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Role of Anderson-Mott 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. Based on a recent relation
between the Polyakov loop and the spectral properties of the Dirac operator we
discuss how the confinement-deconfinement transition may be related to a
metal-insulator transition in the bulk of the spectrum of the Dirac operator.
James Osborn
In collaboration with
PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002
Outline
1. A few words about localization.
2. Disorder in QCD, Dyakonov – Petrov ideas.
3. A few words about QCD phase transitions.
4. Role of localization in the QCD phase transitions. Results
from ILM and lattice.
4.1 The chiral phase transition.
4.2 The deconfinement transition. In progress.
5. What’s next. Quark diffusion in LHC.
A few words on disordered systems
Quantum particle in a random potential
Anderson localization
Quantum destructive interference
can induce a transition to an insulating
state.
Insulator
For d < 3 or, at strong disorder,
in d > 3 all eigenstates are
localized in space.
Metal
d > 2, Weak disorder
Eigenstates delocalized
Mott localization
Interaction can induce a
transition from metal (classical)
to insulator.
Insulator
Metal
Eigenfunction characterization
1. Eigenfunctions moments:
1 Insulator
IPR    (r ) d r ~ 
Metal
V
4
d
1
n
2. Decay of the eigenfunctions:
e

 (r ) ~ 1 / V
 1/ r 

Spectral
 r /
n
characterization ?
Insulator
Metal  d

  d
?
 d

Insulator
Critical
Metal
Spectral characterization
Metal  Σ n  = n  n ~ log n Insulator  Σ n   n


s
( RMT ) 
( Poisson)  P( s )  e
P s  ~ s e
2
2
2
β
2
 As 2
RMT correlations: Weak disorder (d > 2). Up to Thouless.
 2 ( L) ~ Ld / 2
Poisson correlations: Any disorder d < 2, strong disorder d>2
"In the context of QCD the metallic region corresponds with the infrared limit
(constant fields) of the Dirac operator" (Verbaarschot,Shuryak)
QCD vacuum, disorder and instantons
Diakonov, Petrov, later Verbaarschot, Osborn, Zahed, Osborn & AGG
Dirac operator has a zero mode in the field of an instanton

Dψ0 r   0 ψ0 r  1/ r
ins 
μ
D =  μ + gA 
QCD vacuum saturated by weakly interacting (anti) instantons
3
(Shuryak)
Density and size of instantons are fixed phenomenologically
(


)
T   d x ( x  z )iD  ( x  z ) ~ i(u  Rˆ )
R
4
IA


I
I
I
A
A
A
4
Long range hopping in the instanton liquid model (ILM)
Diakonov - Petrov
As a consequence of the long range hopping the QCD vacuum is a metal:
Zero modes initially bounded to an instanton get delocalized due to the
overlapping with the rest of zero modes. By increasing temperature (or
other parameters) the QCD vacuum will eventually undergo a metal
insulator transition. What means a metal?
3
Conductivity versus chiral symmetry breaking
"Spectral properties of the smallest eigenvalues of the Dirac operator
are controled by instantons" Is that important? Yes.
    lim 
m  m0
 (m)
Banks-Casher (Kubo)
V
1 3N N 
   
  (240MeV )
  2 V 
1/ 2
Metallic behavior means chiSB in the ILM
Recent developments:
c
- Thouless energy in QCD. If the QCD vacuum at T= 0 is a metal,
one can predict finite size effects.
Verbaarschot,Osborn, PRL 81 (1998) 268 and Zahed, Janik et.al., PRL. 81 (1998) 264
- The QCD Dirac operator can be described by a random matrix with
long range hopping even beyond the Thouless energy.
AGG and Osborn, PRL, 94 (2005) 244102
3
Phase transitions in QCD
J. Phys. G30 (2004) S1259
Quark- Gluon Plasma
weakly only for T>>Tc
Deconfinement and chiral restoration
They must be related but nobody* knows exactly how
Deconfinement
•Linear confining
potential vanishes.
L
 0
Chiral Restoration
•Matter becomes light
•QCD still non
 ~ 0
perturbative
How to explain these transitions?
1. Effective model of QCD close to the chiral restoration (Wilczek,Pisarski):
Universality, epsilon expansion.... too simple?
2. QCD but only consider certain classical solutions (t'Hooft):
Instantons (chiral), Monopoles (confinement)
No monopoles found, instantons only after lattice cooling, no from QCD
We propose that quantum interference and tunneling, namely, Anderson
localization plays an important role. Nuclear Physics A, 770, 141 (2006)
Localization and chiral transition
Instanton liquid picture
1.The effective QCD coupling constant g(T) decreases as
temperature increases. The density of instantons also
decreases  (Tc )   (0) / 2
2. Zero modes are exponentially localized  ( R)  exp(TR )
in space but oscillatory in time.
3. Amplitude TIA ~ exp( ATR) hopping restricted to
neighboring instantons.
4. Localization will depend strongly on the temperature. There
must exist a T = TLsuch that a MIT takes place. Dyakonov,
5. There must exist a T = Tc such that    0 Petrov
6. This general picture is valid beyond the instanton liquid
approximation (KvBLL, see Ilgenfritz talk) provided that the
hopping induced by topological objects is short range.
Is TL = Tc ?...Yes
Does the MIT occur at the origin? Yes
Main Result
D =  + gA
QCD
μ
 D

QCD

μ
  i 
n
At Tc , Chiral phase transition
A  A , A
lat
n
ins

  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 "
Spectral characterization
Metal  Σ n  = n  n ~ log n Insulator  Σ n   n


s
( RMT ) 
( Poisson)  P( s )  e
P s  ~ s e
2
2
2
β
2
 As 2
 ( L) ~ L
RMT correlations: Weak disorder (d > 2). Up to Thouless.
Poisson correlations: Any disorder d < 2, strong disorder d>2
"In the context of QCD the metallic region corresponds with the infrared limit
(constant fields) of the Dirac operator" (Verbaarschot,Shuryak)
2
d /2
ANDERSON TRANSITION
Main:Non trivial interplay between tunneling and interference leads
to the metal insulator transition (MIT)
Spectral correlations
Wavefunctions
Scale invariance
Multifractals
 (n) ~ n
P( s) ~ s 
s  1
P( s) ~ e
s  1
2
  (r ) d r ~ L
2q
d
 Dq ( q 1)
n
 As
Skolovski, Shapiro, Altshuler
CRITICAL STATISTICS
Kravtsov, Muttalib
97
"Spectral correlations are universal, they depend only on the
dimensionality of the space."
Mobility edge
Anderson transition
Finite size scaling analysis, Dynamical 2+1
var  s  s
2
Massive
2
s n   s n P(s)ds
Massless
Quenched Lattice
IPR versus eigenvalue
Unquenched ILM, 2 m = 0
The transition is located around T =120
Unquenched lattice, close to the origin,
2+1 flavors, N = 200
INSULATOR
METAL
Unquenched ILM, close to the origin,
2+1 flavors, N = 200
Instanton liquid model:,condensate and inverse participation ratio versus
T
Lattice: and inverse participation ratio versus T
Unquenched, massive 2+1
Quenched ( also unquenched masless)
For zero mass, transition sharper with the volume
First order
For finite mass, the condensate is volume independent
Crossover
Localization and order of the chiral phase transition
    lim 
m  m0
 (m)
V
1. Metal insulator transition always occur close to the origin.
2. Systems with chiral symmetry the spectral density is sensitive to
localization.
3. For zero mass localization predicts a first order phase transition.
4. For a non zero mass m, eigenvalues up to m contribute to the
condensate but the metal insulator transition occurs close to the
origin only. Larger eigenvalue are delocalized se we expect a
crossover.
5. Multifractal dimension m=0 should modify susceptibility
exponents.
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 ) N
z1  ( x ) z1

 z1

1  
L( x ) 

8 N   (1  z 2 ) N
z2  ( x ) z2

 z2









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 boundary conditions is a
criterium (Thouless) for localization!
Localization and confinement
The dimensionless conductance, g, a measure of localization, is related to
the sensitivity of eigenstates to a change in boundary conditions.
Metal
g  Ld 2
Insulator
g 0 L 
MI transition
L
g  g c (d ) L  
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?
Accumulated Polyakov loop versus eigenvalue
Confinement is controlled by the ultraviolet part of the spectrum
P

Localization and Confinement
IPR (red), Accumulated Polyakov loop (blue) for T>Tc as a
function of the eigenvalue.
Metal
prediction
MI
transition?
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
Quenched ILM, T =200, bulk
Mobility edge in the Dirac
operator. For T =200 the
transition occurs around the
center of the spectrum
D2~1.5 similar to the
3D Anderson model.
Not related to chiral
symmetry
Unquenched ILM, 2+1 flavors
We have observed a metal-insulator transition at T ~ 125 Mev
Conclusions
●
●
●
Eigenvectors of the QCD Dirac operator becomes more
localized as the temperature is increased.
For a specific temperature we have observed a metalinsulator transition in the QCD Dirac operator.
For lattice and ILM, and for quenched and unquenched
we have found two transitions close to the origin and in the
UV part of the spectrum and.
MAIN
"The Anderson transition occurs at the same T than the chiral
phase transition and in the same spectral region"
“ Confinement-Deconfinemente transition has to do with
localization-delocalization in time direction”
What's next?
1. How critical exponents are affected by
localization?
2. Confinement and localization, analytical
result?
3. How are transport coefficients in the quark
gluon plasma affected by localization?
4. Localization in finite density. Color
superconductivity.
QCD : The Theory of the strong interactions
L   (i   gA )  m    G


QCD
1
4
q
2
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
Analytical information? Instantons , Monopoles, Vortices
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
IPR, two massless flavors
D2 ~ 1.5 (bulk) D2~2.3(origin)
A A
 W  =
A A
RMT
P
RMT
A =  s Ps ds

2
0
How to get information from a bunch of levels
Spectrum
Unfolding
Spectral Correlators
Quenched ILM, Bulk, T=200
Nuclear (quark) matter at finite temperature
1. Cosmology 10-6 sec after Bing Bang, neutron stars (astro)
3 Analytical, 4N=4 super YM ?
2. 1
Lattice QCD finite2size effects.
3. High energy Heavy Ion Collisions. RHIC, LHC
Colliding Nuclei
Hard
Collisions
QG Plasma ?
Hadron Gas &
Freeze-out
sNN = 130, 200 GeV
(center-of-mass energy per nucleon-nucleon collision)
Multifractality
Intuitive: Points in which the modulus of the
wave function is bigger than a (small) cutoff M. If
the fractal dimension depends on the cutoff M,
Kravtsov, Chalker,Aoki,
the wave function is multifractal.
Schreiber,Castellani
IPR  I =  ψ r  d r  L
4
2
Ld
n
d
 D2
Instanton liquid models T = 0
"QCD vacuum saturated by interacting (anti) instantons"
Density and size of (a)instantons are fixed phenomenologically
The Dirac operator D, in a basis of single I,A:
0
iD  
T
T 

0

IA
AI
  200MeV ,
N
 1 fm
V
4
(


)
T   d x ( x  z )iD  ( x  z ) ~ i(u  Rˆ )
R
4
IA


I
I
I
A
A
3
A
4
1. ILM explains the chiSB
2. Describe non perturbative effects in hadronic correlation functions
(Shuryak,Schaefer,dyakonov,petrov,verbaarchot)
QCD Chiral Symmetries
L,R  (1   5 )
Classical
SU A (3)  SUV (3)  UV (1)  U A (1)

Quantum
SUV (3)  UV (1)
U(1)A explicitly broken by the anomaly.
SU(3)A spontaneously broken by the QCD vacuum
qq  (250 MeV )
3
Dynamical mass
Eight light Bosons (,K,), no parity doublets.
Quenched lattice QCD simulations
Symanzik 1-loop glue with asqtad valence