Universality far from equilibrium(Nonthermal fixed points as quantum amplifier)J. Berges
IKP Theory Center
Technische Universität Darmstadt
Institute for Theoretical Physics
Universität Heidelberg

Content
I. Renormalization group fixed points
II. Weak vs. strong wave turbulence
III. Universality far from equilibrium

IV. Nonthermal fixed points as quantum amplifier

Renormalization group fixed points
RG: ‘microscope‘ with varying resolution of length scale

Denote coarse graining length as inverse characteristic momentum scale~ 1/kFixed point: physics looks the same for ‘all‘ resolutions, i.e. becomes
independent of k (in rescaled units)
Typically not for all resolutions:• IR fixed point for k  0• UV fixed point for k  ‘‘Examples in and out of equilibriuma) Thermal equilibrium IR fixed points… associated to second-order phase transitions:effective potential(p)  1/p2-

T > TcT = TcT < Tcorder parameter (p)  pz
power-law exponents• : anomalous dimension• z: dynamical exponentb) Far from equilibrium IR fixed points
Important candidate: stationary transport of conserved charges… IR fixed points associated to diverging time scales in thermalization process• : occupation number exponent

Digression: diverging time scales
Thermalization process in quantum many-body systems, schematically:• Characteristic nonequilibrium time scales? Relaxation? Instabilities?
• Diverging time scales? E.g. associated to- critical slowing down near second-order phase transitions, or- far from equilibrium IR scaling solutions?  nonthermal fixed points

Instability leads to breaking of waves& weak Kolmogorov wave turbulence

Increasing forcing energy injection (higher wave amplitudes)

Experimental example:
Modulation instability andcapillary wave turbulence

Energy injection limited by droplet formation!
Xia, Shats, Punzmann, EPL91 (2010) 14002

Digression: weak wave turbulence
E.g. Boltzmann equation for number conserving 22 scattering

momentum conservationenergy conservation“gain“ term“loss“ term2  1

has different stationary solutions in the (classical) regime n(p)  1:
1. n(p) = 1/(e(p) – 1)

thermal equilibrium

2. n(p)  1/p4/3

turbulent particle cascade

3. n(p)  1/p5/3

energy cascade

Kolmogorov, Proc.
USSR Acad. Sci.,
30 (1941) 299

Solutions 2 and 3 are limited to the “window“ 1/  n(p)  1, since forn(p)  1/ the nm scatterings for n,m=1,.., are as important as 22

Weak vs. strong wave turbulence
Log n(p)n(p)  1/

1/  n(p)  1n(p)  1

nonthermal
fixed point

Kolmogorov
weak wave
turbulencequantum(dissipative)
regime 1/p4/3 , 1/p5/3

 e-p 1/p
1/
1Log pPredicted scaling exponents for nonperturbative regime:a) n(p)  1/p4

strong turbulence particle cascadeb) n(p)  1/p5

energy cascade

Berges, Rothkopf, Schmidt,
PRL 101 (2008) 041603

Experimental verification requires sufficient energy injection!

Experimental candidates for nonperturbatively large densities
1) Relativistic heavy-ion collisions explore strong interaction matterstarting from a transient nonequilibrium state

Schmelling, hep-ex/9701002

CGC: Energy density of gluons with typical momentum Qs (at time 1/Qs)
i.e. occupation numbers

nonperturbative!

 see also last week‘s talks, this conference (jointly with TFT workshop)

2) Early universe at the end of inflation , schematic evolution:• Energy density of matter ( a-3) and radiation ( a-4) dilutes by expansion
 energy at the end of inflation stored in homogeneous field with amplitude

i.e. after preheating
nonperturbatively large in quartic self-coupling   10 -12 !

3) Ultracold quantum gases...atoms at nanokelvins

NIST• mg/4 determined by s-wave scattering length• diluteness condition:• nonperturbatively large occupation number of momentum mode p if

 see also talk by Boris Nowak, this conferenceUniversality far from equilibrium
Early-universe (p)reheating
after inflation (~1016 GeV)

Heavy-ion collisions(~100 MeV)

Cold quantum gas dynamics(~10-13 eV)

WMAP

NIST

Nonequilibrium instabilities

Nonthermal fixed points
Very different microscopic dynamics (instabilities) can lead to same
macroscopic scaling phenomena for bosons (strong turbulence)
Berges, Rothkopf, Schmidt ‘08; Berges, Hoffmeister ‘09; Scheppach, Berges, Gasenzer ‘10;
Carrington, Rebhan ‘10; Nowak, Sexty, Gasenzer ’11; Berges, Sexty ’11; …

Nonperturbative functional descriptions
1.Two-particle
irreducible expansions

2. Functional
renormalization group
kWetterich ‘93

Luttinger, Ward ´60; Baym ´62;
Cornwall, Jackiw, Tomboulis ´74,…

2PI-1/N to NLO: ( (aa)2 )k Real-time functional
renormalization group
Canet, Delamotte, Deloubriere,Wschebor ’04; Mitra, Takei, Kim,
Millis ’06; Gezzi, Pruschke, Meden’07; Jacobs, Meden, Schoeller ’07;
Gasenzer, Pawlowski ’08; Berges,
Hoffmeister ’09; Kloss, Kopietz ’10;…

Berges ´02; Aarts, Ahrensmeier,Baier, Berges, Serreau ´02+ 3. classical lattice simulations

Strong turbulence:

universal scaling exponent• Reheating dynamics after chaotic inflation
Occupation number

nonthermal fixed point(strong turbulence)

Inflation

Quantum
fluctuations

WMAP Science Team

weak wave
turbulence

Berges, Rothkopf, Schmidt,
PRL 101 (2008) 041603• Superfluid turbulence in a cold Bose gas

2-dim case

Tangledvortex lines

Nowak, Sexty, Gasenzer,
PRB84 (2011) 020506(R)Nonthermal fixed points as quantum amplifier
Berges, Gelfand, Pruschke, PRL 107 (2011) 061301• Fermions: n(p)  1 (Pauli principle) no classical-statistical approximation
 dramatic enhancement of genuine quantum effects
if coupled to nonperturbatively occupied bosons!
E.g. dressed fermion self-energy correction in QCD:
~ 1/g2
gg~ O(1)All fermion quantum corrections ’saturate’ parametrically to order one!
Requires real-time lattice simulations with dynamical fermions in QCD!
 test claims with simpler model, where 2PI as well as lattice works

Model• generic interaction of Yukawa type for Nf massless Dirac fermions:g couples left- and right-handed components

i.e. acting like a mass term for     0• we consider Nf = 2 with symmetry group

 N=4 component linear -model with quartic self-interactionBosonicsector(g=0)
parametricresonance

approach to turbulence:occupation number:

d=3n(t,p)  p- with  = 4 for d = 3,
 = 5 for d = 4

d=4IRfor z = 1 (relativistic),  = 0
Berges, Rothkopf, Schmidt, PRL 101 (2008) 041603,Berges, Hoffmeister, NPB 813 (2009) 383,Berges, Sexty, PRD 83 (2011) 085004

Equal-time two-point correlator

Comparing classical to quantum (g=0)

pPractically no bosonic quantum corrections at the end of instability
Accurate nonperturbative description by quantum 2PI-1/N to NLO

Fermions: failure of semi-classical approach
LO:tBaacke, Heitmann, Pätzold, PRD 58 (1998) 125013; Greene, Kofman, PLB 448 (1999) 6;
Giudice, Peloso, Riotto, Tkachev, JHEP 9908 (1999) 014; Garcia-Bellido, Mollerach, Roulet,JHEP 0002 (2000) 034; …
Boson

2PI-NLO:+Fermion

small self-coupling  leads
to large corrections!

Berges, Gelfand, Pruschke, PRL 107 (2011) 061301

Parametric resonancepreheatingm = 0  g2

genuine quantum correction

Occupation number distributions
Fermions

IR fermions thermally occupied

Bosons1/p4

Bosons still far from equilibrium

Nonequilibrium fermion spectral function
vector components
scalar component
quantum field anti-commutation relation:
Wigner transform:( X0 = (t + t‘)/2 )massless fermions‘heavy‘ fermions

Lattice simulations with dynamical fermionsConsider general class of models including lattice gauge theories
with covariant coupling to fermions:

For classical (x) the exact equation for the fermion D(x,y) reads:

Very costly (4 4 N3 N3)! Use low-cost fermions of Borsanyi & Hindmarsh!
Aarts, Smit, NPB 555 (1999) 355

PRD 79 (2009) 065010

Real-time dynamical fermions in 3+1 dimensions!

• Wilson fermions on a 643 lattice

Berges, Gelfand, Pruschke, PRL 107 (2011) 061301• Very good agreement with NLO quantum result (2PI) for   1(differences at larger p depend on Wilson term  larger lattices)• Lattice simulation can be applied to  ~ 1  SU(N) gauge theory

Preliminary results for  = 1• comparison of  = g2/ = 0.1 and  = 1 (relevant for QCD)
log-log plot this time!

with Gelfand, Sexty, in preparation

 substantially enhanced fermion production with early approach
to Fermi-Dirac distribution in wide momentum range!

Conclusions• nonthermal fixed points provide powerful classification for
universal far from equilibrium properties of theories• they show strongly enhanced fluctuations as compared tothermal equilibrium (e.g. 1/p4 as compared to thermal 1/p)
• genuine quantum effects for fermions are dramatically amplified
in the presence of nonthermal fixed points for bosons• heavy ion collisions, cold quantum gases (and maybe early universe
via gravity waves) are promising candidates to discover them

nonequilibrium
instabilities

nonthermal fixed pointn(t,p)  etinitialconditionsn(t=0,p)n(t,p)  p-tnBE
thermal equilibrium

Nonthermal fixed points:• approached from substantial class of initial conditions (no fine tuning!)• properties independent of details of the underlying microscopic theory• critical slowing down can substantially delay thermalization
Fermions:• dramatically enhanced fermion production from quantum corrections(thermally occupied in the IR while bosons are still far from equilibrium)• strongly coupled (1) fermions required to speed-up thermalization of bosons