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