Flow equations and Flow equations and phase transitions phase transitions
Dec 21, 2015
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
Phase diagram for mPhase diagram for mss > m > mu,du,d
deconfinementdeconfinement
liquidliquidvacuumvacuum
Phase diagram for mPhase diagram for mss > m > mu,du,d
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,
wide applicationswide applications
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
wide applicationswide applications
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
wide applicationswide applications
condensed mattercondensed matter
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
wide applicationswide applications
condensed mattercondensed matter
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,…
wide applicationswide applications
condensed mattercondensed matter
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
wide applicationswide applications
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
wide applicationswide applications
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 )
qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics
accessible :accessible :
( 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 !
qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics
accessible :accessible :
( 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
qualitative changes that make qualitative changes that make non-perturbative physics non-perturbative physics
accessible :accessible :
( 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