dynamics of phase transitions: su(3) lattice gauge theorydynamics of phase transitions: su(3)...
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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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 2 / 26
Introduction: RHIC
Central collision of heavy ions followed by formation of fireballs
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AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 3 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 4 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 5 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 6 / 26
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)
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 7 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 8 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 9 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 10 / 26
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)
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 11 / 26
Gauge Systems: The Structure Factor
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 12 / 26
Gauge Systems: The Structure Factor
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 13 / 26
Gauge Systems: The Structure Factor
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 14 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 15 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 16 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 17 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 20 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 21 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 22 / 26
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)
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 23 / 26
Open Questions
I No natural physical time scale
I Quarks: no heatbath dynamics
I Initial heating is not instantaneous
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 24 / 26
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
AB, BB, AV (FSU, UCLA) Dynamics of Phase Transitions: SU(3) July 31, 2006 26 / 26