z ： dynamic exponent ν, β ： static exponent Second moment M (2) (t) AutocorrelationA(t) M(0) = 0 ： high temperature Magnetization M(t) M(0) = m 0 ： small but non- zero (θ, z, ν, β)

5. generalized scaling law Finite size dynamic scaling law in short-time m0=1m0=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

6. ・・ ・ Practical simulation 0 O(10 2 ～ 3 ) … samples Initial states with the condition ・・ ・ sample averaging

7. (2+1)-dim. SU(2) lattice gauge theory at finite temperature Wilson action : Polyakov line asymptotic scaling ―Analysis of deconfinement phase transition ― Heat Bath algorithm Order parameter Physical temperature of the system Relaxation dynamics

8. Universalit y In relaxation process 2-dim.Ising model(2+1)-dim. SU(2) LGT 2-dim.Ising model (2+1)-dim. SU(2) LGT same universality class Same values for （ ν ， β ） In equilibrium Same values for （ θ ， z ） ? (1) Christensen & Damgaard ( NP B348 (1991) 226 ) N=64, N 0 =2 : 4/g c 2 = 3.39, β= 0.120 (8) (2) Teper ( PL B313 (1993) 417 ) N=64, N 0 =2 : 4/g c 2 = 3.47, ν= 0.98 (4) β= 0.125(exact) ν= 1(exact)

9. HOT start ・ includes a new phenomena related to θ ( the anomalous dimension of m 0 ) ・ difficult to prepare the clean Initial state ( with m0≠0 ) Finally one has to extrapolate the result to m 0 → 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

10. 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 ① ② ③ ① ② ③ ① ② ③ ① ② ③ t M(t)M(t) τ ： reduced temperature

11. βc = 3.4505 Fit-range dependence of Inclination parameter [200,800] [250,800] [300,800] [350,800] [400,800]

12. Magnetization (COLD start) M(t)M(t) t βc = 3.4505 Lattice size : 128 2 ×2 Sample : 20000 t mic

13. Cumulant (COLD start) βc = 3.4505 Lattice size : 128 2 ×2 Sample : 20000 t U(t)U(t) t mic

14. Summary of Results (COLD start) ①②③ 2/z1/νzβ/νz SU(2)0.917(4)0.496(17)0.0633(2) Ising0.9280.4640.0580 zνβSU(2)2.182(10)0.925(40)0.128(5) Ising 2.16(2) * 1.0 ** 0.125 ** ① cumulant ② tau-difference of magnetization ③ magnetization From K. Okano, L. Schulke and B. Zheng, Nucl. Phys. B485 (1997) 727 ** exact * 2.155(03) * From several literature

15. Magnetization (HOT start) Linear increase of the magnetization M(t)M(t) t βc = 3.4505 Lattice size : 128 2 ×2 Sample : 20000 t mic

16. Auto-correlation & second moment of magnetization (HOT start) t βc = 3.4505 Lattice size : 128 2 ×2 Sample : 30000 t mic

17. θzνβ hot0.1943(4)2.108(35)1.08(9)0.117(27) cold2.182(10)0.925(40)0.128(5) Ising(hot)0.191(1)2.16(2)11/8 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)

18. Summary (1)Short time scaling behavior is observed near the critical point. (2)It is possible to determine the Critical point β c. (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).