Transcript Slide 1

Role of Anderson localization in the QCD
phase transitions
Antonio M. García-García
[email protected]
Princeton 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.
James Osborn
In collaboration with
PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002
Conclusions:

 D



n
QCD
 n  in  n
At the same T that the
Chiral Phase transition
  0
undergo a metal - insulator transition
n
"A metal-insulator transition in the Dirac operator
induces the QCD chiral phase transition"
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.
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?
Quantum dynamics according to
the one parameter scaling theory
Insulator: For d < 3 or, in d > 3, for strong disorder. Classical diffusion
eventually stops due to destructive interference (Anderson localization).
Metal: For d > 2 and
weak disorder quantum effects do not alter
significantly the classical diffusion. Eigenstates are delocalized.
Metal-Insulator transition: For d > 2 in a certain window of
energies and disorder. Eigenstates are multifractal.
Dclast
<r2>
Dquant
Dquanta
Sridhar,et.al
Insulator
a=?
Metal
Dquan=f(d,W)?
t
How are these different regimes characterized?

 D
QCD
 n  in  n
1. Eigenvector statistics:
2. Eigenvalue statistics:
H n  En n
IPR  L
d
d D
d

(
r
)
d
r
~
L
 n
4
P(s)   s  i 1  i /  
i
Insulator D2 ~ 0

( Poisson) P(s)  es
Metal  D2 ~ d

 As2
(GOE) Ps  ~ se
Altshuler,
Boulder lectures
2
QCD : The Theory of the strong interactions
High Energy g << 1 Perturbative
1. Asymptotic freedom
Quark+gluons, Well understood
Low Energy
g ~ 1 Lattice simulations
The world around us
2. Chiral symmetry breaking
 ~ (240MeV )
3
Massive constituent quark
3. Confinement
Colorless hadrons
V (r )  a / r  r
How to extract analytical information? Instantons , Monopoles, Vortices
QCD at T=0, 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.

ins 
μ
D =  μ + gA 
Dψ0 r   0 ψ0 r  1/ r
3
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
 (m)
1
 ( )
1
   Tr ( D  m)   d
  lim 
mm0
V
m  i
V
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
3
(


)
4


TIA   d x I ( x  z I )iD   A ( x  z A ) ~ i(u  Rˆ ) I 3A
R
1  3N N 

  (240MeV )
  2 V 
1/ 2
1. ILM explains the chiSB
  
c
2. Describe non perturbative effects in hadronic correlation
functions (Shuryak,Schaefer,Verbaarchot)
3 No confinement.
3
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
Differences
Dis.Sys: Exponential decay
QCD vacuum Power law decay
Quarks
Nearest neighbors
Long range hopping!
QCD vacuum as a disordered conductor
Diakonov, Petrov, Verbaarschot, Osborn, Shuryak, Zahed,Janik
Instanton positions and color orientations vary
Impurities
T=0
Instantons
Electron
Quarks
long range hopping 1/Ra, a = 3<4
QCD vacuum is a conductor for any density of instantons
AGG and Osborn, PRL, 94 (2005) 244102
QCD at finite T: Phase transitions
At which temperature does the transition occur ? What is the nature of transition ?
Péter Petreczky
J. Phys. G30 (2004) S1259
Quark- Gluon Plasma
perturbation theory only for
T>>Tc
Deconfinement and chiral restoration
They must be related but nobody* knows exactly how
Deconfinement: Confining potential vanishes. L 
Chiral Restoration:Matter becomes light.
0
 ~ 0
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
Instanton liquid model at finite T
1. Zero modes are localized in space but oscillatory in time.
 ( R)  exp( TR)
2. Hopping amplitude restricted to neighboring instantons.
T ~ exp(  ATR)
IA
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
   0 occurs at T=Tc.
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”
Main Result

 D
At Tc

QCD

 n  in  n
 lim 
m  m0
 ( m)
V
0
but also the low lying,



n
undergo a metal-insulator transition.
n
"A metal-insulator transition in the Dirac operator
induces the chiral phase transition "
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.
Skolovski, Shapiro, Altshuler
2.

P( s ) ~ s
P(s) ~ e  As
s  1
s  1
var
3. Eigenstates are multifractals.
  (r ) d r ~ L
2q
n
d
 Dq ( q 1)
var  s  s
2
2
s n   s n P( s)ds
Mobility edge
Anderson transition
Spectrum is
scale
invariant
ILM with 2+1 massless flavors,
We have observed a metal-insulator transition at T ~ 125 Mev
Metal
insulator
transition
ILM, close to the origin, 2+1
flavors, N = 200
ILM Nf=2 massless. Eigenfunction statistics
AGG and J. Osborn, 2006
Localization versus
chiral transition
Instanton liquid model Nf=2, masless
Chiral and localizzation transition occurs at the same temperature
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 2+1 flavor improved
2. mu= md = ms/10
3. Lattice sizes L3 X 4
RESULTS ARE
THE SAME
AGG, Osborn PRD,75 (2007)
034503
Chiral phase transition and localization
  
 lim 
m  m0
 (m)
V
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.
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


U 4 ( x, N )  zU 4 ( x, N )


 2 N  ( x )  (1  z1 ) Nz1  ( x ) z1 

 z1

1  

L( x ) 

N
8 N   (1  z 2 ) z  ( x ) z

2
2



z
2


Politely Challenged in:
heplat/0703018,
Synatschke, Wipf,
Wozar


1 
N
N
N 
P  L( x ) 
2   (1  z1 ) z1  (1  z2 ) z2

8V  
z1
z2






 ( x )   v  ( x, t )  v  ( x, t )
N
t 1
L,
R,
…. but sensitivity to spatial boundary conditions
is a criterium (Thouless) for localization!
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?
Localization and Confinement
IPR (red), Accumulated Polyakov loop (blue) for T>Tc as a
function of the eigenvalue.
Metal
prediction
MI
transition?
Accumulated Polyakov loop versus eigenvalue
Confinement is controlled by the ultraviolet part of the spectrum
P

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 metalinsulator 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“
What’s 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!
[email protected]
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
Metal
IPR X N= 1
Insulator
IPR X N = N
Multifractal
Similar to overlap prediction
Origin
Morozov,Ilgenfritz,Weinberg, et.al.
Bulk
IPR X N =
D2~2.3(origin)
N
D2