Role of Anderson localization in the QCD phase transitions Antonio M. García-García...
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Transcript of Role of Anderson localization in the QCD phase transitions Antonio M. García-García...
Role of Anderson localization in Role of Anderson localization in the QCD phase transitionsthe QCD phase transitions
Antonio M. García-García
[email protected] University
ICTP, Trieste
We investigate in what situations Anderson localization may be relevant in the We investigate in what situations Anderson localization may be relevant in the context of QCD. At the chiral phase transition we provide compelling evidence context of QCD. At the chiral phase transition we provide compelling evidence from lattice and phenomenological instanton liquid models that the QCD Dirac from lattice and phenomenological instanton liquid models that the QCD Dirac operator undergoes a metal - insulator transition similar to the one observed in operator undergoes a metal - insulator transition similar to the one observed in
a disordered conductor. This suggests that Anderson localization plays a a disordered conductor. This suggests that Anderson localization plays a fundamental role in the chiral phase transition. fundamental role in the chiral phase transition.
In collaboration with In collaboration with James OsbornJames Osborn PRD,75 (2007) 034503 ,PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002NPA, 770, 141 (2006) PRL 93 (2004) 132002
Conclusions:Conclusions:
nnnQCD iD
0At the same T that the Chiral Phase transition
"A metal-insulator transition in the Dirac operator "A metal-insulator transition in the Dirac operator induces the QCD chiral phase transition"induces the QCD chiral phase transition"
n
n
undergo a metal - insulatormetal - insulator transition
Outline:Outline:
1. Introduction to disordered systems and Anderson localization. 1. Introduction to disordered systems and Anderson localization.
2. QCD vacuum as a conductor. QCD vacuum as a disordered 2. QCD vacuum as a conductor. QCD vacuum as a disordered medium. Dyakonov - Petrov ideas. medium. Dyakonov - Petrov ideas.
3. QCD phase transitions.3. QCD phase transitions.
4. Role of localization in the QCD phase transitions. Results from 4. Role of localization in the QCD phase transitions. Results from instanton liquid models and lattice.instanton liquid models and lattice.
V(x)
X
Ea
Eb
Ec
Anderson (1957):Anderson (1957):
1. 1. How does the quantum dynamics depend on How does the quantum dynamics depend on disorder?disorder?
2. How does the quantum dynamics depend on 2. How does the quantum dynamics depend on energy? energy?
0
A five minutes course A five minutes course on disordered systemson disordered systems
The study of the quantum The study of the quantum motion in a random potentialmotion in a random potential
Insulator:Insulator: For d < 3 or, in d > 3, for strong disorder. Classical diffusion For d < 3 or, in d > 3, for strong disorder. Classical diffusion eventually stops due to destructive interference (Anderson localization). eventually stops due to destructive interference (Anderson localization).
Metal:Metal: For For d > 2 and weak disorder quantum effects do not alter d > 2 and weak disorder quantum effects do not alter significantly the classical diffusion. Eigenstates are delocalized.significantly the classical diffusion. Eigenstates are delocalized.
Metal-Insulator transition: Metal-Insulator transition: For d > 2 in a certain window of For d > 2 in a certain window of energies and disorder. Eigenstates are multifractal.energies and disorder. Eigenstates are multifractal.
Quantum dynamics according to Quantum dynamics according to the one the one
parameter scaling theoryparameter scaling theory
<r2
>
a = ?
Dquan=f(d,W)?
t
Dclast
Dquan
t
Dquanta
Sridhar,et.al
Insulator Metal
How are these different regimes characterized?How are these different regimes characterized?
sesP
D
Poisson
Insulator
)(
0~
)(2
22 ~
)( Asse~sP
dD
GOE
Metal
1. Eigenvector statistics:
2. Eigenvalue statistics:
2~)(4 Ddd
nd LrdrLIPR
i
iissP /)( 1
nnnnnnQCD EHiD
Altshuler, Altshuler, Boulder lecturesBoulder lectures
QCD : The Theory of the strong interactionsQCD : The Theory of the strong interactions
HighHigh EnergyEnergy g << 1 Perturbativeg << 1 Perturbative
1. Asymptotic freedom Quark+gluons, Well understoodQuark+gluons, Well understood
Low Energy Low Energy g ~ 1 Lattice simulationsg ~ 1 Lattice simulations The world around usThe world around us
2. Chiral symmetry breaking2. Chiral symmetry breaking
Massive constituent quark Massive constituent quark
3. Confinement3. Confinement Colorless hadronsColorless hadrons
How to extract analytical information?How to extract analytical information? Instantons , Monopoles, Instantons , Monopoles, VorticesVortices
rrarV /)(
3)240(~ MeV
Instantons:Instantons: Non perturbative solutions of the classical Yang Mills Non perturbative solutions of the classical Yang Mills equation. Tunneling between classical vacua. equation. Tunneling between classical vacua.
1. Dirac operator has a zero mode in the field of an instanton1. Dirac operator has a zero mode in the field of an instanton
2. Spectral properties of the smallest eigenvalues of the Dirac operator 2. Spectral properties of the smallest eigenvalues of the Dirac operator are controled by instantons are controled by instantons
3. 3. Spectral properties related to chiSB. Banks-Casher relationSpectral properties related to chiSB. Banks-Casher relation
QCD at T=0, instantons and chiSB QCD at T=0, instantons and chiSB tHooft, Polyakov, Callan, Gross, Shuryak, tHooft, Polyakov, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaalDiakonov, Petrov,VanBaal
300 /10 rrψrDψgA+=D ins
μμ
V
m
imdmDTr
V mm
)(lim
)()(
10
1
Multiinstanton vacuum?Multiinstanton vacuum?
Problem:Problem: Non linear equations Non linear equations No superposition No superposition
Sol:Sol: Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak) Variational principles(Dyakonov,Petrov), Instanton liquid (Shuryak)
Typical size and some aspects of the interactions are fixedTypical size and some aspects of the interactions are fixed
1. ILM explains the chiSB1. ILM explains the chiSB
2. Describe non perturbative effects in hadronic correlation 2. Describe non perturbative effects in hadronic correlation functions functions (Shuryak,Schaefer,Verbaarchot)(Shuryak,Schaefer,Verbaarchot)
3 No confinement.3 No confinement.
3
34 )(
)ˆ(~)()(R
RuizxiDzxxdT AIAAIIIA
3
2/1
)240(2
31MeV
V
NNc
Instanton liquid models T = 0Instanton liquid models T = 0
Metal Metal An electron initially bounded to a single atom gets delocalized due An electron initially bounded to a single atom gets delocalized due
to the overlapping with nearest neighbors.to the overlapping with nearest neighbors.
QCD VacuumQCD Vacuum Zero modes initially bounded to an instanton get delocalized due Zero modes initially bounded to an instanton get delocalized due
to the overlapping with the rest of zero modes. to the overlapping with the rest of zero modes. (Diakonov and (Diakonov and Petrov)Petrov)
Impurities Impurities Instantons Instantons ElectronElectron QuarksQuarks
Differences Differences Dis.Sys:Dis.Sys: Exponential decay N Exponential decay Nearest earest neighborsneighbors QCD vacuumQCD vacuum Power law decayPower law decay Long range hopping! Long range hopping!
QCD vacuum as a conductor (T =0)QCD vacuum as a conductor (T =0)
QCD vacuum as a disordered QCD vacuum as a disordered conductorconductor
Instanton positions and color orientations varyInstanton positions and color orientations vary
Impurities Impurities Instantons Instantons Electron Electron
QuarksQuarksT = 0 long range hopping 1/RT = 0 long range hopping 1/R = 3<4 = 3<4
Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik
AGG and Osborn, AGG and Osborn, PRL, 94 (2005) 244102PRL, 94 (2005) 244102
QCD vacuum is a conductor for any density of instantonsQCD vacuum is a conductor for any density of instantons
QCD at finite T: Phase transitionsQCD at finite T: Phase transitions
Quark- Gluon Plasma perturbation theory only for T>>Tc
J. Phys. G30 (2004) S1259
At which temperature does the transition occur ? What is the nature of transition ?
Péter Petreczky Péter Petreczky
Deconfinement and chiral restorationDeconfinement and chiral restoration
Deconfinement: Confining potential vanishes.
Chiral Restoration:Matter becomes light.
How to explain these transitions?
1. Effective model of QCD close to the phase transition (Wilczek,Pisarski,Yaffe):
Universality, epsilon expansion.... too simple?
2. QCD but only consider certain classical solutions (t'Hooft): Instantons (chiral), Monopoles and vortices (confinement). Instanton do not dissapear at the transiton (Shuryak,Schafer).
We propose that quantum interference and tunneling, namely, Anderson Anderson localization plays an important role. localization plays an important role. Nuclear Physics A, 770, 141 (2006)Nuclear Physics A, 770, 141 (2006)
C. Gattringer, M. Gockeler, et.al. Nucl. Phys. B618, 205 (2001),R.V. Gavai, S. Gupta et.al, PRD 65, 094504 (2002), M.
Golterman and Y. Shamir, Phys. Rev. D 68, 074501 (2003), V. Weinberg, E.-M. Ilgenfritz, et.al, PoS { LAT2005}, 171 (2005), hep-lat 0705.0018, I. Horvath, N. Isgur, J. McCune, and H. B. Thacker, Phys. Rev. D65, 014502 (2002), J. Greensite, S. Olejnik et.al., Phys. Rev. D71, 114507 (2005). V. G. Bornyakov, E.-M. Ilgenfritz, 07064206
They must be related but nobody* knows exactly how
0~0L
1. Zero modes are localized in space but oscillatory in time.1. Zero modes are localized in space but oscillatory in time.
2. Hopping amplitude restricted to neighboring instantons.2. Hopping amplitude restricted to neighboring instantons.
3. Since T3. Since TIAIA is short range there must exist a T = T is short range there must exist a T = TLLsuch that a metal insulator transition takes such that a metal insulator transition takes place. place. (Dyakonov,Petrov)(Dyakonov,Petrov)
4. The chiral phase transition occurs at T=T4. The chiral phase transition occurs at T=Tc.c.
Localization and chiral transition are related if:Localization and chiral transition are related if:
1. T1. TLL = T = Tc . c .
2. The localization transition occurs at the origin 2. The localization transition occurs at the origin (Banks-Casher)(Banks-Casher)
““This is valid beyond the instanton picture provided that TThis is valid beyond the instanton picture provided that TIAIA is short range and the vacuum is is short range and the vacuum is disordered enough”disordered enough”
0
Instanton liquid model at finite T Instanton liquid model at finite T
)exp()( TRR
)exp(~ ATRTIA
nnnQCD iD
At Tc
but also the low lying,
"A metal-insulator transition in the Dirac operator "A metal-insulator transition in the Dirac operator induces the chiral phase transition "induces the chiral phase transition "
n
n
undergo a metal-insulator transition.
Main ResultMain Result
0)(
lim0
V
mmm
Signatures of a metal-insulator transitionSignatures of a metal-insulator transition1. Scale invariance of the spectral correlations.
A finite size scaling analysis is then carried out to determine the transition point.
2.
3. Eigenstates are multifractals.
)1(2
~)( qDdq
n
qLrdr
Skolovski, Shapiro, Altshuler
1~)(
1~)(
sesP
sssPAs
Mobility edge Anderson transition
varvar
dssPssss nn )(var22
ILM with 2+1 massless flavors,
We have observed a metal-insulator transition at T ~ 125 Mev
Spectrum is scale invariant
ILM, close to the origin, 2+1 ILM, close to the origin, 2+1 flavors, N = 200flavors, N = 200
Metal Metal insulator insulator transitiontransition
ILM Nf=2 massless. Eigenfunction ILM Nf=2 massless. Eigenfunction statisticsstatistics
AGG and J. Osborn, 2006 AGG and J. Osborn, 2006
Instanton liquid model Nf=2, maslessInstanton liquid model Nf=2, masless Localization versus Localization versus
chiral transitionchiral transition
Chiral and localizzation transition occurs at the same temperatureChiral and localizzation transition occurs at the same temperature
Lattice QCD Lattice QCD AGG, J. Osborn, PRD, AGG, J. Osborn, PRD,
20072007
1. Simulations around the chiral phase transition 1. Simulations around the chiral phase transition T T
2. Lowest 64 eigenvalues 2. Lowest 64 eigenvalues
QuenchedQuenched
1. Improved gauge action1. Improved gauge action
2. Fixed Polyakov loop in the “real” Z2. Fixed Polyakov loop in the “real” Z33 phase phase
UnquenchedUnquenched
1. MILC colaboration 2+1 flavor improved1. MILC colaboration 2+1 flavor improved
2. m2. muu= m= md d = m= mss/10/10
3. Lattice sizes L3. Lattice sizes L33 X 4 X 4
RESULTS ARE RESULTS ARE THE SAME THE SAME AGG, Osborn AGG, Osborn PRD,75 (2007) PRD,75 (2007) 034503034503
Chiral phase transition and hiral phase transition and localizationlocalization
For massless fermions: For massless fermions: Localization predicts a (first) Localization predicts a (first) order phase transition. Why?order phase transition. Why?
1. Metal insulator transition always occur close to the origin and 1. Metal insulator transition always occur close to the origin and the chiral condensate is determined by the same eigenvalues.the chiral condensate is determined by the same eigenvalues.
2. In chiral systems the spectral density is sensitive to localization2. In chiral systems the spectral density is sensitive to localization..
For nonzero mass:For nonzero mass: Eigenvalues up to m contribute to the Eigenvalues up to m contribute to the condensate but the metal insulator transition occurs close to condensate but the metal insulator transition occurs close to the origin only. Larger eigenvalue are delocalized so we expect the origin only. Larger eigenvalue are delocalized so we expect a crossover.a crossover.
Number of flavors:Number of flavors: Disorder effects diminish with the number Disorder effects diminish with the number of flavours. Vacuum with dynamical fermions is more correlated. of flavours. Vacuum with dynamical fermions is more correlated.
V
mmm
)(lim
0
2
22
1
11
)()1(
)()1()(2
8
1)(
2
1
z
z
zNz
zNz
N
xz
xzx
NxL
),(),()( ,1
, txvtxvx R
N
tL
),(),( 44 NxzUNxU
2
2
1
1)1()1(2
8
1)( 21
zz
Nz
Nz
N zzV
xLP
Confinement and spectral propertiesIdea:Idea: Polyakov loop is expressed as the response of the Dirac operator to a Polyakov loop is expressed as the response of the Dirac operator to a change in time boundary conditionschange in time boundary conditions Gattringer,PRL 97 (2006) 032003, hep-lat/0612020
……. . but sensitivity to but sensitivity to spatialspatial boundary conditions boundary conditions is a criterium (Thouless) for localization!is a criterium (Thouless) for localization!
Politely Challenged in:Politely Challenged in:
heplat/0703018, heplat/0703018,
Synatschke, Wipf, WozarSynatschke, Wipf, Wozar
Localization and confinementLocalization and confinement1.What part of the spectrum contributes the most to the 1.What part of the spectrum contributes the most to the
Polyakov loop?.Does it scale with volume?Polyakov loop?.Does it scale with volume?
2. Does it depend on temperature?2. Does it depend on temperature?
3. Is this region related to a metal-insulator transition at 3. Is this region related to a metal-insulator transition at TTcc??
4. What is the estimation of the P from localization theory?4. What is the estimation of the P from localization theory?
5. Can we define an order parameter for the chiral phase 5. Can we define an order parameter for the chiral phase transition in terms of the sensitivity of the Dirac transition in terms of the sensitivity of the Dirac operator to a change in spatial boundary conditions? operator to a change in spatial boundary conditions?
IPR (red), Accumulated Polyakov loop (blue) for T>TIPR (red), Accumulated Polyakov loop (blue) for T>Tcc as a as a
function of the eigenvalue.function of the eigenvalue.
Localization and ConfinementLocalization and Confinement
MetalMetal
predictionprediction
MI MI transition?transition?
Accumulated Polyakov loop versus eigenvalueAccumulated Polyakov loop versus eigenvalue
Confinement is controlled by the ultraviolet part of the spectrum Confinement is controlled by the ultraviolet part of the spectrum
PP
1. Eigenvectors of the QCD Dirac operator becomes 1. Eigenvectors of the QCD Dirac operator becomes more localized as the temperature is increased. more localized as the temperature is increased.
2. For a specific temperature we have observed a 2. For a specific temperature we have observed a metal-insulator transition in the QCD Dirac operator metal-insulator transition in the QCD Dirac operator in lattice QCD and instanton liquid model.in lattice QCD and instanton liquid model.
3. "The Anderson transition occurs at the same 3. "The Anderson transition occurs at the same T than the chiral phase transition and in the T than the chiral phase transition and in the same spectral region“same spectral region“
What’s next?What’s next?
1. How relevant is localization for confinement? 1. How relevant is localization for confinement?
2. How are transport coefficients in the quark gluon plasma 2. How are transport coefficients in the quark gluon plasma affected by localization?affected by localization?
3 Localization and finite density. Color superconductivity3 Localization and finite density. Color superconductivity..
ConclusionsConclusions
THANKS! THANKS! [email protected]@princeton.edu
Quenched ILM, Origin, N = 2000
For T < 100 MeV we expect (finite size scaling) to see a (slow) convergence to RMT results.
T = 100-140, the metal insulator transition occurs
Quenched ILM, IPR, N = 2000
Similar to overlap prediction
Morozov,Ilgenfritz,Weinberg, et.al.
Metal
IPR X N= 1
Insulator
IPR X N = N
Origin
BulkD2~2.3(origin)
Multifractal
IPR X N = 2DN