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
Strong interaction between hadrons (proton, neutron, pion, ρ-meson)
QCD (quark & gluon)
Chiral symmetry in the u-, d-quark sector
1. Introduction
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
• 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
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
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
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
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
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
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
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
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
?
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!
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.
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
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
ε-expansion• Green func. Green func. for diffusive mode
• Self-energy
• Full Green func.
New parameters ・・・
4. Over-damping near the critical point
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
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
• 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
Ordered phase (Ferroelectric)
Disordered phase
Order parameter fluctuation ・・・ phonon mode
Phonon mode near the ferroelectric transition
4. Over-damping near the critical point
• 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
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
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