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Non-equilibrium critical phenomena in the chiral phase transition

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

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Non-equilibrium critical phenomena in the chiral phase transition

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  1. Non-equilibrium critical phenomenain 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 K.O. & Kunihiro : PLB 632 (2006) 252

  2. 1. Introduction Strong interaction between hadrons (proton, neutron, pion, ρ-meson) QCD (quark & gluon) Chiral symmetry in the u-, d-quark sector

  3. 1. Introduction Spontaneous Breaking of Chiral symmetry pion is the massless Nambu-Goldstone particle Ferromagnet O(3) symmetry is spontaneously broken NG mode = spin wave

  4. 1. Introduction Early universe Heavy Ion Collision (RHIC,LHC) Real world 2nd TCP Quark-Gluon-Plasma phase Hadron phase Color-Superconducting phase 1st • Static (Equilibrium) critical phenomena Lattice simulation, Effective theory, Universality argument, etc. • Dynamic (Non-equilibrium) critical phenomena Heavy Ion Collision, Early universe

  5. 2. Review : Dynamic Critical Phenomena Non-equilibrium, time-dependent Anomalous dynamic critical phenomena • Critical slowing down • Softening of propagating modes • Divergence of transport coefficients • ... Long relaxation time Non-equilibrium state Equilibrium state Relax Slow motion of long wavelength fluctuations of Slow variables

  6. 2. Review : Dynamic Critical Phenomena • 2 kinds of slow variables 1. Order parameter 2. Conserved quantity Flat potential Continuity Eq. Slow variables (Order parameter & Conserved quantities) are the fundamental degrees of freedom in the critical slow dynamics

  7. 2. Review : Dynamic Critical Phenomena • 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 Diffusive mode (Damping mode) Propagating mode (Damped Oscillatory mode)

  8. ( : fixed) ( Dynamic critical exponent) 2. Review : Dynamic Critical Phenomena Spectral func. for slow variables Critical slowing down Softening • Propagating mode pole with Real and Imaginary parts • Diffusive mode pole with only Imaginary part • Dynamic scaling hypothesis

  9. 2. Review : Dynamic Critical Phenomena Universality class Static universality class critical behavior (critical exponents) is identical if symmetry and (spatial) dimension are same. Chiral transition belongs to the same universality class as ferromagnet and anti-ferromagnet Pisarski & Wilczek:PRD29(1984)338 Ferromagnet and anti-ferromagnet belong to the O(3) universality class

  10. 2. Review : Dynamic Critical Phenomena Dynamic universality class Slow variables Hohenberg & Halperin: Rev.Mod.Phys.49 (1977) 435 Classification scheme • Whether the order parameter is conserved or not • 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,....

  11. 2. Review : Dynamic Critical Phenomena Dynamic universality class of chiral transition Rajagopal & Wilczek: NPB 399 (1993) 395 Meson mode is a diffusive mode Chiral transition belongs to anti-ferromagnet • 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

  12. 2. Review : Dynamic Critical Phenomena Meson mode is a propagating mode Hatsuda & Kunihiro: PRL 55 (1985) 158 Meson (particle) is an oscillatory mode of field Diffusive mode Rajagopal & Wilczek Propagating mode Hatsuda & Kunihiro ?

  13. 3. Propagating mode in the O(N) model Meson mode (Propagating mode) Canonical momentum conjugate to order parameter (Koide & Maruyama: NPA 742 (2004) 95) (K.O., Fukushima & Ohta: NPA 748 (2005) 260) Neither Order parameter nor Conserved quantity! Langevin Eq. Brownian particle Zwanzig J.Stat.Phys.9(1973)215 O(N) Ginzburg-Landau potential Square of propagating velocity Damping constant

  14. 4. Over-damping near the critical point K.O. & Kunihiro: PLB 632 (2006) 252 Renormalization Group (RG) analysis of the order parameter fluctuation with canonical momentum Langevin Eq.

  15. 4. Over-damping near the critical point Large damping constant limit of the propagating mode If we impose the large damping condition , then the propagating mode is over-damped. t t Over-damped (diffusive) mode Oscillatory (propagating) mode is the faster degree of freedom is the slower degree of freedom For , we can integrate out explicitly the faster degree of freedom to obtain (Ma: “Modern theory of critical phenomena” (1976)) Langevin eq. for a diffusive mode

  16. 4. Over-damping near the critical point RG analysis of the Langevin Eq. for the propagating mode RG transformation ●Integration of short-wavelength fluctuations ●Scale transformation : Recursion relation :

  17. 4. Over-damping near the critical point ε-expansion • Green func. Green func. for diffusive mode • Self-energy • Full Green func. New parameters ・・・

  18. 4. Over-damping near the critical point Recursion Relation (Hohenberg & Halperin: Rev.Mod.Phys. 49 (1977) 435) Dynamic parameters Usual recursion for the static G-L theory Gaussian & Wilson-Fisher (WF) fixed points We can find fixed points in the space

  19. WF Gaussian • z=1: Propagating mode ( ) ・・・　unstable • z=2: Overdamped mode ( ) ・・・　stable 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

  20. 4. Over-damping near the critical point The fate of meson mode near the chiral transition • Particle (propagating) mode • Hatsuda & Kunihiro (1985) • Overdamped (diffusive) mode • Anti-ferromagnet • Rajagopal & Wilczek (1993) Pion and sigma are not able to propagate and lose a particle-like nature

  21. 4. Over-damping near the critical point Phonon mode near the ferroelectric transition Ordered phase (Ferroelectric) Disordered phase Order parameter fluctuation ・・・ phonon mode

  22. 4. Over-damping near the critical point Phonon mode is over-damped near the critical point Experimental fact Almairac et al. (1977) Over-damping region (z=2) ・・・　Diffusive fixed point Softening with z=1 ・・・ Propagating fixed point • Over-damping as a crossover between the two fixed points • Universality of the propagating behavior

  23. 5. Conclusion • 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 • 2 fixed points for the propagating and diffusive modes • Over-damping near the critical point

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