Transcript Metal insulator transition

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.
James Osborn
In collaboration with
PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002
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
 ~ (240 MeV )
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, Shuryak, Diakonov, Petrov
Instantons (Polyakov,t'Hooft) : Non pertubative solutions of the classical Yang
Mills equation. Tunneling between classical vacua.
D =  μ  + gAμins 
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
QCD vacuum models based on instantons:
1. Density N/V = 1fm-4. Hopping amplitude T ~ i (u  R
ˆ ) (I  A )
IA
R3
Dyakonov,Petrov,
Shuryak
3
2. Describe chiSB and non perturbative effects in hadronic correlation functions.
3 No confinement.
QCD vacuum as a disordered conductor
Conductor
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
Instanton positions and color orientations vary
T=0
long range hopping TIA~aIA
/R,
 = 3<4
AGG and Osborn,
PRL, 94 (2005)
244102
QCD vacuum is a ‘disordered’ conductor for any density of instantons
QCD at finite T: Phase transitions
At which temperature does the transition occur ? What is the nature of transition ?
Deconfinement:
Confining potential
vanishes.
L 0
Chiral
Restoration:Matter
becomes light.
Péter Petreczky
J. Phys. G30 (2004) S1259
 ~ 0
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):
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)
What is Anderson localization?
A particle in a disordered potential. Classical diffusion stops
due to destructive interference.
Insulator: For d < 3 or, in d > 3, for strong disorder. Classical diffusion
eventually stops. Eigenstates are delocalized.
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.
How are these different regimes characterized?
4
1. Eigenvector statistics:
IPR  Ld   n (r ) d d r ~ Ld  D
2. Eigenvalue statistics:
P( s)    s  i 1  i  /  
2
Insulator  D2 ~ 0

( Poisson ) P( s)  e s
i
0  D2  d

 P s  ~ s β s  1
MIT 
 P s  ~ e  As s  1

Metal  D2 ~ d

β  As 2
( RMT ) Ps  ~ s e
Localization and chiral transition
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 picutre provided that TIA is
short range and the vacuum is disordered enough”
Main Result
D =  + gA
QCD
μ
A  A , A
lat
At Tc
μ
ins

 D

QCD

 lim 
m  m0
  i 
n
n
 ( 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 "
Spectrum is
scale invariant
ILM with 2+1 massless flavors,
P(s) of the lowest eigenvalues
We have observed a metal-insulator transition at T ~ 125 Mev
ILM Nf=2 massless. Eigenfunction statistics
AGG and J. Osborn, 2006
Metal insulator
transition
ILM, close to the origin, 2+1
flavors, N = 200
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
Localization and order of the chiral phase transition
  
 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.
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]
Finite size
scaling analysis:
var  s  s
2+1 dynamical fermions
2
2
s   s P(s)ds
n
Quenched
n
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, 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 Lattice QCD
IPR versus eigenvalue
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
Quantum
SU A (3)  SUV (3)  UV (1)  U A (1)

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
3. Spectral characterization:
Spectral correlations in a metal are given by random
matrix theory up to the Thouless energy Ec. Matrix
elements are only constrained by symmetry
2
2
Metal 
n ( E ) = n  n ~ log E

β  As 2
( RMT ) 


P
s
~
s
e

2
Ec  1 / L
2
In units of the mean level spacing,
the Thouless energy,
g
Ec

d 2
L
Eigenvalues in an insulator are not correlated.
Insulator n ( E )  E

( Poisson )  P( s)  e s
2
In the context of QCD the metallic
region corresponds with the infrared
limit (constant fields) of the Dirac
operator" (Verbaarschot,Shuryak)
1. QCD, random matrix theory, Thouless energy:
Spectral correlations of the QCD Dirac operator in the infrared limit are
universal (Verbaarschot, Shuryak Nuclear Physics A 560 306 ,1993). They can be
obtained from a RMT with the symmetries of QCD.
1. The microscopic spectral density is universal, it depends only on the
global symmetries of QCD, and can be computed from random matrix
theory.
2
2

 (s) ~ s J 0 (s)  J1 (s)

2. RMT describes the eigenvalue correlations of the full QCD Dirac
operator up to Ec. This is a finite size effect. In the thermodynamic
limit the spectral window in which RMT applies vanishes but at the
same time the number of eigenvalues, g, described by RMT diverges.
Fπ2
Ec ~ 2  0
L
L
Ec
g  ~ Fπ2 L2  

L
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