non-equilibrium critical phenomena in the chiral phase transition
DESCRIPTION
Non-equilibrium critical phenomena in the chiral phase transition. Kazuaki Ohnishi (NTU). Introduction Review : Dynamic critical phenomena Propagating mode in the O( N ) model Over-damping near the critical point Conclusion. K.O., Fukushima & Ohta : NPA 748 (2005) 260 - PowerPoint PPT PresentationTRANSCRIPT
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Non-equilibrium critical phenomenain the chiral phase transition
1. Introduction
2. Review : Dynamic critical phenomena
3. Propagating mode in the O(N) model
4. Over-damping near the critical point
5. Conclusion
Kazuaki Ohnishi (NTU)
K.O., Fukushima & Ohta : NPA 748 (2005) 260K.O. & Kunihiro : PLB 632 (2006) 252
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Strong interaction between hadrons (proton, neutron, pion, ρ-meson)
QCD (quark & gluon)
Chiral symmetry in the u-, d-quark sector
1. Introduction
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Ferromagnet O(3) symmetry is spontaneously broken NG mode = spin wave
Spontaneous Breaking of Chiral symmetry
pion is the massless Nambu-Goldstone particle
1. Introduction
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• Static (Equilibrium) critical phenomena
• Dynamic (Non-equilibrium) critical phenomena
Heavy Ion Collision, Early universe
Quark-Gluon-Plasma phase
Color-Superconducting phaseHadron phase
Early universeHeavy Ion Collision (RHIC,LHC)
1st
TCP
2nd
1. Introduction
Lattice simulation, Effective theory, Universality argument, etc.
Real world
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Anomalous dynamic critical phenomena
• Critical slowing down
• Softening of propagating modes
• Divergence of transport coefficients
• ...
Long relaxation time
Slow motion of long wavelength fluctuations of Slow variables
2. Review : Dynamic Critical Phenomena
Non-equilibrium, time-dependent
Non-equilibriumstate
EquilibriumstateRelax
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2. Review : Dynamic Critical Phenomena
2 kinds of slow variables
1. Order parameter
2. Conserved quantity
Flat potential
Continuity Eq.
)(V
Slow variables (Order parameter & Conserved quantities) are thefundamental degrees of freedom in the critical slow dynamics
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2 types of Slow modes for slow variables
1. Diffusive (Relaxational) mode
2. Propagating (Oscillatory) mode (Spin wave, Sound wave, Phonon mode, etc)
t
t
Propagating mode (Damped Oscillatory mode)
Diffusive mode (Damping mode)
2. Review : Dynamic Critical Phenomena
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Spectral func. for slow variables
( : fixed)q
)( 0) &( z
z( Dynamic critical exponent)
Critical slowing down
Softening
• Propagating mode pole with Real and Imaginary parts
• Diffusive mode pole with only Imaginary part
• Dynamic scaling hypothesis
2. Review : Dynamic Critical Phenomena
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Static universality class
critical behavior (critical exponents) is identical
if symmetry and (spatial) dimension are same.
Ferromagnet and anti-ferromagnet belong to
the O(3) universality class
Chiral transition belongs to the same universality class asferromagnet and anti-ferromagnet
Pisarski & Wilczek:PRD29(1984)338
2. Review : Dynamic Critical Phenomena
Universality class
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1. Whether the order parameter is conserved or not
2. What kinds of conserved quantities in the system
Whole critical points in condensed matter physics (Ferromagnet, Anti-Ferromagnet, λtransition, Liquid-Gas, etc)have been classified into model A, B, C,....
2. Review : Dynamic Critical Phenomena
Classification scheme
Dynamic universality class
Slow variables Hohenberg & Halperin: Rev.Mod.Phys.49 (1977) 435
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Dynamic universality class of chiral transition
• Slow variables for Chiral phase transition
• Meson field
• Chiral charge
• Energy
• Momentum
Order parameter (Non-conserved)
Conserved quantities
• Slow variables for Anti-Ferromagnet
• Staggered Magnetization
• Magnetization
• Energy
• Momentum
Order parameter (Non-conserved)
Conserved quantities
Rajagopal & Wilczek: NPB 399 (1993) 395
Meson mode is a diffusive mode
2. Review : Dynamic Critical Phenomena
Chiral transition belongs to anti-ferromagnet
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Hatsuda & Kunihiro: PRL 55 (1985) 158
Meson (particle) is an oscillatory mode of field
Diffusive mode Rajagopal & Wilczek
Propagating mode Hatsuda & Kunihiro
2. Review : Dynamic Critical Phenomena
Meson mode is a propagating mode
?
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3. Propagating mode in the O(N) model
Langevin Eq.
Brownian particle Zwanzig J.Stat.Phys.9(1973)215
O(N) Ginzburg-Landaupotential
Meson mode(Propagating mode)
(K.O., Fukushima & Ohta: NPA 748 (2005) 260)(Koide & Maruyama: NPA 742 (2004) 95)
Square of propagating velocity
Damping constant
Canonical momentum conjugate to order parameter
Neither Order parameter nor Conserved quantity!
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Renormalization Group (RG) analysis of the order parameter fluctuationwith canonical momentum
K.O. & Kunihiro: PLB 632 (2006) 252
4. Over-damping near the critical point
Langevin Eq.
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Large damping constant limit of the propagating mode
If we impose the large damping condition ,then the propagating mode is over-damped.
For , we can integrate out explicitly the faster degree of freedomto obtain (Ma: “Modern theory of critical phenomena” (1976))
is the faster degree of freedom
is the slower degree of freedom
t
)/1(/1
t
)/1(/1 /1
Oscillatory (propagating) mode Over-damped (diffusive) mode
4. Over-damping near the critical point
Langevin eq. fora diffusive mode
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RG analysis of the Langevin Eq. for the propagating mode
RG transformation
● Integration of short-wavelength fluctuations
● Scale transformation :
Recursion relation :
4. Over-damping near the critical point
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ε-expansion• Green func. Green func. for diffusive mode
• Self-energy
• Full Green func.
New parameters ・・・
4. Over-damping near the critical point
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Recursion Relation
We can find fixed points in the space
Usual recursion for the static G-L theory
Dynamic parameters
Gaussian & Wilson-Fisher (WF) fixed points
(Hohenberg & Halperin: Rev.Mod.Phys. 49 (1977) 435)
4. Over-damping near the critical point
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Two fixed points with respect to Wilson-Fisher fixed point
Crossover between the two fixed points Propagating mode becomes over-damped near the critical point
4. Over-damping near the critical point
Gaussian WF
• z=1: Propagating mode ( ) ・・・ unstable• z=2: Overdamped mode ( ) ・・・ stable
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• Overdamped (diffusive) mode
• Anti-ferromagnet Rajagopal & Wilczek (1993)
• Particle (propagating) mode Hatsuda & Kunihiro (1985)
The fate of meson mode near the chiral transition
4. Over-damping near the critical point
Pion and sigma are not able to propagate and lose a particle-like nature
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Ordered phase (Ferroelectric)
Disordered phase
Order parameter fluctuation ・・・ phonon mode
Phonon mode near the ferroelectric transition
4. Over-damping near the critical point
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• Over-damping as a crossover between the two fixed points• Universality of the propagating behavior
Phonon mode is over-damped near the critical point Experimental fact
Almairac et al. (1977)
Softening with z=1・・・ Propagating fixed
point
Over-damping region (z=2)・・・ Diffusive fixed
point
4. Over-damping near the critical point
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Propagating mode in the O(N) model
Meson mode at chiral transition
Phonon mode at ferroelectric transition
Canonical momentum is necessary as a slow variable
RG analysis of the propagating mode
Meson mode near chiral transition is over-damped!
Anti-ferromagnet (Rajagopal & Wilczek)
Phonon mode near ferroelectric transition
5. Conclusion
• 2 fixed points for the propagating and diffusive modes
• Over-damping near the critical point