Role of Anderson localization in the QCD phase transitions

1
Role of Anderson localization in the QCD phase
transitions
Antonio M. García-García ag3_at_princeton.edu Prin
ceton University ICTP, Trieste We investigate in
what situations Anderson localization may be
relevant in the context of QCD. At the chiral
phase transition we provide compelling evidence
from lattice and phenomenological instanton
liquid models that the QCD Dirac operator
undergoes a metal - insulator transition similar
to the one observed in a disordered conductor.
This suggests that Anderson localization plays a
fundamental role in the chiral phase transition.
In collaboration with
James Osborn PRD,75 (2007) 034503 ,NPA, 770, 141
(2006) PRL 93 (2004) 132002
2
Conclusions
At the same T that the Chiral Phase transition
undergo a metal - insulator transition
"A metal-insulator transition in the Dirac
operator induces the QCD chiral phase transition"
3
Outline

• 1. Introduction to disordered systems and
  Anderson localization.
• 2. QCD vacuum as a conductor. QCD vacuum as a
  disordered medium. Dyakonov - Petrov ideas.
• 3. QCD phase transitions.
• 4. Role of localization in the QCD phase
  transitions. Results from instanton liquid
  models and lattice.

4
A five minutes course on disordered systems
The study of the quantum motion in a random
potential
Ea
V(x)
Eb
0
Ec
X
Anderson (1957) 1. How does the quantum
dynamics depend on disorder? 2. How does the
quantum dynamics depend on energy?
5
Quantum dynamics according to
the one
parameter scaling theory

• Insulator For d lt 3 or, in d gt 3, for strong
  disorder. Classical diffusion eventually stops
  due to destructive interference (Anderson
  localization).
  
  
• Metal For d gt 2 and weak disorder quantum
  effects do not alter
• significantly the classical diffusion.
  Eigenstates are delocalized.
• Metal-Insulator transition For d gt 2 in a
  certain window of energies and disorder.
  Eigenstates are multifractal.

Dclast
Dquant
ltr2gt
Dquanta
t
Sridhar,et.al
a ? Dquanf(d,W)?
Insulator
Metal
6
How are these different regimes characterized?
1. Eigenvector statistics 2. Eigenvalue
statistics
Altshuler, Boulder lectures
7
QCD The Theory of the strong interactions

• High
  Energy g ltlt 1 Perturbative
• 
  1. Asymptotic freedom
• 
  Quarkgluons, Well understood
• 
  Low Energy g 1 Lattice simulations
• 
  The world around us
• 
  2. Chiral symmetry breaking
• 
  
• 
  Massive constituent quark
• 
  3. Confinement
• 
  Colorless hadrons
  
  
• 
  
  
  
• How to extract analytical information?
  Instantons , Monopoles, Vortices

8

QCD at T0, instantons and chiSB
tHooft, Polyakov, Callan, Gross, Shuryak,
Diakonov, Petrov,VanBaal

• Instantons Non perturbative solutions of the
  classical Yang Mills equation. Tunneling between
  classical vacua.
• 1. Dirac operator has a zero mode in the field of
  an instanton
• 2. Spectral properties of the smallest
  eigenvalues of the Dirac operator are controled
  by instantons
• 3. Spectral properties related to chiSB.
  Banks-Casher relation
• 
  

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Instanton liquid models T 0

• Multiinstanton
  vacuum?
• Problem Non linear equations
  No superposition
• Sol Variational principles(Dyakonov,Petrov),
  Instanton liquid (Shuryak)
• Typical size and some aspects of the
  interactions are fixed
• 1. ILM explains the chiSB
• 2. Describe non perturbative effects in hadronic
  correlation functions (Shuryak,Schaefer,Verbaarcho
  t)
• 3 No confinement.

10
QCD vacuum as a conductor (T 0)

Metal An electron initially
bounded to a single atom gets delocalized due to
the overlapping with nearest neighbors.

QCD Vacuum Zero modes initially
bounded to an instanton get delocalized due to
the overlapping with the rest of zero modes.
(Diakonov and Petrov)


Impurities Instantons Electron
Quarks



Differences
Dis.Sys Exponential
decay Nearest neighbors
QCD vacuum Power law decay
Long range hopping!

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QCD vacuum as a disordered conductor
Diakonov, Petrov, Verbaarschot,
Osborn, Shuryak, Zahed,Janik
Instanton positions and color
orientations vary
Impurities Instantons
Electron Quarks
T 0 long range hopping 1/Ra, ? 3lt4
QCD vacuum is a conductor for any density of
instantons
AGG and Osborn, PRL, 94 (2005) 244102
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QCD at finite T Phase transitions
At which temperature does the transition occur ?
What is the nature of transition ?
J. Phys. G30 (2004) S1259
Péter Petreczky
Quark- Gluon Plasma perturbation theory only
for TgtgtTc
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Deconfinement and chiral restoration
They must be related but nobody knows
exactly how
Deconfinement Confining potential vanishes.
Chiral RestorationMatter 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 localization
  plays an important role. 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

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Instanton liquid model at finite T

• 1. Zero modes are localized in space but
  oscillatory in time.
• 2. Hopping amplitude restricted to neighboring
  instantons.
• 3. Since TIA is short range there must exist a T
  TLsuch that a metal insulator transition takes
  place. (Dyakonov,Petrov)
• 4. The chiral phase transition
  occurs at TTc.
• Localization and chiral transition are related
  if
• 1. TL Tc .
• 2. The localization transition occurs at the
  origin (Banks-Casher)
• This is valid beyond the instanton picture
  provided that TIA is short range and the vacuum
  is disordered enough

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Main Result
At Tc but also the low lying,
undergo a metal-insulator transition.
"A metal-insulator transition in the Dirac
operator induces the chiral phase transition "
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Signatures of a metal-insulator transition
1. 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.






Skolovski, Shapiro, Altshuler
var
Mobility edge Anderson transition
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Spectrum is scale invariant
ILM with 21 massless flavors,
We have observed a metal-insulator transition at
T 125 Mev
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ILM, close to the origin, 21 flavors, N 200
Metal insulator transition
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ILM Nf2 massless. Eigenfunction statistics
AGG and J. Osborn, 2006
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Localization versus chiral transition
Instanton liquid model Nf2, masless
Chiral and localizzation transition occurs at the
same temperature
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Lattice QCD AGG, J. Osborn, PRD, 2007

• 1. Simulations around the chiral phase transition
  T
• 2. Lowest 64 eigenvalues
• Quenched
• 1. Improved gauge action
• 2. Fixed Polyakov loop in the real Z3 phase
• Unquenched
• 1. MILC colaboration 21 flavor improved
• 2. mu md ms/10
• 3. Lattice sizes L3 X 4

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RESULTS ARE THE SAME
AGG, Osborn PRD,75 (2007) 034503
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Chiral phase transition and localization

• For massless fermions Localization predicts a
  (first) order phase transition. Why?
• 1. Metal insulator transition always occur close
  to the origin and the chiral condensate is
  determined by the same eigenvalues.
• 2. In chiral systems the spectral density is
  sensitive to localization.
• For nonzero mass Eigenvalues up to m contribute
  to the condensate but the metal insulator
  transition occurs close to the origin only.
  Larger eigenvalue are delocalized so we expect a
  crossover.
• Number of flavors Disorder effects diminish
  with the number of flavours. Vacuum with
  dynamical fermions is more correlated.

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Confinement and spectral properties Idea
Polyakov loop is expressed as the response of the
Dirac operator to a change in time boundary
conditions Gattringer,PRL 97 (2006) 032003,
hep-lat/0612020
Politely Challenged in heplat/0703018,
Synatschke, Wipf, Wozar
. but sensitivity to spatial boundary conditions
is a criterium (Thouless) for localization!
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Localization and confinement

• 1.What part of the spectrum contributes the most
  to the Polyakov loop?.Does it scale with volume?
• 2. Does it depend on temperature?
• 3. Is this region related to a metal-insulator
  transition at Tc?
• 4. What is the estimation of the P from
  localization theory?
• 5. Can we define an order parameter for the
  chiral phase transition in terms of the
  sensitivity of the Dirac operator to a change in
  spatial boundary conditions?

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Localization and
Confinement
IPR (red), Accumulated Polyakov loop (blue) for
TgtTc as a function of the eigenvalue.
MI transition?
Metal prediction
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Accumulated Polyakov loop versus
eigenvalue Confinement is controlled by the
ultraviolet part of the spectrum
P
?
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Conclusions

• 1. Eigenvectors of the QCD Dirac operator becomes
  more localized as the temperature is increased.
• 2. For a specific temperature we have observed a
  metal-insulator transition in the QCD Dirac
  operator in lattice QCD and instanton liquid
  model.
• 3. "The Anderson transition occurs at the same T
  than the chiral phase transition and in the same
  spectral region
• Whats next?
• 1. How relevant is localization for
  confinement?
• 2. How are transport coefficients in the quark
  gluon plasma affected by localization?
• 3 Localization and finite density. Color
  superconductivity.

THANKS! ag3_at_princeton.edu
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Quenched ILM, Origin, N 2000
For T lt 100 MeV we expect (finite size scaling)
to see a (slow) convergence to RMT results. T
100-140, the metal insulator transition occurs
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Quenched ILM, IPR, N 2000
Metal IPR X N 1 Insulator IPR X N N
Multifractal IPR X N
Similar to overlap prediction Morozov,Ilgenfritz,
Weinberg, et.al.
Origin
D22.3(origin)
Bulk