Dynamics of Phase Transitions: SU(3) Lattice Gauge TheoryDynamics of Phase Transitions: SU(3)...

Dynamics of Phase Transitions:SU(3) Lattice Gauge Theory

Alexei Bazavov1 Bernd A. Berg1 Alexander Velytsky2

1Department of Physics, FSU

2Department of Physics and Astronomy, UCLA

July 31, 2006.

Reference: Phys. Rev. D 74, 014501 (2006)

AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 1 / 26

IntroductionRHICThe Order ParameterThe Structure Factor

Dynamics of Phase TransitionScenarios of Phase TransitionsSpin SystemsGauge SystemsThe Linear TheoryThe Debye Screening MassThe Energy and Pressure DensityPolyakov loop correlations

Open Questions and Conclusion


Introduction: RHIC

Central collision of heavy ions followed by formation of fireballs

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Fire-tunnel evolution (target rest frame)

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Introduction: Setup

I SU(3) pure gauge theory on a Nτ N3s lattice

I For a system in equilibrium notion of time is lost, so it hasto be reintroduced

I We study the Glauber (heatbath) dynamics under a heatingquench driving the system from the disordered into theordered phase


Introduction: Dynamical Studies

Non-equilibrium studies performed along these lines include:

I Pioneering study of SU(2) and SU(3) pure gauge

I q-state Potts models, scaling in the infinite volume limit

I SU(3) pure gauge theory, scaling in the infinite volume limit

I SU(3) pure gauge theory, spatial expansion

I SU(3) pure gauge theory, the finite volume continuum limit


The Order ParameterThe Polyakov loop is defined as

l(~x) = trNt−1∏m=0

U~x+mt,0 (1)

where U are SU(3) matrices on the links of a hypercubic lattice.Its average value serves as the order parameter of the theory.

I The symmetry group of the order parameter in the SU(3)gauge theory is Z3

I Symmetric (confined) phase 〈l〉 = 0

I Broken (deconfined) phase 〈l〉 6= 0

I The transition is weak 1st order

I Quarks smooth out this behavior, the transition becomes arapid crossover


The Structure Factor

Two-point correlation function of the Polyakov loops (〈...〉L isthe lattice average)

〈l(0)l †(~j)〉L =1

N3σ

∑~i

l(~i)l †(~i +~j) (2)

The structure factor F (~p) is a Fourier transform of (2). Afterdiscretization and using periodicity of the boundary conditionsone arrives at the expression

F (~p) =a3

N3σ

∣∣∣∣∣∣∑

~i

e−i ~k~i l(~i )

∣∣∣∣∣∣2

(3)


Scenarios of Phase Transitions

I Nucleation:I instability against finite amplitudeI localized fluctuationsI has an activation barrierI metastable regionI dominated by the growth of the largest clusters

I Spinodal decomposition:I instability against infinitesimal amplitudeI nonlocalized fluctuationsI no activation energyI unstable regionI signaled by exponential growth of the structure functions


Spin Systems: Domains

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400

Vd

t

FKgeometrical

Geometrical vs. Fortuin-Kasteleyn clusters in 3-state 3D Pottsmodel on a 403 lattice


Spin Systems: The Structure Factor

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 200 400 600 800 1000

F 1/V

t

Ns=020Ns=040Ns=060Ns=080

The first structure factor mode on N3σ lattices for 3-state 3D

Potts model


Gauge Systems: The Structure Factor

0

0.0005

0.001

0.0015

0.002

0.0025

0 400 800 1200 1600 2000

F 1/V

t

Nσ=16Nσ=32Nσ=64

The first structure factor mode on 4× N3σ lattices in SU(3)



I Infinite volume limit: Nτ = const, Nσ →∞I Finite volume continuum limit: Nτ/Nσ = const, Nσ →∞,

we study Nτ = 4, 6, 8

I Rescale time axis so that all maxima coincide

t ′ =t

λt(Nτ , Tf /Tc), (4)

corresponding ratios of λt(Nτ , 1.25) are 1:2.655:5.457 andλt(Nτ , 1.57) are 1:2.768:6.362

I To overcome the renormalization problem of (bare)Polyakov loop correlations divide all structure factors bytheir equilibrium values at Tf



0

1

2

3

4

5

6

7

0 200 400 600

F 1/F

1,f

t’

4×163

6×243

8×323

The first structure factor mode on different lattices with thesame physical volume for Tf = 1.25Tc



0

2

4

6

8

10

12

0 200 400 600

F 1/F

1,f

t’

4×163

6×243

8×323

The first structure factor mode on different lattices with thesame physical volume for Tf = 1.57Tc


The Linear Theory

Dynamical generalization of the Landau-Ginzburg theory in thelinear approximation results in the following equation for astructure factor:

∂F (~p, t)

∂t= 2ω(~p) F (~p, t) (5)

with the solution

F (~p, t) = F (~p, 0) exp (2ω(~p)t) , (6)

ω(~p) > 0 for |~p| > pc

Rescale ω′(~p) = λt(Nτ , Tf /Tc) ω(~p) so ω′(~p)t ′ = ω(~p)t


The Critical Momentum

-0.1

-0.05

0

0.05

0.1

0 1 2 3 4 5 6

2ω’(p

)

p2/Tc2

Nτ=4Nτ=6Nτ=8

SU(3) determination of pc for Tf /Tc = 1.25


The Critical Momentum

-0.1

-0.05

0

0.05

0.1

0 2 4 6 8 10

2ω’(p

)

p2/Tc2

Nτ=4Nτ=6Nτ=8

SU(3) determination of pc for Tf /Tc = 1.57


The Debye Screening MassFit results for pc/Tc

Tf /Tc Nτ = 4 Nτ = 6 Nτ = 8 Nτ = ∞1.25 1.613 (18) 1.424 (26) 1.37 (10) 1.058 (79)1.568 2.098 (19) 2.058 (22) 2.29 (15) 2.006 (73)

The critical momentum pc is related1 by

mD =√

3 pc (7)

to the Debye screening mass at the final temperature Tf afterthe quench:

mD = 1.83 (14) Tc for Tf /Tc = 1.25 , (8)

mD = 3.47 (13) Tc for Tf /Tc = 1.568 (9)

1T.R. Miller and M.C. Ogilvie, Nucl. Phys. B (Proc. Suppl.) 106(2002) 537; Phys. Lett. B 488 (2000) 313AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 18 / 26

Energy and Pressure Density

0

1

2

3

4

5

6

200 300 400 500 600

ε/T4 , 3

p/T4

t’

ε/T4, Nτ=4

ε/T4, Nτ=6

ε/T4, Nτ=83p/T4, Nτ=4

3p/T4, Nτ=6

3p/T4, Nτ=8

The gluonic energy and pressure2 density, Tf = 1.25Tc

2G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M.Lutgemeier, B. Peterson, Nucl. Phys. B469, 419 (1996)AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 19 / 26

The Order Parameter

The histogram for the order parameter (Polyakov loop) at thefirst time step on 6× 243 lattice, Tf = 1.57Tc

lattice, Tf = 1.568Tc


The Order Parameter

The histogram for the order parameter (Polyakov loop) at thetime step where the structure function reaches maximum on

6× 243 lattice, Tf = 1.57Tc


The Order Parameter

The histogram for the order parameter (Polyakov loop) at thelast time step (equilibrium) on 6× 243 lattice, Tf = 1.57Tc

lattice, Tf = 1.568Tc


Polyakov loop correlations

0

0.001

0.002

1 2 3 4 5 6

Co

d

0.5 tmaxtmax

5 tmax

Co(d , t) = 〈l(0, t)l(d , t)〉L − (〈|l(0, t)|〉L)2 (10)


Open Questions

I No natural physical time scale

I Quarks: no heatbath dynamics

I Initial heating is not instantaneous


The Structure Factor in Effective Model3

0

0.01

0.02

0.03

0.04

0 100 200 300 400

F 1/V

t

The first structure factor mode on 643 lattice, Tf = 1.25Tc

3R.D. Pisarski, Phys. Rev. D62, 111501(R)(2000); A. Dumitru and R.Pisarski, Phys. Lett. B504, 282 (2001)AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 25 / 26

Conclusion

I The phase transition proceeds through the spinodaldecomposition scenario

I Domains of different triality slow down the equilibration

I The energy and pressure density evolve to equilibrium values

I The critical momentum is related to the Debye screeningmass

I Correlations in equilibrium are weaker than in theout-of-equilibrium state

I Physical time scale can be set in effective models


Dynamics of Phase Transitions: SU(3) Lattice Gauge TheoryDynamics of Phase Transitions: SU(3)...

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