Random Matrix Theory for QCD

Random Matrix Theory for QCD – p. 1/30
Random Matrix Theory for QCD
Kim Splittorff
Jac Verbaarschot
Poul Henrik Damgaard
Gernot Akemann
Fall Course of the International Graduate School
Bielefeld - Paris - Helsinki, GRK 881 & PACO
Helsinki, 25 October 2010

LECTURE 3: Dirac spectra
Random Matrix Theory for QCD – p. 2/30

Appetizer !
Random Matrix Theory for QCD – p. 3/30

Random Matrix Theory for QCD – p. 4/30
The spectrum of the Dirac operator
0.5
0.4
0.3
ρ S
(ζ)
0.2
0.1
0
0 10 20 30
ζ
Damgaard, Heller, Niclasen, Rummukainen, PLB 495 (2000) 263

Random Matrix Theory for QCD – p. 5/30
What: Spectrum of the Dirac operator
Why: Spontaneous Breaking of Chiral symmetry
Understand: Finite volume effects
Effects of lattice spacing

Random Matrix Theory for QCD – p. 5/30
What: Spectrum of the Dirac operator
Why: Spontaneous Breaking of Chiral symmetry
Understand: Finite volume effects
Effects of lattice spacing
Learning Goal: be able to derive the Banks-Casher relation

Random Matrix Theory for QCD – p. 6/30
The Banks Casher relation
- why the small eigenvalues of the Dirac operator are particularly interesting
Σ ∼ ρ(λ = 0)

Random Matrix Theory for QCD – p. 6/30
The Banks Casher relation
- why the small eigenvalues of the Dirac operator are particularly interesting
Σ ∼ ρ(λ = 0)
you will derive this relation shortly

Random Matrix Theory for QCD – p. 7/30
Properties of the QCD Dirac operator
Anti-Hermitian
(iD µ γ µ ) † = −iD µ γ µ
Axial-Symmetry
{iD µ γ µ , γ 5 } = 0
No additional symmetries.

Random Matrix Theory for QCD – p. 8/30
The QCD partition function with source
Path integral
Z QCD =
Z
dA det(iD µ γ µ + m) N f
e −S YM(A)
U(N f ) × U(N f ) chiral symmetry broken to U(N f ) by the quark mass.

The QCD partition function with source
Path integral
Z QCD =
Z
dA det(iD µ γ µ + m) N f
e −S YM(A)
U(N f ) × U(N f ) chiral symmetry broken to U(N f ) by the quark mass.
The chRMT partition function
Matrix integral
Z ≡
Z
dW det(D(m)) N f
e − N 2 TrW † W
where
D(m) =
0
@ m iW
iW † m
1
A
same explicit symmetry breaking Same effective theory !
Shuryak, Verbaarschot, NPA 560, 306 (1993)
Random Matrix Theory for QCD – p. 8/30

Mini-Activity:
What can we say about the eigenvalues of iD µ γ µ
Random Matrix Theory for QCD – p. 9/30

Random Matrix Theory for QCD – p. 9/30
Mini-Activity:
What can we say about the eigenvalues of iD µ γ µ
1) The eigenvalues must be purely imaginary
2) If iDψ = iλψ then iDγ 5 ψ = −iλγ 5 ψ
Eigenvalues in pairs (iλ, −iλ) unless λ = 0.

Activity: Banks Casher relation
Random Matrix Theory for QCD – p. 10/30

Random Matrix Theory for QCD – p. 11/30
Banks Casher
Im(z)
_
< ψ ψ >(m)
X
X
X
X
X
X
Re(z)
_
< ψ ψ >
m
X
X
〈 ¯ψψ〉 = π V ρ(0) Banks Casher NPB 169 (1980) 103

Random Matrix Theory for QCD – p. 12/30
The spectrum of the Dirac operator
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 5 10
How do we read of ρ(λ ∼ 0)
Berg, Bittner, Lombardo, Markum, Pullirsch, Wettig hep-lat/0007008

How do we read of ρ(λ ∼ 0)
Analytic prediction from chRMT
only depends on λΣV
ρ(λ) = λΣV
2 (J 0(λΣV ) 2 + J 1 (λΣV ) 2 )
Verbaarschot Zahed PRL 70 (1993) 3852
Random Matrix Theory for QCD – p. 13/30

How do we read of ρ(λ ∼ 0)
Analytic prediction from chRMT
ρ(λ) = λΣV
2 (J 0(λΣV ) 2 + J 1 (λΣV ) 2 )
only depends on λΣV
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 5 10
Extract Σ from simulation at finite volume !
Verbaarschot Zahed PRL 70 (1993) 3852
Random Matrix Theory for QCD – p. 13/30

Random Matrix Theory for QCD – p. 14/30
The spectrum of the Dirac operator
0.5
0.4
0.3
ρ S
(ζ)
0.2
0.1
0
0 10 20 30
ζ
Get Σ from the first eigenvalue distribution !
Damgaard, Heller, Niclasen, Rummukainen, PLB 495 (2000) 263

New: non zero lattice spacing a
Random Matrix Theory for QCD – p. 15/30

Goal: analytic predictions for the Dirac spectrum with a ≠ 0
Random Matrix Theory for QCD – p. 16/30

Random Matrix Theory for QCD – p. 17/30
Recall: Properties of the QCD Dirac operator
Anti-Hermitian
Axial-Symmetry
(iD µ γ µ ) † = −iD µ γ µ
{iD µ γ µ , γ 5 } = 0

Random Matrix Theory for QCD – p. 17/30
Recall: Properties of the QCD Dirac operator
Anti-Hermitian
Axial-Symmetry
(iD µ γ µ ) † = −iD µ γ µ
{iD µ γ µ , γ 5 } = 0
No go theorem: This is not possible on the lattice (doublers)
Nielsen Ninomiya NPB 185 (1981) 20

Random Matrix Theory for QCD – p. 18/30
Discretization effects depend on the discretization:
Here: Wilson fermions
iD W = γ µ [iD + µ + iD − µ ] − aD + µ D − µ
The Wilson term aD + µ D − µ cures fermion doubling but violates chiral symmetry
Nielsen, Ninomiya, NPB 185 (1981) 20

Random Matrix Theory for QCD – p. 18/30
Discretization effects depend on the discretization:
Here: Wilson fermions
iD W = γ µ [iD + µ + iD − µ ] − aD + µ D − µ
The Wilson term aD + µ D − µ cures fermion doubling but violates chiral symmetry
γ 5 iD W ≠ −iD W γ 5
(iD W ) † ≠ −iD W
Nielsen, Ninomiya, NPB 185 (1981) 20

Random Matrix Theory for QCD – p. 19/30
Wilson fermions
iD W = γ µ [iD + µ + iD − µ ] − aD + µ D − µ
γ 5 -hermiticity
(iD W ) † = γ 5 iD W γ 5
Itho, Iwasaki, Yoshie, PRD 36 (1987) 527

Random Matrix Theory for QCD – p. 19/30
Wilson fermions
iD W = γ µ [iD + µ + iD − µ ] − aD + µ D − µ
γ 5 -hermiticity
(iD W ) † = γ 5 iD W γ 5
Eigenvalues, z, of iD W
- complex conjugate pairs (z,z ∗ )
- exact real eigenvalues
Itho, Iwasaki, Yoshie, PRD 36 (1987) 527

Random Matrix Theory for QCD – p. 20/30
Eigenvalues, z, of D W
(illustration)
2
1
Im(z)
0
-1
-2
-0.2 -0.1 0 0.1 0.2
Re(z)

RMT for Wilson Lattice QCD
Random Matrix Theory for QCD – p. 21/30

Random Matrix Theory for QCD – p. 22/30
Properties of the Wilson Dirac operator
γ 5 -Hermiticity
(iD W ) † = γ 5 iD W γ 5
What is the structure of the matrix iD W

Random Matrix Theory for QCD – p. 22/30
Properties of the Wilson Dirac operator
γ 5 -Hermiticity
(iD W ) † = γ 5 iD W γ 5
What is the structure of the matrix iD W
iD W =
0
@
aA iW
iW †
aB
1
A
A (n × n) and B ((n + ν) × (n + ν)) are hermitian
W is a general complex matrix

Random Matrix Theory for QCD – p. 23/30
The spectrum of one Random Matrix
2
1
Im(z)
0
-1
-2
-0.2 -0.1 0 0.1 0.2
Re(z)
Damgaard Splittorff Verbaarschot PRL 105, 162002 (2010)

Random Matrix Theory for QCD – p. 24/30
The Wilson RMT partition function
Z ν N f
≡
Z
dW det(iD W + m) N f
e − N 2 Tr(A2 +B 2 )−NTrW † W
where
iD W + m =
0
@ aA + m
iW †
iW
aB + m
1
A
same flavor symmetries as Wilson QCD and same breaking by m and a
Damgaard Splittorff Verbaarschot PRL 105, 162002 (2010)

Random Matrix Theory for QCD – p. 24/30
The Wilson RMT partition function
Z ν N f
≡
Z
dW det(iD W + m) N f
e − N 2 Tr(A2 +B 2 )−NTrW † W
where
iD W + m =
0
@ aA + m
iW †
iW
aB + m
1
A
same flavor symmetries as Wilson QCD and same breaking by m and a
Same low energy theory
Damgaard Splittorff Verbaarschot PRL 105, 162002 (2010)

Random Matrix Theory for QCD – p. 25/30
The Wilson RMT partition function
Z ν N f
≡
Z
dW det(iD W + m) N f
e − N 2 Tr(A2 +B 2 )−NTrW † W
where
iD W + m =
0
@ aA + m
iW †
iW
aB + m
1
A
same flavor symmetries as Wilson QCD and same breaking by m and a
Z ν N f
=
∫
U(N f )
dU det ν (U) e NmTr(U+U † )− Na2
2 Tr(U2 +U † 2
)
for N → ∞ with mN and a 2 N fixed
Damgaard Splittorff Verbaarschot PRL 105, 162002 (2010)

Random Matrix Theory for QCD – p. 26/30
Eigenvalue density of D 5 = γ 5 (iD W + m)
Sector ν = 0
mΣV = 3
a √ W 8 V = 0
a √ W 8 V = 0.03
a √ W 8 V = 0.250
ρ 5
ν=0
(x;a)
1.5
1
0.5
0
0 2 4 6 8 10
xΣV
- Aoki phase when gap closes
Damgaard Splittorff Verbaarschot PRL 105, 162002 (2010)

Random Matrix Theory for QCD – p. 27/30
Lattice
Spectrum of D 5
6 12 18 24 30 36 42 48 54 60 66
ρ(λ)
0.03
Z A
m sea
0.02
0.01
λ [MeV]
- Aoki phase when gap closes
Lüscher JHEP0707:081,2007
Del Debbio Giusti Lüscher Petronzio Tantalo JHEP0702:082,2007
Aoki PRD 30 (1984) 2653
Bitar Heller Narayanan PLB 418 167 (1998)

Random Matrix Theory for QCD – p. 28/30
a ≠ 0
Aoki phase (parity broken phase)
X
Im(z)
X
_
< ψ ψ >(m)
X
X
X
X
X
X
X
X
X
Re(z)
_
< ψ ψ >=0
m
X
X
X
X
X
Electrostatic analogy:
Eigenvalues = charges, quark mass = test charge
Aoki PRD 30 2653 (1984)
Barbour et al. NPB 275 (1986) 296 (nonzero µ)

Random Matrix Theory for QCD – p. 29/30
Conclusions
Interplay between lattice QCD and analytic QCD is essential
to understand chiral symmetry breaking in QCD

Random Matrix Theory for QCD – p. 29/30
Conclusions
Interplay between lattice QCD and analytic QCD is essential
to understand chiral symmetry breaking in QCD
Here:
Link between CPT and RMT
Σ from simulations at finite V
Σ from simulations at finite V and nonzero a

Random Matrix Theory for QCD – p. 29/30
Conclusions
Interplay between lattice QCD and analytic QCD is essential
to understand chiral symmetry breaking in QCD
Here:
Link between CPT and RMT
Σ from simulations at finite V
Σ from simulations at finite V and nonzero a
Outlook: apply Random Matrix Theory in your own research

Additional slides
Random Matrix Theory for QCD – p. 30/30