Flow equations and phase transitions. phase transitions.

Flow equations and Flow equations and phase transitionsphase transitions

phase transitionsphase transitions

QCD – phase transitionQCD – phase transition

Quark –gluon plasmaQuark –gluon plasma

Gluons : 8 x 2 = 16Gluons : 8 x 2 = 16 Quarks : 9 x 7/2 =12.5Quarks : 9 x 7/2 =12.5 Dof : 28.5Dof : 28.5

Chiral symmetryChiral symmetry

Hadron gasHadron gas

Light mesons : 8Light mesons : 8 (pions : 3 )(pions : 3 ) Dof : 8Dof : 8

Chiral sym. brokenChiral sym. broken

Large difference in number of degrees of freedom !Large difference in number of degrees of freedom !Strong increase of density and energy density at TStrong increase of density and energy density at Tcc !!

Understanding the phase Understanding the phase diagramdiagram

quark-gluon quark-gluon plasmaplasma““deconfinement”deconfinement”

quark matter : quark matter : superfluidsuperfluidB spontaneously B spontaneously brokenbroken

nuclear matter : B,isospin (Inuclear matter : B,isospin (I33) spontaneously ) spontaneously broken, broken, S conservedS conserved

Phase diagram for mPhase diagram for mss > m > mu,du,d

vacuumvacuum

Order parametersOrder parameters

Nuclear matter and quark matter are Nuclear matter and quark matter are separated from other phases by true separated from other phases by true critical linescritical lines

Different realizations of Different realizations of globalglobal symmetriessymmetries

Quark matter: SSB of baryon number BQuark matter: SSB of baryon number B Nuclear matter: SSB of combination of B Nuclear matter: SSB of combination of B

and isospin Iand isospin I3 3 neutron-neutron condensateneutron-neutron condensate

quark-gluon plasmaquark-gluon plasma““deconfinement”deconfinement”

quark matter : superfluidquark matter : superfluidB spontaneously brokenB spontaneously broken

nuclear matter : superfluidnuclear matter : superfluidB, isospin (IB, isospin (I33) spontaneously broken, S conserved) spontaneously broken, S conserved

vacuumvacuum


deconfinementdeconfinement

liquidliquidvacuumvacuum


nuclear matternuclear matterB,isospin (IB,isospin (I33) spontaneously broken, S conserved) spontaneously broken, S conserved

quark matter : superfluidquark matter : superfluidB spontaneously brokenB spontaneously broken

quark-gluon plasmaquark-gluon plasma““deconfinement”deconfinement”

Phase diagram for mPhase diagram for mss > > mmu,du,d

Phase diagramPhase diagram

<<φφ>= >= σσ ≠≠ 0 0

<<φφ>≈>≈00

quark-gluon quark-gluon plasmaplasma““deconfinement”deconfinement”

vacuumvacuum quark matter ,quark matter ,nuclear nuclear mater: mater: superfluidsuperfluidB B spontaneously spontaneously brokenbroken

First order phase transition First order phase transition lineline

hadrons hadrons

quarks and gluonsquarks and gluons

μμ=923MeV=923MeVtransition totransition to nuclearnuclear matter matter

Exclusion argumentExclusion argument

hadronic phasehadronic phasewith sufficient with sufficient production of production of ΩΩ : : excluded !!excluded !!

P.Braun-Munzinger, J.Stachel ,…

Tc ≈ Tch

““minimal” phase minimal” phase diagramdiagram

for equal nonzero quark for equal nonzero quark massesmasses

strong interactions in high strong interactions in high T phaseT phase

presence of strong presence of strong interactions:interactions:

crossovercrossover or or first orderfirst order phase phase transition transition

if no global order parameter distinguishes if no global order parameter distinguishes phasesphases

( second order only at critical endpoint )( second order only at critical endpoint )

use flow non-perturbative use flow non-perturbative equationsequations

Electroweak phase Electroweak phase transition ?transition ?

1010-12 -12 s after big bangs after big bang fermions and W-,Z-bosons get massfermions and W-,Z-bosons get mass standard model : crossoverstandard model : crossover baryogenesis if first orderbaryogenesis if first order ( only for some SUSY – models )( only for some SUSY – models ) bubble formation of “ our vacuum “bubble formation of “ our vacuum “

Reuter,Wetterich ‘93Reuter,Wetterich ‘93

Kuzmin,Rubakov,Shaposhnikov ‘85 , Shaposhnikov ‘87Kuzmin,Rubakov,Shaposhnikov ‘85 , Shaposhnikov ‘87

strong electroweak strong electroweak interactions interactions

responsible for crossoverresponsible for crossover

Electroweak phase Electroweak phase diagramdiagram

M.Reuter,C.WetterichM.Reuter,C.WetterichNucl.Phys.B408,91(1993)Nucl.Phys.B408,91(1993)

Masses of excitations Masses of excitations (d=3)(d=3)

O.Philipsen,M.Teper,H.Wittig ‘97

small MH large MH

ContinuityContinuity

BEC – BCS crossoverBEC – BCS crossover

Bound molecules of two atoms Bound molecules of two atoms on microscopic scale:on microscopic scale:

Bose-Einstein condensate (BEC ) for low TBose-Einstein condensate (BEC ) for low T

Fermions with attractive interactionsFermions with attractive interactions (molecules play no role ) :(molecules play no role ) :

BCS – superfluidity at low T BCS – superfluidity at low T by condensation of Cooper pairsby condensation of Cooper pairs

CrossoverCrossover by by Feshbach resonanceFeshbach resonance as a transition in terms of external as a transition in terms of external magnetic fieldmagnetic field

scattering lengthscattering length

BCS

BEC

concentrationconcentration c = a kF , a(B) : scattering length needs computation of density n=kF

3/(3π2)

BCS

BEC

dilute dilutedense

T = 0

non-interactingFermi gas

non-interactingBose gas

Floerchinger, Scherer , Diehl,…see also Diehl, Gies, Pawlowski,…

BCS BEC

free bosons

interacting bosons

BCS

Gorkov

BCS – BEC crossoverBCS – BEC crossover

Flow equationsFlow equations

Unification fromUnification fromFunctional Renormalization Functional Renormalization

fluctuations in d=0,1,2,3,...fluctuations in d=0,1,2,3,... linear and non-linear sigma modelslinear and non-linear sigma models vortices and perturbation theoryvortices and perturbation theory bosonic and fermionic modelsbosonic and fermionic models relativistic and non-relativistic physicsrelativistic and non-relativistic physics classical and quantum statisticsclassical and quantum statistics non-universal and universal aspectsnon-universal and universal aspects homogenous systems and local disorderhomogenous systems and local disorder equilibrium and out of equilibriumequilibrium and out of equilibrium

unificationunification

complexity

abstract laws

quantum gravity

grand unification

standard model

electro-magnetism

gravity

Landau universal functionaltheory critical physics renormalization

unified description of unified description of scalar models for all d scalar models for all d

and Nand N

Scalar field theoryScalar field theory

Flow equation for average Flow equation for average potentialpotential

Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact

Infrared cutoffInfrared cutoff

Wave function renormalization Wave function renormalization and anomalous dimensionand anomalous dimension

for Zfor Zk k ((φφ,q,q22) : flow equation is) : flow equation is exact !exact !

unified approachunified approach

choose Nchoose N choose dchoose d choose initial form of potentialchoose initial form of potential run !run !

( quantitative results : systematic derivative ( quantitative results : systematic derivative expansion in second order in derivatives )expansion in second order in derivatives )

Flow of effective potentialFlow of effective potential

Ising modelIsing model CO2

TT** =304.15 K =304.15 K

pp** =73.8.bar =73.8.bar

ρρ** = 0.442 g cm-2 = 0.442 g cm-2

Experiment :

S.Seide …

Critical exponents

critical exponents , BMW critical exponents , BMW approximationapproximation

Blaizot, Benitez , … , Wschebor

Solution of partial differential Solution of partial differential equation :equation :

yields highly nontrivial non-perturbative results despite the one loop structure !

Example: Kosterlitz-Thouless phase transition

Essential scaling : d=2,N=2Essential scaling : d=2,N=2

Flow equation Flow equation contains contains correctly the correctly the non-non-perturbative perturbative information !information !

(essential (essential scaling usually scaling usually described by described by vortices)vortices)

Von Gersdorff …

Kosterlitz-Thouless phase Kosterlitz-Thouless phase transition (d=2,N=2)transition (d=2,N=2)

Correct description of phase Correct description of phase withwith

Goldstone boson Goldstone boson

( infinite correlation ( infinite correlation length ) length )

for T<Tfor T<Tcc

Exact renormalization Exact renormalization group equationgroup equation

wide applicationswide applications

particle physicsparticle physics

gauge theories, QCDgauge theories, QCD Reuter,…, Marchesini et al, Ellwanger et al, Litim, Reuter,…, Marchesini et al, Ellwanger et al, Litim,

Pawlowski, Gies ,Freire, Morris et al., Braun , many othersPawlowski, Gies ,Freire, Morris et al., Braun , many others

electroweak interactions, gauge hierarchyelectroweak interactions, gauge hierarchy problemproblem

Jaeckel,Gies,…Jaeckel,Gies,…

electroweak phase transitionelectroweak phase transition Reuter,Tetradis,…Bergerhoff,Reuter,Tetradis,…Bergerhoff,


gravitygravity

asymptotic safetyasymptotic safety Reuter, Lauscher, Schwindt et al, Percacci et al, Reuter, Lauscher, Schwindt et al, Percacci et al,

Litim, Fischer,Litim, Fischer,

SaueressigSaueressig


condensed mattercondensed matter

unified description for classical bosons unified description for classical bosons CW , Tetradis , Aoki , Morikawa , Souma, Sumi , CW , Tetradis , Aoki , Morikawa , Souma, Sumi ,

Terao , Morris , Graeter , v.Gersdorff , Litim , Terao , Morris , Graeter , v.Gersdorff , Litim , Berges , Mouhanna , Delamotte , Canet , Bervilliers Berges , Mouhanna , Delamotte , Canet , Bervilliers , Blaizot , Benitez , Chatie , Mendes-Galain , , Blaizot , Benitez , Chatie , Mendes-Galain , Wschebor Wschebor

Hubbard model Hubbard model Baier , Bick,…, Metzner et al, Salmhofer et al, Baier , Bick,…, Metzner et al, Salmhofer et al,

Honerkamp et al, Krahl , Kopietz et al, Katanin , Honerkamp et al, Krahl , Kopietz et al, Katanin , Pepin , Tsai , Strack ,Pepin , Tsai , Strack ,

Husemann , Lauscher Husemann , Lauscher



quantum criticalityquantum criticality Floerchinger , Dupuis , Sengupta , Jakubczyk ,Floerchinger , Dupuis , Sengupta , Jakubczyk ,

sine- Gordon model sine- Gordon model Nagy , PolonyiNagy , Polonyi

disordered systems disordered systems Tissier , Tarjus , Delamotte , CanetTissier , Tarjus , Delamotte , Canet



equation of state for COequation of state for CO2 2 Seide,…Seide,…

liquid Heliquid He44 Gollisch,…Gollisch,… and He and He3 3 Kindermann,…Kindermann,…

frustrated magnets frustrated magnets Delamotte, Mouhanna, TissierDelamotte, Mouhanna, Tissier

nucleation and first order phase transitionsnucleation and first order phase transitions Tetradis, Strumia,…, Berges,…Tetradis, Strumia,…, Berges,…



crossover phenomena crossover phenomena Bornholdt , Tetradis ,…Bornholdt , Tetradis ,…

superconductivity ( scalar QEDsuperconductivity ( scalar QED3 3 )) Bergerhoff , Lola , Litim , Freire,…Bergerhoff , Lola , Litim , Freire,… non equilibrium systemsnon equilibrium systems Delamotte , Tissier , Canet , Pietroni , Meden , Delamotte , Tissier , Canet , Pietroni , Meden ,

Schoeller , Gasenzer , Pawlowski , Berges , Schoeller , Gasenzer , Pawlowski , Berges , Pletyukov , Reininghaus Pletyukov , Reininghaus


nuclear physicsnuclear physics

effective NJL- type modelseffective NJL- type models Ellwanger , Jungnickel , Berges , Tetradis,…, Ellwanger , Jungnickel , Berges , Tetradis,…,

Pirner , Schaefer , Wambach , Kunihiro , Pirner , Schaefer , Wambach , Kunihiro , Schwenk Schwenk

di-neutron condensatesdi-neutron condensates Birse, Krippa, Birse, Krippa, equation of state for nuclear matterequation of state for nuclear matter Berges, Jungnickel …, Birse, Krippa Berges, Jungnickel …, Birse, Krippa nuclear interactionsnuclear interactions SchwenkSchwenk


ultracold atomsultracold atoms

Feshbach resonances Feshbach resonances Diehl, Krippa, Birse , Gies, Pawlowski , Diehl, Krippa, Birse , Gies, Pawlowski ,

Floerchinger , Scherer , Krahl , Floerchinger , Scherer , Krahl ,

BEC BEC Blaizot, Wschebor, Dupuis, Sengupta, Blaizot, Wschebor, Dupuis, Sengupta,

FloerchingerFloerchinger

conclusionsconclusions

There is still a lot of interesting There is still a lot of interesting theory to learn for an understanding theory to learn for an understanding of phase transitionsof phase transitions

Perhaps non-perturbative low Perhaps non-perturbative low equations can helpequations can help

qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics

accessible :accessible :

( 1 ) basic object is ( 1 ) basic object is simplesimple

average action ~ classical actionaverage action ~ classical action

~ generalized Landau ~ generalized Landau theorytheory

direct connection to thermodynamicsdirect connection to thermodynamics

(coarse grained free energy )(coarse grained free energy )



( 2 ) Infrared scale k ( 2 ) Infrared scale k

instead of Ultraviolet instead of Ultraviolet cutoff cutoff ΛΛ

short distance memory not lostshort distance memory not lost

no modes are integrated out , but only part no modes are integrated out , but only part of the fluctuations is includedof the fluctuations is included

simple one-loop form of flowsimple one-loop form of flow

simple comparison with perturbation theorysimple comparison with perturbation theory

infrared cutoff kinfrared cutoff k

cutoff on momentum cutoff on momentum resolution resolution

or frequency or frequency resolutionresolution

e.g. distance from pure anti-ferromagnetic e.g. distance from pure anti-ferromagnetic momentum or from Fermi surfacemomentum or from Fermi surface

intuitive interpretation of k by intuitive interpretation of k by association with physical IR-cutoff , i.e. association with physical IR-cutoff , i.e. finite size of system :finite size of system :

arbitrarily small momentum arbitrarily small momentum differences cannot be resolved !differences cannot be resolved !



( 3 ) only physics in small ( 3 ) only physics in small momentum range around k momentum range around k matters for the flowmatters for the flow

ERGE regularizationERGE regularization

simple implementation on latticesimple implementation on lattice

artificial non-analyticities can be avoidedartificial non-analyticities can be avoided



( 4 ) ( 4 ) flexibilityflexibility

change of fields change of fields

microscopic or composite variablesmicroscopic or composite variables

simple description of collective degrees of freedom simple description of collective degrees of freedom and bound statesand bound states

many possible choices of “cutoffs”many possible choices of “cutoffs”

some history …some history … exact RG equations :exact RG equations : Symanzik eq. , Wilson eq. , Wegner-Houghton eq. , Symanzik eq. , Wilson eq. , Wegner-Houghton eq. ,

Polchinski eq. ,Polchinski eq. , mathematical physicsmathematical physics

1PI :1PI : RG for 1PI-four-point function and hierarchy RG for 1PI-four-point function and hierarchy WeinbergWeinberg

formal Legendre transform of Wilson eq.formal Legendre transform of Wilson eq. Nicoll, ChangNicoll, Chang

non-perturbative flow :non-perturbative flow : d=3 : sharp cutoff , d=3 : sharp cutoff , no wave function renormalization or momentum no wave function renormalization or momentum

dependencedependence HasenfratzHasenfratz22

Flow equations and phase transitions. phase transitions.

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