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. In collaboration with
James Osborn
PRD,75 (2007) 034503 ,NPA, 770, 141 (2006) PRL 93 (2004) 132002
Conclusions:
D QCD
n
i
n
n
n n
At the same T that the Chiral Phase transition
undergo a
metal - insulator
transition 0
"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 E a V(x) E b 0 X Anderson (1957): 1. How does the quantum dynamics depend on disorder?
2. How does the quantum dynamics depend on energy? E c
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.
D clas t
Insulator Metal
a = ?
D quan =f(d,W)?
t
How are these different regimes characterized?
D QCD
n
i
n
n H
n
E n
n
1. Eigenvector statistics: 2. Eigenvalue statistics:
IPR P
(
s
)
L d
i
s n
(
r
) 4
d
i
1
d r
i
~ /
L d
D
2
Insulator
(
Poisson
)
D
2 ~ 0
P
(
s
)
e
s Metal
(
GOE
)
D
2
P
~
~
d se
As
2 Altshuler, Boulder lectures
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, Callan, Gross, Shuryak, Diakonov, Petrov,VanBaal Instantons: Non perturbative solutions of the classical Yang Mills equation. Tunneling between classical vacua.
D =
μ
+ gA ins μ
Dψ
0 0
ψ
0 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 1
V Tr
(
D
m
) 1
d
m
( )
i
lim
m
m
0 (
m
)
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
T IA
d
4
x
I
(
x
z I
)
iD
A
(
x
z A
) ~
i
(
u
R
ˆ ) (
I
A
) 3
R
3 1 3
N c N V
1 / ( 240
MeV
) 3 3 No confinement.
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 N earest neighbors QCD vacuum Power law decay Long range hopping!
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/R a , 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>>T c
Deconfinement and chiral restoration
They must be related but nobody* knows exactly how
Deconfinement:
Confining potential vanishes.
L
0
Chiral Restoration:
Matter becomes light.
~ 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 IA
~ exp(
ATR
) 3. Since T IA is short range there must exist a T = T L such that a metal insulator transition takes place. (Dyakonov,Petrov) 4. The chiral phase transition occurs at T=T c.
Localization and chiral transition are related if:
1. T L = T c . 2. The localization transition occurs at the origin (Banks-Casher) “This is valid beyond the instanton picture provided that T IA short range and the vacuum is disordered enough” is
Main Result
D QCD
n
i
n
n
At T
c
m
lim
m
0 (
m
)
V
but also the low lying,
n n
undergo a metal-insulator transition.
"A metal-insulator transition in the Dirac operator induces the chiral phase transition "
0
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
var
P s s
s
1
P
(
s
) ~
e
As s
1 3. Eigenstates are multifractals.
n
(
r
) 2
q d d r
~
L
D q
(
q
1 ) var
s
2
s
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” Z 3 Unquenched phase 1. MILC colaboration 2+1 flavor improved 2. m u = m d = m s /10 3. Lattice sizes L 3 X 4
RESULTS ARE THE SAME
AGG, Osborn PRD,75 (2007) 034503
Chiral phase transition and localization
m
lim
m
0 (
V m
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: crossover.
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 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 Politely Challenged in:
L
(
x
) 1 8
N
2
U
4 ( 1 (
x
,
N
N
( )
x
)
z
2 )
z
2
N z
2
zU
4 ( 1 (
x
,
N z
1 (
x
)
z
2 )
z
1 )
N z
1 (
x
)
z
1
heplat/0703018, Synatschke, Wipf, Wozar
P
L
(
x
) 1 8
V
2 (
x
)
N
( 1
t N
1
v L z
1 , )
z
1 (
x
N z
1 ,
t
) ( 1
v R
,
z
( 2 )
z
2
x
,
t
)
N z
2 …. 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 T c ?
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>T c function of the eigenvalue.
as a Metal prediction MI transition?
P
Accumulated Polyakov loop versus eigenvalue Confinement is controlled by the ultraviolet part of the spectrum
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“ 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!
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
Origin
Metal IPR X N= 1 Insulator IPR X N = N
Bulk Similar to overlap prediction Morozov,Ilgenfritz,Weinberg, et.al.
Multifractal IPR X N =
N D
2
D2~2.3(origin)